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0 facts-divide-by-8 8 ÷ 8 1 facts-divide-by-8 16 ÷ 8 2 facts-divide-by-8 24 ÷ 8 3 facts-divide-by-8 32 ÷ 8 4 facts-divide-by-8 40 ÷ 8 5 facts-divide-by-8 48 ÷ 8 6 facts-divide-by-8 56 ÷ 8 7 facts-divide-by-8 64 ÷ 8 8 facts-divide-by-8 72 ÷ 8 9 facts-divide-by-8 80 ÷ 8 10 facts-divide-by-8 88 ÷ 8 11 facts-divide-by-8 96 ÷ 8 12 facts-divide-by-8 0 ÷ 9 0 facts-divide-by-9 9 ÷ 9 1 facts-divide-by-9 18 ÷ 9 2 facts-divide-by-9 27 ÷ 9 3 facts-divide-by-9 36 ÷ 9 4 facts-divide-by-9 45 ÷ 9 5 facts-divide-by-9 54 ÷ 9 6 facts-divide-by-9 63 ÷ 9 7 facts-divide-by-9 72 ÷ 9 8 facts-divide-by-9 81 ÷ 9 9 facts-divide-by-9 90 ÷ 9 10 facts-divide-by-9 99 ÷ 9 11 facts-divide-by-9 108 ÷ 9 12 facts-divide-by-9 1 + 1 2 facts-doubles Half of 2 1 facts-doubles 2 + 2 4 facts-doubles Half of 4 2 facts-doubles 3 + 3 6 facts-doubles Half of 6 3 facts-doubles 4 + 4 8 facts-doubles Half of 8 4 facts-doubles 5 + 5 10 facts-doubles Half of 10 5 facts-doubles 6 + 6 12 facts-doubles Half of 12 6 facts-doubles 7 + 7 14 facts-doubles Half of 14 7 facts-doubles 8 + 8 16 facts-doubles Half of 16 8 facts-doubles 9 + 9 18 facts-doubles Half of 18 9 facts-doubles 10 + 10 20 facts-doubles Half of 20 10 facts-doubles 11 + 11 22 facts-doubles Half of 22 11 facts-doubles 12 + 12 24 facts-doubles Half of 24 12 facts-doubles 13 + 13 26 facts-doubles Half of 26 13 facts-doubles 14 + 14 28 facts-doubles Half of 28 14 facts-doubles 15 + 15 30 facts-doubles Half of 30 15 facts-doubles 16 + 16 32 facts-doubles Half of 32 16 facts-doubles 17 + 17 34 facts-doubles Half of 34 17 facts-doubles 18 + 18 36 facts-doubles Half of 36 18 facts-doubles 19 + 19 38 facts-doubles Half of 38 19 facts-doubles 20 + 20 40 facts-doubles Half of 40 20 facts-doubles 0 + ? = 10 10 facts-make-ten 1 + ? = 10 9 facts-make-ten 2 + ? = 10 8 facts-make-ten 3 + ? = 10 7 facts-make-ten 4 + ? = 10 6 facts-make-ten 5 + ? = 10 5 facts-make-ten 6 + ? = 10 4 facts-make-ten 7 + ? = 10 3 facts-make-ten 8 + ? = 10 2 facts-make-ten 9 + ? = 10 1 facts-make-ten 10 + ? = 10 0 facts-make-ten 13 × 2 26 facts-multiply-teens 13 × 3 39 facts-multiply-teens 13 × 4 52 facts-multiply-teens 13 × 5 65 facts-multiply-teens 13 × 6 78 facts-multiply-teens 13 × 7 91 facts-multiply-teens 13 × 8 104 facts-multiply-teens 13 × 9 117 facts-multiply-teens 14 × 2 28 facts-multiply-teens 14 × 3 42 facts-multiply-teens 14 × 4 56 facts-multiply-teens 14 × 5 70 facts-multiply-teens 14 × 6 84 facts-multiply-teens 14 × 7 98 facts-multiply-teens 14 × 8 112 facts-multiply-teens 14 × 9 126 facts-multiply-teens 15 × 2 30 facts-multiply-teens 15 × 3 45 facts-multiply-teens 15 × 4 60 facts-multiply-teens 15 × 5 75 facts-multiply-teens 15 × 6 90 facts-multiply-teens 15 × 7 105 facts-multiply-teens 15 × 8 120 facts-multiply-teens 15 × 9 135 facts-multiply-teens 16 × 2 32 facts-multiply-teens 16 × 3 48 facts-multiply-teens 16 × 4 64 facts-multiply-teens 16 × 5 80 facts-multiply-teens 16 × 6 96 facts-multiply-teens 16 × 7 112 facts-multiply-teens 16 × 8 128 facts-multiply-teens 16 × 9 144 facts-multiply-teens 17 × 2 34 facts-multiply-teens 17 × 3 51 facts-multiply-teens 17 × 4 68 facts-multiply-teens 17 × 5 85 facts-multiply-teens 17 × 6 102 facts-multiply-teens 17 × 7 119 facts-multiply-teens 17 × 8 136 facts-multiply-teens 17 × 9 153 facts-multiply-teens 18 × 2 36 facts-multiply-teens 18 × 3 54 facts-multiply-teens 18 × 4 72 facts-multiply-teens 18 × 5 90 facts-multiply-teens 18 × 6 108 facts-multiply-teens 18 × 7 126 facts-multiply-teens 18 × 8 144 facts-multiply-teens 18 × 9 162 facts-multiply-teens 19 × 2 38 facts-multiply-teens 19 × 3 57 facts-multiply-teens 19 × 4 76 facts-multiply-teens 19 × 5 95 facts-multiply-teens 19 × 6 114 facts-multiply-teens 19 × 7 133 facts-multiply-teens 19 × 8 152 facts-multiply-teens 19 × 9 171 facts-multiply-teens 20 × 2 40 facts-multiply-teens 20 × 3 60 facts-multiply-teens 20 × 4 80 facts-multiply-teens 20 × 5 100 facts-multiply-teens 20 × 6 120 facts-multiply-teens 20 × 7 140 facts-multiply-teens 20 × 8 160 facts-multiply-teens 20 × 9 180 facts-multiply-teens 10⁰ 1 facts-powers-of-10 10¹ 10 (ten) facts-powers-of-10 10² 100 (hundred) facts-powers-of-10 10³ 1,000 (thousand) facts-powers-of-10 10⁴ 10,000 (ten thousand) facts-powers-of-10 10⁵ 100,000 (hundred thousand) facts-powers-of-10 10⁶ 1,000,000 (million) facts-powers-of-10 10⁹ 1,000,000,000 (billion) facts-powers-of-10 10¹² 1,000,000,000,000 (trillion) facts-powers-of-10 10⁻¹ 0.1 (tenth) facts-powers-of-10 10⁻² 0.01 (hundredth) facts-powers-of-10 10⁻³ 0.001 (thousandth) facts-powers-of-10 2^0 1 facts-powers-of-2 2^1 2 facts-powers-of-2 2^2 4 facts-powers-of-2 2^3 8 facts-powers-of-2 2^4 16 facts-powers-of-2 2^5 32 facts-powers-of-2 2^6 64 facts-powers-of-2 2^7 128 facts-powers-of-2 2^8 256 facts-powers-of-2 2^9 512 facts-powers-of-2 2^10 1024 facts-powers-of-2 2^11 2048 facts-powers-of-2 2^12 4096 facts-powers-of-2 2^13 8192 facts-powers-of-2 2^14 16384 facts-powers-of-2 2^15 32768 facts-powers-of-2 2^16 65536 facts-powers-of-2 2^17 131072 facts-powers-of-2 2^18 262144 facts-powers-of-2 2^19 524288 facts-powers-of-2 2^20 1048576 facts-powers-of-2 Is 2 prime? Yes facts-primes-100 Is 3 prime? Yes facts-primes-100 Is 4 prime? No: 4 = 2 × 2 facts-primes-100 Is 5 prime? Yes facts-primes-100 Is 6 prime? No: 6 = 2 × 3 facts-primes-100 Is 7 prime? Yes facts-primes-100 Is 8 prime? No: 8 = 2 × 4 facts-primes-100 Is 9 prime? No: 9 = 3 × 3 facts-primes-100 Is 10 prime? No: 10 = 2 × 5 facts-primes-100 Is 11 prime? Yes facts-primes-100 Is 12 prime? No: 12 = 2 × 6 facts-primes-100 Is 13 prime? Yes facts-primes-100 Is 14 prime? No: 14 = 2 × 7 facts-primes-100 Is 15 prime? No: 15 = 3 × 5 facts-primes-100 Is 16 prime? No: 16 = 2 × 8 facts-primes-100 Is 17 prime? Yes facts-primes-100 Is 18 prime? No: 18 = 2 × 9 facts-primes-100 Is 19 prime? Yes facts-primes-100 Is 20 prime? No: 20 = 2 × 10 facts-primes-100 Is 21 prime? No: 21 = 3 × 7 facts-primes-100 Is 22 prime? No: 22 = 2 × 11 facts-primes-100 Is 23 prime? Yes facts-primes-100 Is 24 prime? No: 24 = 2 × 12 facts-primes-100 Is 25 prime? No: 25 = 5 × 5 facts-primes-100 Is 26 prime? No: 26 = 2 × 13 facts-primes-100 Is 27 prime? No: 27 = 3 × 9 facts-primes-100 Is 28 prime? No: 28 = 2 × 14 facts-primes-100 Is 29 prime? Yes facts-primes-100 Is 30 prime? No: 30 = 2 × 15 facts-primes-100 Is 31 prime? Yes facts-primes-100 Is 32 prime? No: 32 = 2 × 16 facts-primes-100 Is 33 prime? No: 33 = 3 × 11 facts-primes-100 Is 34 prime? No: 34 = 2 × 17 facts-primes-100 Is 35 prime? No: 35 = 5 × 7 facts-primes-100 Is 36 prime? No: 36 = 2 × 18 facts-primes-100 Is 37 prime? Yes facts-primes-100 Is 38 prime? No: 38 = 2 × 19 facts-primes-100 Is 39 prime? No: 39 = 3 × 13 facts-primes-100 Is 40 prime? No: 40 = 2 × 20 facts-primes-100 Is 41 prime? Yes facts-primes-100 Is 42 prime? No: 42 = 2 × 21 facts-primes-100 Is 43 prime? Yes facts-primes-100 Is 44 prime? No: 44 = 2 × 22 facts-primes-100 Is 45 prime? No: 45 = 3 × 15 facts-primes-100 Is 46 prime? No: 46 = 2 × 23 facts-primes-100 Is 47 prime? Yes facts-primes-100 Is 48 prime? No: 48 = 2 × 24 facts-primes-100 Is 49 prime? No: 49 = 7 × 7 facts-primes-100 Is 50 prime? No: 50 = 2 × 25 facts-primes-100 Is 51 prime? No: 51 = 3 × 17 facts-primes-100 Is 52 prime? No: 52 = 2 × 26 facts-primes-100 Is 53 prime? Yes facts-primes-100 Is 54 prime? No: 54 = 2 × 27 facts-primes-100 Is 55 prime? No: 55 = 5 × 11 facts-primes-100 Is 56 prime? No: 56 = 2 × 28 facts-primes-100 Is 57 prime? No: 57 = 3 × 19 facts-primes-100 Is 58 prime? No: 58 = 2 × 29 facts-primes-100 Is 59 prime? Yes facts-primes-100 Is 60 prime? No: 60 = 2 × 30 facts-primes-100 1st number counting by 10s 10 facts-skip-count-10 2nd number counting by 10s 20 facts-skip-count-10 3rd number counting by 10s 30 facts-skip-count-10 4th number counting by 10s 40 facts-skip-count-10 5th number counting by 10s 50 facts-skip-count-10 6th number counting by 10s 60 facts-skip-count-10 7th number counting by 10s 70 facts-skip-count-10 8th number counting by 10s 80 facts-skip-count-10 9th number counting by 10s 90 facts-skip-count-10 10th number counting by 10s 100 facts-skip-count-10 11th number counting by 10s 110 facts-skip-count-10 12th number counting by 10s 120 facts-skip-count-10 1st number counting by 11s 11 facts-skip-count-11 2nd number counting by 11s 22 facts-skip-count-11 3rd number counting by 11s 33 facts-skip-count-11 4th number counting by 11s 44 facts-skip-count-11 5th number counting by 11s 55 facts-skip-count-11 6th number counting by 11s 66 facts-skip-count-11 7th number counting by 11s 77 facts-skip-count-11 8th number counting by 11s 88 facts-skip-count-11 9th number counting by 11s 99 facts-skip-count-11 10th number counting by 11s 110 facts-skip-count-11 11th number counting by 11s 121 facts-skip-count-11 12th number counting by 11s 132 facts-skip-count-11 1st number counting by 12s 12 facts-skip-count-12 2nd number counting by 12s 24 facts-skip-count-12 3rd number counting by 12s 36 facts-skip-count-12 4th number counting by 12s 48 facts-skip-count-12 5th number counting by 12s 60 facts-skip-count-12 6th number counting by 12s 72 facts-skip-count-12 7th number counting by 12s 84 facts-skip-count-12 8th number counting by 12s 96 facts-skip-count-12 9th number counting by 12s 108 facts-skip-count-12 10th number counting by 12s 120 facts-skip-count-12 11th number counting by 12s 132 facts-skip-count-12 12th number counting by 12s 144 facts-skip-count-12 1st number counting by 15s 15 facts-skip-count-15 2nd number counting by 15s 30 facts-skip-count-15 3rd number counting by 15s 45 facts-skip-count-15 4th number counting by 15s 60 facts-skip-count-15 5th number counting by 15s 75 facts-skip-count-15 6th number counting by 15s 90 facts-skip-count-15 7th number counting by 15s 105 facts-skip-count-15 8th number counting by 15s 120 facts-skip-count-15 9th number counting by 15s 135 facts-skip-count-15 10th number counting by 15s 150 facts-skip-count-15 11th number counting by 15s 165 facts-skip-count-15 12th number counting by 15s 180 facts-skip-count-15 1st number counting by 2s 2 facts-skip-count-2 2nd number counting by 2s 4 facts-skip-count-2 3rd number counting by 2s 6 facts-skip-count-2 4th number counting by 2s 8 facts-skip-count-2 5th number counting by 2s 10 facts-skip-count-2 6th number counting by 2s 12 facts-skip-count-2 7th number counting by 2s 14 facts-skip-count-2 8th number counting by 2s 16 facts-skip-count-2 9th number counting by 2s 18 facts-skip-count-2 10th number counting by 2s 20 facts-skip-count-2 11th number counting by 2s 22 facts-skip-count-2 12th number counting by 2s 24 facts-skip-count-2 1st number counting by 20s 20 facts-skip-count-20 2nd number counting by 20s 40 facts-skip-count-20 3rd number counting by 20s 60 facts-skip-count-20 4th number counting by 20s 80 facts-skip-count-20 5th number counting by 20s 100 facts-skip-count-20 6th number counting by 20s 120 facts-skip-count-20 7th number counting by 20s 140 facts-skip-count-20 8th number counting by 20s 160 facts-skip-count-20 9th number counting by 20s 180 facts-skip-count-20 10th number counting by 20s 200 facts-skip-count-20 11th number counting by 20s 220 facts-skip-count-20 12th number counting by 20s 240 facts-skip-count-20 1st number counting by 25s 25 facts-skip-count-25 2nd number counting by 25s 50 facts-skip-count-25 3rd number counting by 25s 75 facts-skip-count-25 4th number counting by 25s 100 facts-skip-count-25 5th number counting by 25s 125 facts-skip-count-25 6th number counting by 25s 150 facts-skip-count-25 7th number counting by 25s 175 facts-skip-count-25 8th number counting by 25s 200 facts-skip-count-25 9th number counting by 25s 225 facts-skip-count-25 10th number counting by 25s 250 facts-skip-count-25 11th number counting by 25s 275 facts-skip-count-25 12th number counting by 25s 300 facts-skip-count-25 1st number counting by 3s 3 facts-skip-count-3 2nd number counting by 3s 6 facts-skip-count-3 3rd number counting by 3s 9 facts-skip-count-3 4th number counting by 3s 12 facts-skip-count-3 5th number counting by 3s 15 facts-skip-count-3 6th number counting by 3s 18 facts-skip-count-3 7th number counting by 3s 21 facts-skip-count-3 8th number counting by 3s 24 facts-skip-count-3 9th number counting by 3s 27 facts-skip-count-3 10th number counting by 3s 30 facts-skip-count-3 11th number counting by 3s 33 facts-skip-count-3 12th number counting by 3s 36 facts-skip-count-3 1st number counting by 4s 4 facts-skip-count-4 2nd number counting by 4s 8 facts-skip-count-4 3rd number counting by 4s 12 facts-skip-count-4 4th number counting by 4s 16 facts-skip-count-4 5th number counting by 4s 20 facts-skip-count-4 6th number counting by 4s 24 facts-skip-count-4 7th number counting by 4s 28 facts-skip-count-4 8th number counting by 4s 32 facts-skip-count-4 9th number counting by 4s 36 facts-skip-count-4 10th number counting by 4s 40 facts-skip-count-4 11th number counting by 4s 44 facts-skip-count-4 12th number counting by 4s 48 facts-skip-count-4 1st number counting by 5s 5 facts-skip-count-5 2nd number counting by 5s 10 facts-skip-count-5 3rd number counting by 5s 15 facts-skip-count-5 4th number counting by 5s 20 facts-skip-count-5 5th number counting by 5s 25 facts-skip-count-5 6th number counting by 5s 30 facts-skip-count-5 7th number counting by 5s 35 facts-skip-count-5 8th number counting by 5s 40 facts-skip-count-5 9th number counting by 5s 45 facts-skip-count-5 10th number counting by 5s 50 facts-skip-count-5 11th number counting by 5s 55 facts-skip-count-5 12th number counting by 5s 60 facts-skip-count-5 1st number counting by 6s 6 facts-skip-count-6 2nd number counting by 6s 12 facts-skip-count-6 3rd number counting by 6s 18 facts-skip-count-6 4th number counting by 6s 24 facts-skip-count-6 5th number counting by 6s 30 facts-skip-count-6 6th number counting by 6s 36 facts-skip-count-6 7th number counting by 6s 42 facts-skip-count-6 8th number counting by 6s 48 facts-skip-count-6 9th number counting by 6s 54 facts-skip-count-6 10th number counting by 6s 60 facts-skip-count-6 11th number counting by 6s 66 facts-skip-count-6 12th number counting by 6s 72 facts-skip-count-6 1st number counting by 7s 7 facts-skip-count-7 2nd number counting by 7s 14 facts-skip-count-7 3rd number counting by 7s 21 facts-skip-count-7 4th number counting by 7s 28 facts-skip-count-7 5th number counting by 7s 35 facts-skip-count-7 6th number counting by 7s 42 facts-skip-count-7 7th number counting by 7s 49 facts-skip-count-7 8th number counting by 7s 56 facts-skip-count-7 9th number counting by 7s 63 facts-skip-count-7 10th number counting by 7s 70 facts-skip-count-7 11th number counting by 7s 77 facts-skip-count-7 12th number counting by 7s 84 facts-skip-count-7 1st number counting by 8s 8 facts-skip-count-8 2nd number counting by 8s 16 facts-skip-count-8 3rd number counting by 8s 24 facts-skip-count-8 4th number counting by 8s 32 facts-skip-count-8 5th number counting by 8s 40 facts-skip-count-8 6th number counting by 8s 48 facts-skip-count-8 7th number counting by 8s 56 facts-skip-count-8 8th number counting by 8s 64 facts-skip-count-8 9th number counting by 8s 72 facts-skip-count-8 10th number counting by 8s 80 facts-skip-count-8 11th number counting by 8s 88 facts-skip-count-8 12th number counting by 8s 96 facts-skip-count-8 1st number counting by 9s 9 facts-skip-count-9 2nd number counting by 9s 18 facts-skip-count-9 3rd number counting by 9s 27 facts-skip-count-9 4th number counting by 9s 36 facts-skip-count-9 5th number counting by 9s 45 facts-skip-count-9 6th number counting by 9s 54 facts-skip-count-9 7th number counting by 9s 63 facts-skip-count-9 8th number counting by 9s 72 facts-skip-count-9 9th number counting by 9s 81 facts-skip-count-9 10th number counting by 9s 90 facts-skip-count-9 11th number counting by 9s 99 facts-skip-count-9 12th number counting by 9s 108 facts-skip-count-9 √1 1 facts-square-roots √4 2 facts-square-roots √9 3 facts-square-roots √16 4 facts-square-roots √25 5 facts-square-roots √36 6 facts-square-roots √49 7 facts-square-roots √64 8 facts-square-roots √81 9 facts-square-roots √100 10 facts-square-roots √121 11 facts-square-roots √144 12 facts-square-roots √169 13 facts-square-roots √196 14 facts-square-roots √225 15 facts-square-roots √256 16 facts-square-roots √289 17 facts-square-roots √324 18 facts-square-roots √361 19 facts-square-roots √400 20 facts-square-roots √441 21 facts-square-roots √484 22 facts-square-roots √529 23 facts-square-roots √576 24 facts-square-roots √625 25 facts-square-roots √676 26 facts-square-roots √729 27 facts-square-roots √784 28 facts-square-roots √841 29 facts-square-roots √900 30 facts-square-roots 1² 1 facts-squares-1-30 2² 4 facts-squares-1-30 3² 9 facts-squares-1-30 4² 16 facts-squares-1-30 5² 25 facts-squares-1-30 6² 36 facts-squares-1-30 7² 49 facts-squares-1-30 8² 64 facts-squares-1-30 9² 81 facts-squares-1-30 10² 100 facts-squares-1-30 11² 121 facts-squares-1-30 12² 144 facts-squares-1-30 13² 169 facts-squares-1-30 14² 196 facts-squares-1-30 15² 225 facts-squares-1-30 16² 256 facts-squares-1-30 17² 289 facts-squares-1-30 18² 324 facts-squares-1-30 19² 361 facts-squares-1-30 20² 400 facts-squares-1-30 21² 441 facts-squares-1-30 22² 484 facts-squares-1-30 23² 529 facts-squares-1-30 24² 576 facts-squares-1-30 25² 625 facts-squares-1-30 26² 676 facts-squares-1-30 27² 729 facts-squares-1-30 28² 784 facts-squares-1-30 29² 841 facts-squares-1-30 30² 900 facts-squares-1-30 0 − 0 0 facts-subtract-from-20 1 − 0 1 facts-subtract-from-20 2 − 0 2 facts-subtract-from-20 3 − 0 3 facts-subtract-from-20 4 − 0 4 facts-subtract-from-20 5 − 0 5 facts-subtract-from-20 6 − 0 6 facts-subtract-from-20 7 − 0 7 facts-subtract-from-20 8 − 0 8 facts-subtract-from-20 9 − 0 9 facts-subtract-from-20 10 − 0 10 facts-subtract-from-20 1 − 1 0 facts-subtract-from-20 2 − 1 1 facts-subtract-from-20 3 − 1 2 facts-subtract-from-20 4 − 1 3 facts-subtract-from-20 5 − 1 4 facts-subtract-from-20 6 − 1 5 facts-subtract-from-20 7 − 1 6 facts-subtract-from-20 8 − 1 7 facts-subtract-from-20 9 − 1 8 facts-subtract-from-20 10 − 1 9 facts-subtract-from-20 11 − 1 10 facts-subtract-from-20 2 − 2 0 facts-subtract-from-20 3 − 2 1 facts-subtract-from-20 4 − 2 2 facts-subtract-from-20 5 − 2 3 facts-subtract-from-20 6 − 2 4 facts-subtract-from-20 7 − 2 5 facts-subtract-from-20 8 − 2 6 facts-subtract-from-20 9 − 2 7 facts-subtract-from-20 10 − 2 8 facts-subtract-from-20 11 − 2 9 facts-subtract-from-20 12 − 2 10 facts-subtract-from-20 3 − 3 0 facts-subtract-from-20 4 − 3 1 facts-subtract-from-20 5 − 3 2 facts-subtract-from-20 6 − 3 3 facts-subtract-from-20 7 − 3 4 facts-subtract-from-20 8 − 3 5 facts-subtract-from-20 9 − 3 6 facts-subtract-from-20 10 − 3 7 facts-subtract-from-20 11 − 3 8 facts-subtract-from-20 12 − 3 9 facts-subtract-from-20 13 − 3 10 facts-subtract-from-20 4 − 4 0 facts-subtract-from-20 5 − 4 1 facts-subtract-from-20 6 − 4 2 facts-subtract-from-20 7 − 4 3 facts-subtract-from-20 8 − 4 4 facts-subtract-from-20 9 − 4 5 facts-subtract-from-20 10 − 4 6 facts-subtract-from-20 11 − 4 7 facts-subtract-from-20 12 − 4 8 facts-subtract-from-20 13 − 4 9 facts-subtract-from-20 14 − 4 10 facts-subtract-from-20 5 − 5 0 facts-subtract-from-20 6 − 5 1 facts-subtract-from-20 7 − 5 2 facts-subtract-from-20 8 − 5 3 facts-subtract-from-20 9 − 5 4 facts-subtract-from-20 10 − 5 5 facts-subtract-from-20 11 − 5 6 facts-subtract-from-20 12 − 5 7 facts-subtract-from-20 13 − 5 8 facts-subtract-from-20 14 − 5 9 facts-subtract-from-20 15 − 5 10 facts-subtract-from-20 6 − 6 0 facts-subtract-from-20 7 − 6 1 facts-subtract-from-20 8 − 6 2 facts-subtract-from-20 9 − 6 3 facts-subtract-from-20 10 − 6 4 facts-subtract-from-20 11 − 6 5 facts-subtract-from-20 12 − 6 6 facts-subtract-from-20 13 − 6 7 facts-subtract-from-20 14 − 6 8 facts-subtract-from-20 15 − 6 9 facts-subtract-from-20 16 − 6 10 facts-subtract-from-20 7 − 7 0 facts-subtract-from-20 8 − 7 1 facts-subtract-from-20 9 − 7 2 facts-subtract-from-20 10 − 7 3 facts-subtract-from-20 11 − 7 4 facts-subtract-from-20 12 − 7 5 facts-subtract-from-20 13 − 7 6 facts-subtract-from-20 14 − 7 7 facts-subtract-from-20 15 − 7 8 facts-subtract-from-20 16 − 7 9 facts-subtract-from-20 17 − 7 10 facts-subtract-from-20 8 − 8 0 facts-subtract-from-20 9 − 8 1 facts-subtract-from-20 10 − 8 2 facts-subtract-from-20 11 − 8 3 facts-subtract-from-20 12 − 8 4 facts-subtract-from-20 13 − 8 5 facts-subtract-from-20 14 − 8 6 facts-subtract-from-20 15 − 8 7 facts-subtract-from-20 16 − 8 8 facts-subtract-from-20 17 − 8 9 facts-subtract-from-20 18 − 8 10 facts-subtract-from-20 9 − 9 0 facts-subtract-from-20 10 − 9 1 facts-subtract-from-20 11 − 9 2 facts-subtract-from-20 12 − 9 3 facts-subtract-from-20 13 − 9 4 facts-subtract-from-20 14 − 9 5 facts-subtract-from-20 15 − 9 6 facts-subtract-from-20 16 − 9 7 facts-subtract-from-20 17 − 9 8 facts-subtract-from-20 18 − 9 9 facts-subtract-from-20 19 − 9 10 facts-subtract-from-20 10 − 10 0 facts-subtract-from-20 11 − 10 1 facts-subtract-from-20 12 − 10 2 facts-subtract-from-20 13 − 10 3 facts-subtract-from-20 14 − 10 4 facts-subtract-from-20 15 − 10 5 facts-subtract-from-20 16 − 10 6 facts-subtract-from-20 17 − 10 7 facts-subtract-from-20 18 − 10 8 facts-subtract-from-20 19 − 10 9 facts-subtract-from-20 20 − 10 10 facts-subtract-from-20 0 × 0 0 facts-times-0 0 × 1 0 facts-times-0 0 × 2 0 facts-times-0 0 × 3 0 facts-times-0 0 × 4 0 facts-times-0 0 × 5 0 facts-times-0 0 × 6 0 facts-times-0 0 × 7 0 facts-times-0 0 × 8 0 facts-times-0 0 × 9 0 facts-times-0 0 × 10 0 facts-times-0 0 × 11 0 facts-times-0 0 × 12 0 facts-times-0 1 × 0 0 facts-times-1 1 × 1 1 facts-times-1 1 × 2 2 facts-times-1 1 × 3 3 facts-times-1 1 × 4 4 facts-times-1 1 × 5 5 facts-times-1 1 × 6 6 facts-times-1 1 × 7 7 facts-times-1 1 × 8 8 facts-times-1 1 × 9 9 facts-times-1 1 × 10 10 facts-times-1 1 × 11 11 facts-times-1 1 × 12 12 facts-times-1 10 × 0 0 facts-times-10 10 × 1 10 facts-times-10 10 × 2 20 facts-times-10 10 × 3 30 facts-times-10 10 × 4 40 facts-times-10 10 × 5 50 facts-times-10 10 × 6 60 facts-times-10 10 × 7 70 facts-times-10 10 × 8 80 facts-times-10 10 × 9 90 facts-times-10 10 × 10 100 facts-times-10 10 × 11 110 facts-times-10 10 × 12 120 facts-times-10 11 × 0 0 facts-times-11 11 × 1 11 facts-times-11 11 × 2 22 facts-times-11 11 × 3 33 facts-times-11 11 × 4 44 facts-times-11 11 × 5 55 facts-times-11 11 × 6 66 facts-times-11 11 × 7 77 facts-times-11 11 × 8 88 facts-times-11 11 × 9 99 facts-times-11 11 × 10 110 facts-times-11 11 × 11 121 facts-times-11 11 × 12 132 facts-times-11 12 × 0 0 facts-times-12 12 × 1 12 facts-times-12 12 × 2 24 facts-times-12 12 × 3 36 facts-times-12 12 × 4 48 facts-times-12 12 × 5 60 facts-times-12 12 × 6 72 facts-times-12 12 × 7 84 facts-times-12 12 × 8 96 facts-times-12 12 × 9 108 facts-times-12 12 × 10 120 facts-times-12 12 × 11 132 facts-times-12 12 × 12 144 facts-times-12 2 × 0 0 facts-times-2 2 × 1 2 facts-times-2 2 × 2 4 facts-times-2 2 × 3 6 facts-times-2 2 × 4 8 facts-times-2 2 × 5 10 facts-times-2 2 × 6 12 facts-times-2 2 × 7 14 facts-times-2 2 × 8 16 facts-times-2 2 × 9 18 facts-times-2 2 × 10 20 facts-times-2 2 × 11 22 facts-times-2 2 × 12 24 facts-times-2 3 × 0 0 facts-times-3 3 × 1 3 facts-times-3 3 × 2 6 facts-times-3 3 × 3 9 facts-times-3 3 × 4 12 facts-times-3 3 × 5 15 facts-times-3 3 × 6 18 facts-times-3 3 × 7 21 facts-times-3 3 × 8 24 facts-times-3 3 × 9 27 facts-times-3 3 × 10 30 facts-times-3 3 × 11 33 facts-times-3 3 × 12 36 facts-times-3 4 × 0 0 facts-times-4 4 × 1 4 facts-times-4 4 × 2 8 facts-times-4 4 × 3 12 facts-times-4 4 × 4 16 facts-times-4 4 × 5 20 facts-times-4 4 × 6 24 facts-times-4 4 × 7 28 facts-times-4 4 × 8 32 facts-times-4 4 × 9 36 facts-times-4 4 × 10 40 facts-times-4 4 × 11 44 facts-times-4 4 × 12 48 facts-times-4 5 × 0 0 facts-times-5 5 × 1 5 facts-times-5 5 × 2 10 facts-times-5 5 × 3 15 facts-times-5 5 × 4 20 facts-times-5 5 × 5 25 facts-times-5 5 × 6 30 facts-times-5 5 × 7 35 facts-times-5 5 × 8 40 facts-times-5 5 × 9 45 facts-times-5 5 × 10 50 facts-times-5 5 × 11 55 facts-times-5 5 × 12 60 facts-times-5 6 × 0 0 facts-times-6 6 × 1 6 facts-times-6 6 × 2 12 facts-times-6 6 × 3 18 facts-times-6 6 × 4 24 facts-times-6 6 × 5 30 facts-times-6 6 × 6 36 facts-times-6 6 × 7 42 facts-times-6 6 × 8 48 facts-times-6 6 × 9 54 facts-times-6 6 × 10 60 facts-times-6 6 × 11 66 facts-times-6 6 × 12 72 facts-times-6 7 × 0 0 facts-times-7 7 × 1 7 facts-times-7 7 × 2 14 facts-times-7 7 × 3 21 facts-times-7 7 × 4 28 facts-times-7 7 × 5 35 facts-times-7 7 × 6 42 facts-times-7 7 × 7 49 facts-times-7 7 × 8 56 facts-times-7 7 × 9 63 facts-times-7 7 × 10 70 facts-times-7 7 × 11 77 facts-times-7 7 × 12 84 facts-times-7 8 × 0 0 facts-times-8 8 × 1 8 facts-times-8 8 × 2 16 facts-times-8 8 × 3 24 facts-times-8 8 × 4 32 facts-times-8 8 × 5 40 facts-times-8 8 × 6 48 facts-times-8 8 × 7 56 facts-times-8 8 × 8 64 facts-times-8 8 × 9 72 facts-times-8 8 × 10 80 facts-times-8 8 × 11 88 facts-times-8 8 × 12 96 facts-times-8 9 × 0 0 facts-times-9 9 × 1 9 facts-times-9 9 × 2 18 facts-times-9 9 × 3 27 facts-times-9 9 × 4 36 facts-times-9 9 × 5 45 facts-times-9 9 × 6 54 facts-times-9 9 × 7 63 facts-times-9 9 × 8 72 facts-times-9 9 × 9 81 facts-times-9 9 × 10 90 facts-times-9 9 × 11 99 facts-times-9 9 × 12 108 facts-times-9 0 × 0 0 facts-times-all 0 × 1 0 facts-times-all 0 × 2 0 facts-times-all 0 × 3 0 facts-times-all 0 × 4 0 facts-times-all 0 × 5 0 facts-times-all 0 × 6 0 facts-times-all 0 × 7 0 facts-times-all 0 × 8 0 facts-times-all 0 × 9 0 facts-times-all 0 × 10 0 facts-times-all 0 × 11 0 facts-times-all 0 × 12 0 facts-times-all 1 × 0 0 facts-times-all 1 × 1 1 facts-times-all 1 × 2 2 facts-times-all 1 × 3 3 facts-times-all 1 × 4 4 facts-times-all 1 × 5 5 facts-times-all 1 × 6 6 facts-times-all 1 × 7 7 facts-times-all 1 × 8 8 facts-times-all 1 × 9 9 facts-times-all 1 × 10 10 facts-times-all 1 × 11 11 facts-times-all 1 × 12 12 facts-times-all 2 × 0 0 facts-times-all 2 × 1 2 facts-times-all 2 × 2 4 facts-times-all 2 × 3 6 facts-times-all 2 × 4 8 facts-times-all 2 × 5 10 facts-times-all 2 × 6 12 facts-times-all 2 × 7 14 facts-times-all 2 × 8 16 facts-times-all 2 × 9 18 facts-times-all 2 × 10 20 facts-times-all 2 × 11 22 facts-times-all 2 × 12 24 facts-times-all 3 × 0 0 facts-times-all 3 × 1 3 facts-times-all 3 × 2 6 facts-times-all 3 × 3 9 facts-times-all 3 × 4 12 facts-times-all 3 × 5 15 facts-times-all 3 × 6 18 facts-times-all 3 × 7 21 facts-times-all 3 × 8 24 facts-times-all 3 × 9 27 facts-times-all 3 × 10 30 facts-times-all 3 × 11 33 facts-times-all 3 × 12 36 facts-times-all 4 × 0 0 facts-times-all 4 × 1 4 facts-times-all 4 × 2 8 facts-times-all 4 × 3 12 facts-times-all 4 × 4 16 facts-times-all 4 × 5 20 facts-times-all 4 × 6 24 facts-times-all 4 × 7 28 facts-times-all 4 × 8 32 facts-times-all 4 × 9 36 facts-times-all 4 × 10 40 facts-times-all 4 × 11 44 facts-times-all 4 × 12 48 facts-times-all 5 × 0 0 facts-times-all 5 × 1 5 facts-times-all 5 × 2 10 facts-times-all 5 × 3 15 facts-times-all 5 × 4 20 facts-times-all 5 × 5 25 facts-times-all 5 × 6 30 facts-times-all 5 × 7 35 facts-times-all 5 × 8 40 facts-times-all 5 × 9 45 facts-times-all 5 × 10 50 facts-times-all 5 × 11 55 facts-times-all 5 × 12 60 facts-times-all 6 × 0 0 facts-times-all 6 × 1 6 facts-times-all 6 × 2 12 facts-times-all 6 × 3 18 facts-times-all 6 × 4 24 facts-times-all 6 × 5 30 facts-times-all 6 × 6 36 facts-times-all 6 × 7 42 facts-times-all 6 × 8 48 facts-times-all 6 × 9 54 facts-times-all 6 × 10 60 facts-times-all 6 × 11 66 facts-times-all 6 × 12 72 facts-times-all 7 × 0 0 facts-times-all 7 × 1 7 facts-times-all 7 × 2 14 facts-times-all 7 × 3 21 facts-times-all 7 × 4 28 facts-times-all 7 × 5 35 facts-times-all 7 × 6 42 facts-times-all 7 × 7 49 facts-times-all 7 × 8 56 facts-times-all 7 × 9 63 facts-times-all 7 × 10 70 facts-times-all 7 × 11 77 facts-times-all 7 × 12 84 facts-times-all 8 × 0 0 facts-times-all 8 × 1 8 facts-times-all 8 × 2 16 facts-times-all 8 × 3 24 facts-times-all 8 × 4 32 facts-times-all 8 × 5 40 facts-times-all 8 × 6 48 facts-times-all 8 × 7 56 facts-times-all 8 × 8 64 facts-times-all 8 × 9 72 facts-times-all 8 × 10 80 facts-times-all 8 × 11 88 facts-times-all 8 × 12 96 facts-times-all 9 × 0 0 facts-times-all 9 × 1 9 facts-times-all 9 × 2 18 facts-times-all 9 × 3 27 facts-times-all 9 × 4 36 facts-times-all 9 × 5 45 facts-times-all 9 × 6 54 facts-times-all 9 × 7 63 facts-times-all 9 × 8 72 facts-times-all 9 × 9 81 facts-times-all 9 × 10 90 facts-times-all 9 × 11 99 facts-times-all 9 × 12 108 facts-times-all 10 × 0 0 facts-times-all 10 × 1 10 facts-times-all 10 × 2 20 facts-times-all 10 × 3 30 facts-times-all 10 × 4 40 facts-times-all 10 × 5 50 facts-times-all 10 × 6 60 facts-times-all 10 × 7 70 facts-times-all 10 × 8 80 facts-times-all 10 × 9 90 facts-times-all 10 × 10 100 facts-times-all 10 × 11 110 facts-times-all 10 × 12 120 facts-times-all 11 × 0 0 facts-times-all 11 × 1 11 facts-times-all 11 × 2 22 facts-times-all 11 × 3 33 facts-times-all 11 × 4 44 facts-times-all 11 × 5 55 facts-times-all 11 × 6 66 facts-times-all 11 × 7 77 facts-times-all 11 × 8 88 facts-times-all 11 × 9 99 facts-times-all 11 × 10 110 facts-times-all 11 × 11 121 facts-times-all 11 × 12 132 facts-times-all 12 × 0 0 facts-times-all 12 × 1 12 facts-times-all 12 × 2 24 facts-times-all 12 × 3 36 facts-times-all 12 × 4 48 facts-times-all 12 × 5 60 facts-times-all 12 × 6 72 facts-times-all 12 × 7 84 facts-times-all 12 × 8 96 facts-times-all 12 × 9 108 facts-times-all 12 × 10 120 facts-times-all 12 × 11 132 facts-times-all 12 × 12 144 facts-times-all Triangle 3 sides, 3 angles math-2d-shapes Quadrilateral 4 sides, 4 angles math-2d-shapes Pentagon 5 sides math-2d-shapes Hexagon 6 sides math-2d-shapes Heptagon 7 sides (Also called a septagon) math-2d-shapes Octagon 8 sides (A stop sign) math-2d-shapes Nonagon 9 sides math-2d-shapes Decagon 10 sides math-2d-shapes Hendecagon 11 sides (Also called an undecagon) math-2d-shapes Dodecagon 12 sides math-2d-shapes Circle All points the same distance from the center; no sides or corners math-2d-shapes Oval (ellipse) A stretched circle math-2d-shapes Square 4 equal sides and 4 right angles math-2d-shapes Rectangle 4 right angles; opposite sides equal math-2d-shapes Rhombus 4 equal sides; opposite angles equal math-2d-shapes Parallelogram Both pairs of opposite sides are parallel and equal math-2d-shapes Trapezoid A quadrilateral with at least one pair of parallel sides math-2d-shapes Kite A quadrilateral with two pairs of equal sides that meet side by side math-2d-shapes Equilateral triangle All 3 sides equal; all angles 60 degrees math-2d-shapes Isosceles triangle At least 2 equal sides and 2 equal angles math-2d-shapes Scalene triangle No equal sides math-2d-shapes Right triangle Has one 90-degree angle math-2d-shapes Acute triangle All three angles are less than 90 degrees math-2d-shapes Obtuse triangle Has one angle greater than 90 degrees math-2d-shapes Regular polygon A polygon with all sides equal and all angles equal math-2d-shapes Irregular polygon A polygon whose sides or angles are not all equal math-2d-shapes Convex polygon A polygon with no angle pointing inward math-2d-shapes Concave polygon A polygon with at least one angle pointing inward math-2d-shapes Sum of angles in a triangle 180 degrees math-2d-shapes Sum of angles in a quadrilateral 360 degrees math-2d-shapes Sum of angles in a pentagon 540 degrees math-2d-shapes Sum of angles in a hexagon 720 degrees math-2d-shapes Sum of angles in an n-sided polygon (n − 2) × 180 degrees math-2d-shapes Each angle of a regular hexagon 120 degrees math-2d-shapes Each angle of a regular pentagon 108 degrees math-2d-shapes Each angle of a regular octagon 135 degrees math-2d-shapes Number of diagonals of a quadrilateral 2 math-2d-shapes Number of diagonals of a pentagon 5 math-2d-shapes Number of diagonals of a hexagon 9 math-2d-shapes Diagonal A segment joining two corners of a polygon that are not next to each other math-2d-shapes Is a square a rectangle? Yes. A square is a rectangle with all sides equal. math-2d-shapes Is a square a rhombus? Yes. A square is a rhombus with right angles. math-2d-shapes Is a rectangle a parallelogram? Yes. Its opposite sides are parallel. math-2d-shapes Do the diagonals of a rectangle have equal lengths? Yes math-2d-shapes Do the diagonals of a rhombus meet at right angles? Yes, and they bisect each other math-2d-shapes Crescent A moon shape formed by two arcs math-2d-shapes Semicircle Half of a circle math-2d-shapes Quarter circle A quarter of a circle; a 90-degree sector math-2d-shapes Star shape (pentagram) A five-pointed star drawn with five straight lines math-2d-shapes Cube 6 square faces, 12 edges, 8 vertices math-3d-solids Rectangular prism 6 rectangular faces, 12 edges, 8 vertices (A shoebox) math-3d-solids Triangular prism 5 faces, 9 edges, 6 vertices math-3d-solids Square pyramid 5 faces, 8 edges, 5 vertices math-3d-solids Triangular pyramid (tetrahedron) 4 faces, 6 edges, 4 vertices math-3d-solids Pentagonal prism 7 faces, 15 edges, 10 vertices math-3d-solids Hexagonal prism 8 faces, 18 edges, 12 vertices math-3d-solids Cylinder 2 flat circular faces and 1 curved surface; no vertices math-3d-solids Cone 1 flat circular face, 1 curved surface, 1 vertex math-3d-solids Sphere A perfectly round solid; every surface point is the same distance from the center math-3d-solids Hemisphere Half of a sphere math-3d-solids Face A flat surface of a solid math-3d-solids Edge A line where two faces meet math-3d-solids Vertex A corner where edges meet math-3d-solids Base The face a solid sits on, or the matching ends of a prism math-3d-solids Prism A solid with two parallel identical bases and flat rectangular sides math-3d-solids Pyramid A solid with one base and triangular sides meeting at an apex math-3d-solids Apex The top point of a pyramid or cone math-3d-solids Euler's formula for polyhedra Vertices − Edges + Faces = 2 (V − E + F = 2) math-3d-solids Polyhedron A solid with only flat faces math-3d-solids Platonic solid A polyhedron whose faces are all the same regular polygon, with the same number meeting at each vertex math-3d-solids How many Platonic solids are there? Five math-3d-solids Tetrahedron Platonic solid with 4 triangular faces math-3d-solids Cube (hexahedron) Platonic solid with 6 square faces math-3d-solids Octahedron Platonic solid with 8 triangular faces math-3d-solids Dodecahedron Platonic solid with 12 pentagonal faces math-3d-solids Icosahedron Platonic solid with 20 triangular faces math-3d-solids Net of a cube A flat pattern of 6 connected squares that folds into a cube math-3d-solids Lateral face A side face of a prism or pyramid (not a base) math-3d-solids Slant height The distance along the sloped side of a cone or pyramid from apex to base edge math-3d-solids Which solid rolls and slides? A cylinder math-3d-solids Which solid rolls but does not stack? A sphere math-3d-solids Cuboid Another name for a rectangular prism math-3d-solids Torus A donut shape math-3d-solids Ellipsoid A stretched sphere, like a rugby ball math-3d-solids Frustum A cone or pyramid with its top sliced off parallel to the base math-3d-solids Commutative property of addition a + b = b + a math-algebra-rules Commutative property of multiplication ab = ba math-algebra-rules Associative property of addition (a + b) + c = a + (b + c) math-algebra-rules Associative property of multiplication (ab)c = a(bc) math-algebra-rules Distributive property a(b + c) = ab + ac math-algebra-rules Additive identity a + 0 = a math-algebra-rules Multiplicative identity a × 1 = a math-algebra-rules Additive inverse a + (−a) = 0 math-algebra-rules Multiplicative inverse a × (1/a) = 1, for a not zero math-algebra-rules Zero property of multiplication a × 0 = 0 math-algebra-rules Reflexive property a = a math-algebra-rules Symmetric property If a = b, then b = a math-algebra-rules Transitive property If a = b and b = c, then a = c math-algebra-rules Addition property of equality If a = b, then a + c = b + c math-algebra-rules Multiplication property of equality If a = b, then ac = bc math-algebra-rules Substitution property If a = b, then b can replace a in any expression math-algebra-rules Rule for multiplying two negatives The product is positive math-algebra-rules Rule for multiplying a negative and a positive The product is negative math-algebra-rules Rule for dividing two negatives The quotient is positive math-algebra-rules Subtracting a negative a − (−b) = a + b math-algebra-rules Solve x + 7 = 12 x = 5 math-algebra-rules Solve 3x = 21 x = 7 math-algebra-rules Solve 2x − 5 = 9 x = 7 math-algebra-rules Solve x/4 = 6 x = 24 math-algebra-rules Solve 5x + 3 = 2x + 15 x = 4 math-algebra-rules Rule for inequalities Multiplying or dividing by a negative number flips the inequality sign math-algebra-rules Solve −2x > 6 x < −3 math-algebra-rules Solve |x| = 5 x = 5 or x = −5 math-algebra-rules Solve |x| < 3 −3 < x < 3 math-algebra-rules Solve |x| > 3 x < −3 or x > 3 math-algebra-rules FOIL First, Outer, Inner, Last: multiplying two binomials math-algebra-rules Expand (x + 3)(x + 2) x² + 5x + 6 math-algebra-rules Expand (x + 4)² x² + 8x + 16 math-algebra-rules Expand (x − 5)² x² − 10x + 25 math-algebra-rules Expand (x + 3)(x − 3) x² − 9 math-algebra-rules Factor x² + 7x + 12 (x + 3)(x + 4) math-algebra-rules Factor x² − 9 (x − 3)(x + 3) math-algebra-rules Factor x² − 5x + 6 (x − 2)(x − 3) math-algebra-rules Factor x² + 2x + 1 (x + 1)² math-algebra-rules Factor 2x² + 6x 2x(x + 3) math-algebra-rules Sum of cubes a³ + b³ = (a + b)(a² − ab + b²) math-algebra-rules Difference of cubes a³ − b³ = (a − b)(a² + ab + b²) math-algebra-rules Binomial cube (a + b)³ = a³ + 3a²b + 3ab² + b³ math-algebra-rules Solve x² = 49 x = 7 or x = −7 math-algebra-rules Solve x² − 5x + 6 = 0 x = 2 or x = 3 math-algebra-rules Midpoint of two numbers Their average: (a + b)/2 math-algebra-rules Cross-multiplication If a/b = c/d, then ad = bc math-algebra-rules Adding fractions with unlike denominators Use a common denominator first math-algebra-rules Slope between two points (y₂ − y₁)/(x₂ − x₁) math-algebra-rules Perpendicular to slope 2 Slope −1/2 math-algebra-rules Distance, rate, time d = r × t math-algebra-rules Percent change (new − old) / old × 100% math-algebra-rules Simple interest I = P r t math-algebra-rules Average (mean) formula Sum of values divided by number of values math-algebra-rules Order of solving with variables on both sides Collect variable terms on one side, constants on the other, then divide math-algebra-rules Check a solution Substitute it back into the original equation math-algebra-rules Variable A letter that stands for a number math-algebra-terms Constant A fixed number that does not change math-algebra-terms Coefficient The number multiplying a variable (In −4y, it is −4) math-algebra-terms Term A part of an expression separated by + or − math-algebra-terms Like terms Terms with the same variables raised to the same powers math-algebra-terms Expression A combination of numbers, variables and operations without an equals sign math-algebra-terms Equation A statement that two expressions are equal math-algebra-terms Inequality A statement comparing two expressions using <, >, ≤ or ≥ math-algebra-terms Solve Find the value of the variable that makes the statement true math-algebra-terms Evaluate Replace variables with numbers and compute the result math-algebra-terms Simplify Combine like terms and reduce an expression math-algebra-terms Factor Write an expression as a product of simpler expressions math-algebra-terms Expand Multiply out brackets to remove them math-algebra-terms Substitution Replacing a variable with a value or another expression math-algebra-terms Linear equation An equation whose graph is a straight line; the variable has degree 1 math-algebra-terms Slope-intercept form y = mx + b math-algebra-terms Point-slope form y − y₁ = m(x − x₁) math-algebra-terms Standard form of a line Ax + By = C math-algebra-terms Slope Rise over run; change in y divided by change in x math-algebra-terms Rise The vertical change between two points math-algebra-terms Run The horizontal change between two points math-algebra-terms Y-intercept Where a graph crosses the y-axis (x = 0) math-algebra-terms X-intercept Where a graph crosses the x-axis (y = 0) math-algebra-terms Parallel lines (slopes) Have equal slopes math-algebra-terms Perpendicular lines (slopes) Slopes are negative reciprocals; their product is −1 math-algebra-terms Slope of a horizontal line 0 math-algebra-terms Slope of a vertical line Undefined math-algebra-terms Function A rule that gives each input exactly one output math-algebra-terms Function notation f(x) is read 'f of x'; the output for input x math-algebra-terms Domain All allowed input values math-algebra-terms Range All resulting output values math-algebra-terms Independent variable The input variable, usually x math-algebra-terms Dependent variable The output variable, usually y; it depends on x math-algebra-terms Vertical line test A graph is a function if no vertical line crosses it twice math-algebra-terms Absolute value The distance of a number from zero, written |x| math-algebra-terms System of equations Two or more equations solved together math-algebra-terms Substitution method Solve one equation for a variable and substitute into the other math-algebra-terms Elimination method Add or subtract equations to cancel a variable math-algebra-terms Consistent system A system with at least one solution math-algebra-terms Inconsistent system A system with no solution, such as parallel lines math-algebra-terms Dependent system A system with infinitely many solutions; the same line math-algebra-terms Polynomial An expression of terms with whole-number powers of variables math-algebra-terms Degree The highest power of the variable in a polynomial math-algebra-terms Standard form of a polynomial Terms written from the highest power to the lowest math-algebra-terms Quadratic equation An equation of the form ax² + bx + c = 0 math-algebra-terms Quadratic formula x = (−b ± √(b² − 4ac)) / 2a math-algebra-terms Completing the square Rewriting a quadratic so a perfect square trinomial appears math-algebra-terms Vertex form y = a(x − h)² + k math-algebra-terms Difference of squares a² − b² = (a − b)(a + b) math-algebra-terms Perfect square trinomial a² + 2ab + b² = (a + b)² math-algebra-terms Greatest common factor (GCF) The largest factor shared by all the terms math-algebra-terms Zero product property If ab = 0, then a = 0 or b = 0 math-algebra-terms Root, zero or solution An x value where the function equals 0 math-algebra-terms Radical A root symbol, such as √ math-algebra-terms Rationalize the denominator Rewrite a fraction so the denominator has no radical math-algebra-terms Rational expression A fraction whose numerator and denominator are polynomials math-algebra-terms Inverse variation y = k/x; as one quantity goes up, the other goes down math-algebra-terms Direct variation y = kx; the ratio y/x is constant math-algebra-terms Constant of variation The constant k in a variation equation math-algebra-terms Arithmetic sequence formula aₙ = a₁ + (n − 1)d math-algebra-terms Geometric sequence formula aₙ = a₁ · rⁿ⁻¹ math-algebra-terms Exponential growth y = a(1 + r)^t with r > 0 math-algebra-terms Exponential decay y = a(1 − r)^t with 0 < r < 1 math-algebra-terms Compound interest formula A = P(1 + r/n)^(nt) math-algebra-terms Continuous compounding formula A = Pe^(rt) math-algebra-terms Acute angle Between 0° and 90° math-angles-lines Right angle Exactly 90° math-angles-lines Obtuse angle Between 90° and 180° math-angles-lines Straight angle Exactly 180° math-angles-lines Reflex angle Between 180° and 360° math-angles-lines Full rotation 360° math-angles-lines Complementary angles Add up to 90° math-angles-lines Supplementary angles Add up to 180° math-angles-lines Complement of 30° 60° math-angles-lines Supplement of 110° 70° math-angles-lines Vertical angles Opposite angles made by two crossing lines; they are equal math-angles-lines Adjacent angles Share a vertex and a side, with no overlap math-angles-lines Linear pair Two adjacent angles that form a straight line; add up to 180° math-angles-lines Corresponding angles Angles in matching positions at two different crossings of a transversal math-angles-lines Alternate interior angles Angles inside the two lines on opposite sides of the transversal math-angles-lines Alternate exterior angles Angles outside the two lines on opposite sides of the transversal math-angles-lines Same-side interior angles Angles inside the two lines on the same side of the transversal math-angles-lines Vertex of an angle The point where the two rays meet math-angles-lines Angle bisector A ray that splits an angle into two equal angles math-angles-lines Protractor A tool for measuring angles in degrees math-angles-lines Line A straight path that goes on forever in both directions math-angles-lines Line segment A piece of a line between two endpoints math-angles-lines Ray A part of a line with one endpoint, going on forever one way math-angles-lines Collinear points Points that lie on the same line math-angles-lines Coplanar Lying in the same flat plane math-angles-lines Parallel lines Lines in a plane that never meet math-angles-lines Perpendicular lines Lines that meet at 90° math-angles-lines Skew lines Lines in 3D that do not meet and are not parallel math-angles-lines Midpoint The point exactly halfway along a segment math-angles-lines Perpendicular bisector A line that cuts a segment in half at a right angle math-angles-lines Transversal A line that crosses two or more lines math-angles-lines Plane A flat surface that extends forever in every direction math-angles-lines Center of a circle The point equally far from every point on the circle math-angles-lines Radius A segment from the center to the circle math-angles-lines Diameter A chord through the center; the longest chord math-angles-lines Chord A segment joining two points on a circle math-angles-lines Secant A line that cuts through a circle at two points math-angles-lines Tangent A line that touches a circle at one point math-angles-lines Arc A piece of the circle's outline math-angles-lines Minor arc An arc smaller than a semicircle math-angles-lines Major arc An arc larger than a semicircle math-angles-lines Semicircle Half a circle; an arc of 180° math-angles-lines Sector A region bounded by two radii and an arc math-angles-lines Segment of a circle A region bounded by a chord and an arc math-angles-lines Central angle An angle with its vertex at the center of a circle math-angles-lines Inscribed angle An angle with its vertex on the circle math-angles-lines Concentric circles Circles that share the same center math-angles-lines Congruent circles Circles with equal radii math-angles-lines Degrees in a circle 360 math-angles-lines Radians in a full circle 2π math-angles-lines Degrees in one radian About 57.3° math-angles-lines Convert 180° to radians π radians math-angles-lines Convert 90° to radians π/2 math-angles-lines Convert 60° to radians π/3 math-angles-lines Convert π/4 to degrees 45° math-angles-lines Convert π/6 to degrees 30° math-angles-lines Limit The value a function approaches as the input approaches a number math-calculus-concepts One-sided limit A limit approached only from the left or only from the right math-calculus-concepts Limit at infinity The value a function approaches as x grows without bound math-calculus-concepts Continuous at a point The limit equals the function value there math-calculus-concepts Removable discontinuity A hole in the graph that could be filled in math-calculus-concepts Jump discontinuity The left and right limits exist but differ math-calculus-concepts Infinite discontinuity The function grows without bound near that point (vertical asymptote) math-calculus-concepts Intermediate value theorem A continuous function takes every value between f(a) and f(b) math-calculus-concepts Extreme value theorem A continuous function on a closed interval has a maximum and a minimum math-calculus-concepts Differentiable Having a derivative at a point; the graph is smooth there math-calculus-concepts Differentiable implies Continuous math-calculus-concepts Local maximum A point higher than nearby points math-calculus-concepts Local minimum A point lower than nearby points math-calculus-concepts Absolute (global) extremum The highest or lowest point over the whole domain math-calculus-concepts Optimization Using derivatives to find the best (largest or smallest) value math-calculus-concepts Linear approximation Using the tangent line to estimate nearby values math-calculus-concepts Differential dy = f'(x) dx; a small change in y math-calculus-concepts Antiderivative A function whose derivative is the given function math-calculus-concepts Riemann sum A sum of rectangle areas that approximates an area math-calculus-concepts Definite integral A number: signed area under a curve between limits math-calculus-concepts Net change The integral of a rate of change math-calculus-concepts Slope field A picture of tangent slopes for a differential equation math-calculus-concepts Differential equation An equation involving a function and its derivatives math-calculus-concepts Separable differential equation One where the variables can be moved to opposite sides math-calculus-concepts Exponential growth equation dy/dt = ky has solution y = Ce^(kt) math-calculus-concepts Sequence limit The value the terms approach as n grows math-calculus-concepts Geometric series test Converges if |r| < 1, with sum a/(1 − r) math-calculus-concepts p-series Σ 1/nᵖ converges if p > 1 and diverges if p ≤ 1 math-calculus-concepts Ratio test Compare consecutive terms; convergence if the limit of |aₙ₊₁/aₙ| is less than 1 math-calculus-concepts Alternating series test Converges if terms alternate sign, decrease in size and go to 0 math-calculus-concepts Taylor series A power series that matches a function's derivatives at a point math-calculus-concepts Maclaurin series A Taylor series centered at 0 math-calculus-concepts Maclaurin series of eˣ 1 + x + x²/2! + x³/3! + ... math-calculus-concepts Maclaurin series of sin x x − x³/3! + x⁵/5! − ... math-calculus-concepts Maclaurin series of cos x 1 − x²/2! + x⁴/4! − ... math-calculus-concepts Maclaurin series of 1/(1 − x) 1 + x + x² + x³ + ... for |x| < 1 math-calculus-concepts Radius of convergence How far from the center a power series converges math-calculus-concepts lim (x→0) sin x / x 1 math-calculus-concepts lim (x→0) (1 − cos x) / x 0 math-calculus-concepts lim (x→0) (eˣ − 1) / x 1 math-calculus-concepts lim (x→∞) (1 + 1/x)ˣ e math-calculus-concepts lim (x→∞) 1/x 0 math-calculus-concepts lim (x→1) (x² − 1)/(x − 1) 2 math-calculus-concepts lim (x→∞) (3x² + 1)/(x² + 5) 3 math-calculus-concepts lim (x→0) ln(1 + x)/x 1 math-calculus-concepts Squeeze theorem If a function is trapped between two functions with the same limit, it has that limit math-calculus-concepts Vertical asymptote A vertical line the graph approaches where values blow up math-calculus-concepts Horizontal asymptote A horizontal line the graph approaches as x goes to ±∞ math-calculus-concepts Indeterminate form A limit form such as 0/0 or ∞/∞ that needs more work math-calculus-concepts Partial derivative A derivative with respect to one variable, others held constant math-calculus-concepts Gradient A vector of partial derivatives pointing in the direction of steepest increase math-calculus-concepts Double integral Integral over a 2D region, finding volume under a surface math-calculus-concepts Multivariable calculus Calculus of functions with more than one input math-calculus-concepts Differential vs. derivative The derivative is a rate; the differential is a small change math-calculus-concepts Leibniz notation dy/dx math-calculus-concepts Lagrange notation f'(x) math-calculus-concepts Newton's notation ẏ, a dot over the variable math-calculus-concepts Who developed calculus independently? Isaac Newton and Gottfried Leibniz math-calculus-concepts Horizontal axis The x-axis math-coordinate-graphing Vertical axis The y-axis math-coordinate-graphing Origin The point (0, 0) math-coordinate-graphing Ordered pair (x, y) First number is across, second is up or down math-coordinate-graphing Quadrant I x positive, y positive (top right) math-coordinate-graphing Quadrant II x negative, y positive (top left) math-coordinate-graphing Quadrant III x negative, y negative (bottom left) math-coordinate-graphing Quadrant IV x positive, y negative (bottom right) math-coordinate-graphing Where is (0, 5)? On the y-axis, 5 units above the origin math-coordinate-graphing Where is (−3, 0)? On the x-axis, 3 units left of the origin math-coordinate-graphing Which quadrant is (3, −2)? IV math-coordinate-graphing Which quadrant is (−4, −1)? III math-coordinate-graphing Which quadrant is (−2, 6)? II math-coordinate-graphing Reflect (3, 4) over the x-axis (3, −4) math-coordinate-graphing Reflect (3, 4) over the y-axis (−3, 4) math-coordinate-graphing Reflect (3, 4) through the origin (−3, −4) math-coordinate-graphing Reflect (a, b) over the line y = x (b, a) math-coordinate-graphing Translate (2, 1) by 3 right and 2 up (5, 3) math-coordinate-graphing Rotate (a, b) 90° counterclockwise about the origin (−b, a) math-coordinate-graphing Rotate (a, b) 180° about the origin (−a, −b) math-coordinate-graphing Dilate (2, 3) by scale factor 2 about the origin (4, 6) math-coordinate-graphing Distance from (0, 0) to (3, 4) 5 math-coordinate-graphing Distance from (1, 2) to (4, 6) 5 math-coordinate-graphing Midpoint of (2, 4) and (6, 8) (4, 6) math-coordinate-graphing Midpoint of (−2, 3) and (4, −1) (1, 1) math-coordinate-graphing Slope through (0, 0) and (2, 4) 2 math-coordinate-graphing Slope through (1, 5) and (3, 1) −2 math-coordinate-graphing Slope of y = 3x + 2 3 math-coordinate-graphing Y-intercept of y = 3x + 2 2 math-coordinate-graphing X-intercept of y = 2x − 6 3 math-coordinate-graphing Graph of y = 5 A horizontal line math-coordinate-graphing Graph of x = −2 A vertical line math-coordinate-graphing Graph of y = x A line through the origin with slope 1 math-coordinate-graphing Graph of y = x² A parabola opening up with vertex at the origin math-coordinate-graphing Graph of y = −x² A parabola opening down math-coordinate-graphing Graph of y = |x| A V shape with vertex at the origin math-coordinate-graphing Graph of y = 1/x A hyperbola with branches in quadrants I and III math-coordinate-graphing Graph of y = √x Half of a sideways parabola starting at the origin math-coordinate-graphing Graph of y = 2ˣ A curve that rises quickly and passes through (0, 1) math-coordinate-graphing Graph of x² + y² = 9 A circle with center (0, 0) and radius 3 math-coordinate-graphing Shift y = f(x) up 3 y = f(x) + 3 math-coordinate-graphing Shift y = f(x) right 2 y = f(x − 2) math-coordinate-graphing Reflect y = f(x) over the x-axis y = −f(x) math-coordinate-graphing Reflect y = f(x) over the y-axis y = f(−x) math-coordinate-graphing Stretch y = f(x) vertically by 3 y = 3f(x) math-coordinate-graphing Vertex of y = (x − 2)² + 5 (2, 5) math-coordinate-graphing Axis of symmetry of y = (x − 2)² + 5 x = 2 math-coordinate-graphing Parallel to y = 2x + 1 Any line with slope 2 math-coordinate-graphing Perpendicular to y = 2x + 1 Any line with slope −1/2 math-coordinate-graphing Rise over run Slope math-coordinate-graphing Positive slope The line goes up from left to right math-coordinate-graphing Negative slope The line goes down from left to right math-coordinate-graphing Where lines of a system meet The solution of the system math-coordinate-graphing Line graph Plots points joined by lines to show change over time math-coordinate-graphing Scale of a graph The value each grid mark stands for math-coordinate-graphing Interval of an axis The step between marked numbers math-coordinate-graphing Latitude and longitude A coordinate system for locations on Earth math-coordinate-graphing Battleship grid A game that uses coordinates like B7 math-coordinate-graphing Plotting order Start at the origin, move across x, then up or down y math-coordinate-graphing René Descartes Introduced the Cartesian coordinate system math-coordinate-graphing Polar coordinates (r, θ) Distance from the origin and angle from the positive x-axis math-coordinate-graphing 3D coordinates Add a z-axis: (x, y, z) math-coordinate-graphing Number of axes in a 3D grid 3 math-coordinate-graphing ∛1 1 math-cube-roots ∛8 2 math-cube-roots ∛27 3 math-cube-roots ∛64 4 math-cube-roots ∛125 5 math-cube-roots ∛216 6 math-cube-roots ∛343 7 math-cube-roots ∛512 8 math-cube-roots ∛729 9 math-cube-roots ∛1000 10 math-cube-roots ∛1331 11 math-cube-roots ∛1728 12 math-cube-roots ∛2197 13 math-cube-roots ∛2744 14 math-cube-roots ∛3375 15 math-cube-roots ∛4096 16 math-cube-roots ∛4913 17 math-cube-roots ∛5832 18 math-cube-roots ∛6859 19 math-cube-roots ∛8000 20 math-cube-roots Constant: d/dx [c] 0 math-derivative-rules Power rule: d/dx [xⁿ] n xⁿ⁻¹ math-derivative-rules Constant multiple rule d/dx [c f(x)] = c f'(x) math-derivative-rules Sum rule d/dx [f + g] = f' + g' math-derivative-rules Difference rule d/dx [f − g] = f' − g' math-derivative-rules Product rule d/dx [fg] = f'g + fg' math-derivative-rules Quotient rule d/dx [f/g] = (f'g − fg') / g² math-derivative-rules Chain rule d/dx [f(g(x))] = f'(g(x)) · g'(x) math-derivative-rules d/dx [x] 1 math-derivative-rules d/dx [x²] 2x math-derivative-rules d/dx [x³] 3x² math-derivative-rules d/dx [√x] 1/(2√x) math-derivative-rules d/dx [1/x] −1/x² math-derivative-rules d/dx [eˣ] eˣ math-derivative-rules d/dx [aˣ] aˣ ln a math-derivative-rules d/dx [ln x] 1/x math-derivative-rules d/dx [log_a x] 1/(x ln a) math-derivative-rules d/dx [sin x] cos x math-derivative-rules d/dx [cos x] −sin x math-derivative-rules d/dx [tan x] sec² x math-derivative-rules d/dx [cot x] −csc² x math-derivative-rules d/dx [sec x] sec x tan x math-derivative-rules d/dx [csc x] −csc x cot x math-derivative-rules d/dx [arcsin x] 1/√(1 − x²) math-derivative-rules d/dx [arccos x] −1/√(1 − x²) math-derivative-rules d/dx [arctan x] 1/(1 + x²) math-derivative-rules d/dx [sinh x] cosh x math-derivative-rules d/dx [cosh x] sinh x math-derivative-rules d/dx [e^(2x)] 2e^(2x) math-derivative-rules d/dx [sin(3x)] 3 cos(3x) math-derivative-rules d/dx [(x² + 1)³] 6x(x² + 1)² math-derivative-rules d/dx [x eˣ] eˣ(x + 1) math-derivative-rules d/dx [x sin x] sin x + x cos x math-derivative-rules d/dx [ln(x²)] 2/x math-derivative-rules d/dx [5x⁴ − 3x + 7] 20x³ − 3 math-derivative-rules Derivative as slope The slope of the tangent line at a point math-derivative-rules Limit definition of derivative f'(x) = lim (h→0) [f(x + h) − f(x)] / h math-derivative-rules Second derivative The derivative of the derivative; tells concavity math-derivative-rules Critical point A point where f' = 0 or f' does not exist math-derivative-rules First derivative test f' changes from + to − at a local maximum, and − to + at a local minimum math-derivative-rules Second derivative test f'' > 0 gives a local minimum; f'' < 0 gives a local maximum math-derivative-rules Inflection point A point where concavity changes math-derivative-rules Concave up f'' > 0; the graph curves like a cup math-derivative-rules Concave down f'' < 0; the graph curves like a cap math-derivative-rules Velocity from position v(t) = s'(t) math-derivative-rules Acceleration from position a(t) = s''(t) math-derivative-rules Mean value theorem A smooth curve has a point where the slope equals the average slope over the interval math-derivative-rules L'Hôpital's rule For 0/0 or ∞/∞ forms, lim f/g = lim f'/g' (if that limit exists) math-derivative-rules Implicit differentiation Differentiate both sides with respect to x, using dy/dx for y terms math-derivative-rules Related rates Differentiate an equation in time to link rates of change math-derivative-rules Tangent line equation y − f(a) = f'(a)(x − a) math-derivative-rules Rolle's theorem If f(a) = f(b) and f is smooth, there is a point between where f' = 0 math-derivative-rules Product rule aᵐ · aⁿ = aᵐ⁺ⁿ (Same base: add exponents) math-exponent-rules Quotient rule aᵐ / aⁿ = aᵐ⁻ⁿ (Same base: subtract exponents) math-exponent-rules Power of a power (aᵐ)ⁿ = aᵐⁿ (Multiply exponents) math-exponent-rules Power of a product (ab)ⁿ = aⁿbⁿ math-exponent-rules Power of a quotient (a/b)ⁿ = aⁿ/bⁿ math-exponent-rules Zero exponent a⁰ = 1, for a not zero math-exponent-rules Negative exponent a⁻ⁿ = 1/aⁿ math-exponent-rules Fractional exponent a^(1/n) = the nth root of a math-exponent-rules Rational exponent a^(m/n) = (nth root of a)^m math-exponent-rules Exponent of 1 a¹ = a math-exponent-rules 2³ · 2⁴ 2⁷ = 128 math-exponent-rules 5⁶ / 5² 5⁴ = 625 math-exponent-rules (3²)³ 3⁶ = 729 math-exponent-rules (2x)³ 8x³ math-exponent-rules (x²)⁴ x⁸ math-exponent-rules x³ · x⁵ x⁸ math-exponent-rules x⁹ / x³ x⁶ math-exponent-rules (−2)² 4 math-exponent-rules −2² −4 (The exponent applies only to the 2) math-exponent-rules (−2)³ −8 math-exponent-rules 10⁰ 1 math-exponent-rules 0⁰ Usually left undefined (or treated as 1 in some settings) math-exponent-rules 2⁻³ 1/8 math-exponent-rules 10⁻² 0.01 math-exponent-rules 4^(1/2) 2 math-exponent-rules 8^(1/3) 2 math-exponent-rules 27^(2/3) 9 math-exponent-rules 16^(3/4) 8 math-exponent-rules 9^(−1/2) 1/3 math-exponent-rules Is (a + b)² equal to a² + b²? No. (a + b)² = a² + 2ab + b² math-exponent-rules Is aᵐ + aⁿ = aᵐ⁺ⁿ? No. The product rule works for multiplying, not adding math-exponent-rules Negative base with even power The result is positive math-exponent-rules Negative base with odd power The result is negative math-exponent-rules Scientific notation of 45,000 4.5 × 10⁴ math-exponent-rules Scientific notation of 0.0007 7 × 10⁻⁴ math-exponent-rules (3 × 10²)(2 × 10³) 6 × 10⁵ math-exponent-rules Square root as an exponent √a = a^(1/2) math-exponent-rules Cube root as an exponent ∛a = a^(1/3) math-exponent-rules Reciprocal of x⁻² x² math-exponent-rules Simplify (x³y²)² x⁶y⁴ math-exponent-rules Simplify x⁻²y³ with positive exponents y³/x² math-exponent-rules 0! 1 (By definition) math-factorials 1! 1 math-factorials 2! 2 math-factorials 3! 6 math-factorials 4! 24 math-factorials 5! 120 math-factorials 6! 720 math-factorials 7! 5040 math-factorials 8! 40320 math-factorials 9! 362880 math-factorials 10! 3628800 math-factorials 11! 39916800 math-factorials 12! 479001600 math-factorials 1/2 as a decimal 0.5 math-fraction-decimal-percent 1/2 as a percent 50% math-fraction-decimal-percent 1/3 as a decimal 0.333... (3 repeats) math-fraction-decimal-percent 1/3 as a percent about 33.3%, exactly 33 and 1/3% math-fraction-decimal-percent 2/3 as a decimal 0.666... (6 repeats) math-fraction-decimal-percent 2/3 as a percent about 66.7%, exactly 66 and 2/3% math-fraction-decimal-percent 1/4 as a decimal 0.25 math-fraction-decimal-percent 1/4 as a percent 25% math-fraction-decimal-percent 3/4 as a decimal 0.75 math-fraction-decimal-percent 3/4 as a percent 75% math-fraction-decimal-percent 1/5 as a decimal 0.2 math-fraction-decimal-percent 1/5 as a percent 20% math-fraction-decimal-percent 2/5 as a decimal 0.4 math-fraction-decimal-percent 2/5 as a percent 40% math-fraction-decimal-percent 3/5 as a decimal 0.6 math-fraction-decimal-percent 3/5 as a percent 60% math-fraction-decimal-percent 4/5 as a decimal 0.8 math-fraction-decimal-percent 4/5 as a percent 80% math-fraction-decimal-percent 1/6 as a decimal 0.1666... (6 repeats) math-fraction-decimal-percent 1/6 as a percent about 16.7%, exactly 16 and 2/3% math-fraction-decimal-percent 5/6 as a decimal 0.8333... (3 repeats) math-fraction-decimal-percent 5/6 as a percent about 83.3%, exactly 83 and 1/3% math-fraction-decimal-percent 1/8 as a decimal 0.125 math-fraction-decimal-percent 1/8 as a percent 12.5% math-fraction-decimal-percent 3/8 as a decimal 0.375 math-fraction-decimal-percent 3/8 as a percent 37.5% math-fraction-decimal-percent 5/8 as a decimal 0.625 math-fraction-decimal-percent 5/8 as a percent 62.5% math-fraction-decimal-percent 7/8 as a decimal 0.875 math-fraction-decimal-percent 7/8 as a percent 87.5% math-fraction-decimal-percent 1/10 as a decimal 0.1 math-fraction-decimal-percent 1/10 as a percent 10% math-fraction-decimal-percent 3/10 as a decimal 0.3 math-fraction-decimal-percent 3/10 as a percent 30% math-fraction-decimal-percent 7/10 as a decimal 0.7 math-fraction-decimal-percent 7/10 as a percent 70% math-fraction-decimal-percent 9/10 as a decimal 0.9 math-fraction-decimal-percent 9/10 as a percent 90% math-fraction-decimal-percent 1/20 as a decimal 0.05 math-fraction-decimal-percent 1/20 as a percent 5% math-fraction-decimal-percent 1/25 as a decimal 0.04 math-fraction-decimal-percent 1/25 as a percent 4% math-fraction-decimal-percent 3/20 as a decimal 0.15 math-fraction-decimal-percent 3/20 as a percent 15% math-fraction-decimal-percent 1/50 as a decimal 0.02 math-fraction-decimal-percent 1/50 as a percent 2% math-fraction-decimal-percent 1/100 as a decimal 0.01 math-fraction-decimal-percent 1/100 as a percent 1% math-fraction-decimal-percent 1/12 as a decimal 0.08333... (3 repeats) math-fraction-decimal-percent 1/12 as a percent about 8.3%, exactly 8 and 1/3% math-fraction-decimal-percent 2/8 as a decimal 0.25 math-fraction-decimal-percent 2/8 as a percent 25% math-fraction-decimal-percent 9/20 as a decimal 0.45 math-fraction-decimal-percent 9/20 as a percent 45% math-fraction-decimal-percent 11/20 as a decimal 0.55 math-fraction-decimal-percent 11/20 as a percent 55% math-fraction-decimal-percent 13/20 as a decimal 0.65 math-fraction-decimal-percent 13/20 as a percent 65% math-fraction-decimal-percent 17/20 as a decimal 0.85 math-fraction-decimal-percent 17/20 as a percent 85% math-fraction-decimal-percent 19/20 as a decimal 0.95 math-fraction-decimal-percent 19/20 as a percent 95% math-fraction-decimal-percent 3/25 as a decimal 0.12 math-fraction-decimal-percent 3/25 as a percent 12% math-fraction-decimal-percent 7/20 as a decimal 0.35 math-fraction-decimal-percent 7/20 as a percent 35% math-fraction-decimal-percent What does the denominator tell you? How many equal parts the whole is divided into math-fractions-skills What does the numerator tell you? How many of those parts are being counted math-fractions-skills Proper fraction Numerator is smaller than denominator, like 3/5 math-fractions-skills Improper fraction Numerator is equal to or larger than denominator, like 9/4 math-fractions-skills Mixed number A whole number plus a fraction, like 2 1/4 math-fractions-skills Equivalent fractions Different fractions with the same value math-fractions-skills How to make an equivalent fraction Multiply or divide top and bottom by the same number math-fractions-skills Simplify 6/8 3/4 math-fractions-skills Simplify 10/15 2/3 math-fractions-skills Simplify 12/18 2/3 math-fractions-skills Simplify 24/36 2/3 math-fractions-skills Simplify 15/45 1/3 math-fractions-skills Simplify 18/24 3/4 math-fractions-skills Simplify 20/50 2/5 math-fractions-skills Convert 7/4 to a mixed number 1 3/4 math-fractions-skills Convert 11/3 to a mixed number 3 2/3 math-fractions-skills Convert 2 1/2 to an improper fraction 5/2 math-fractions-skills Convert 3 2/5 to an improper fraction 17/5 math-fractions-skills Convert 1 3/4 to an improper fraction 7/4 math-fractions-skills 1/2 + 1/4 3/4 math-fractions-skills 1/3 + 1/6 1/2 math-fractions-skills 2/5 + 1/10 1/2 math-fractions-skills 3/8 + 1/4 5/8 math-fractions-skills 1/2 + 1/3 5/6 math-fractions-skills 3/4 + 2/3 1 5/12 math-fractions-skills 3/4 − 1/4 1/2 math-fractions-skills 5/6 − 1/3 1/2 math-fractions-skills 7/8 − 1/2 3/8 math-fractions-skills 2/3 − 1/4 5/12 math-fractions-skills 1 − 3/5 2/5 math-fractions-skills 2/3 × 3/4 1/2 math-fractions-skills 1/2 × 1/2 1/4 math-fractions-skills 3/5 × 5/6 1/2 math-fractions-skills 2/7 × 14 4 math-fractions-skills 1/3 of 12 4 math-fractions-skills 3/4 of 20 15 math-fractions-skills 2/5 of 25 10 math-fractions-skills 1/2 ÷ 1/4 2 math-fractions-skills 3/4 ÷ 3/8 2 math-fractions-skills 2/3 ÷ 4 1/6 math-fractions-skills 4 ÷ 1/2 8 math-fractions-skills How to divide fractions Multiply by the reciprocal of the second fraction math-fractions-skills Reciprocal of 5/7 7/5 math-fractions-skills Reciprocal of 4 1/4 math-fractions-skills Common denominator A shared multiple of the denominators, needed for adding and subtracting math-fractions-skills Least common denominator of 1/4 and 1/6 12 math-fractions-skills Least common denominator of 2/3 and 3/5 15 math-fractions-skills Which is greater, 3/4 or 2/3? 3/4 math-fractions-skills Which is greater, 5/8 or 3/5? 5/8 math-fractions-skills Which is smaller, 1/3 or 2/7? 2/7 math-fractions-skills Compare 4/9 and 1/2 4/9 < 1/2 math-fractions-skills Order 1/2, 1/3, 1/4 from least to greatest 1/4, 1/3, 1/2 math-fractions-skills Unit fractions get smaller as the denominator Gets bigger math-fractions-skills Benchmark fractions 0, 1/2 and 1, used for comparing and estimating math-fractions-skills Is 5/9 more or less than 1/2? More math-fractions-skills Is 3/8 more or less than 1/2? Less math-fractions-skills Fraction of a day that is 6 hours 1/4 math-fractions-skills Fraction of an hour that is 20 minutes 1/3 math-fractions-skills Fraction of a dollar that is a quarter 1/4 math-fractions-skills Fraction as division a/b means a ÷ b math-fractions-skills 1/0 Undefined; you cannot divide by zero math-fractions-skills 0/5 0 math-fractions-skills Fraction bar also means Divide math-fractions-skills Fraction of a circle in a half turn 1/2 math-fractions-skills Number line fraction: halfway between 0 and 1 1/2 math-fractions-skills Complex fraction A fraction with a fraction in the numerator or denominator math-fractions-skills Area of a rectangle A = l × w math-geometry-formulas Perimeter of a rectangle P = 2l + 2w math-geometry-formulas Area of a square A = s² math-geometry-formulas Perimeter of a square P = 4s math-geometry-formulas Area of a triangle A = ½ × b × h math-geometry-formulas Perimeter of a triangle Add the three sides math-geometry-formulas Area of a parallelogram A = b × h math-geometry-formulas Area of a trapezoid A = ½ × (b₁ + b₂) × h math-geometry-formulas Area of a rhombus A = ½ × d₁ × d₂ (The two diagonals) math-geometry-formulas Area of a kite A = ½ × d₁ × d₂ math-geometry-formulas Area of a circle A = πr² math-geometry-formulas Circumference of a circle C = 2πr = πd math-geometry-formulas Diameter in terms of radius d = 2r math-geometry-formulas Length of an arc (θ/360) × 2πr, with θ in degrees math-geometry-formulas Area of a sector (θ/360) × πr², with θ in degrees math-geometry-formulas Area of an equilateral triangle A = (√3/4) × s² math-geometry-formulas Area of a regular polygon A = ½ × apothem × perimeter math-geometry-formulas Heron's formula A = √[s(s−a)(s−b)(s−c)], where s is half the perimeter math-geometry-formulas Area of a triangle with two sides and the included angle A = ½ ab sin C math-geometry-formulas Area of an ellipse A = πab (a and b are the semi-axes) math-geometry-formulas Volume of a cube V = s³ math-geometry-formulas Surface area of a cube SA = 6s² math-geometry-formulas Volume of a rectangular prism V = l × w × h math-geometry-formulas Surface area of a rectangular prism SA = 2(lw + lh + wh) math-geometry-formulas Volume of any prism V = base area × height math-geometry-formulas Volume of a cylinder V = πr²h math-geometry-formulas Lateral surface area of a cylinder 2πrh math-geometry-formulas Total surface area of a cylinder SA = 2πr² + 2πrh math-geometry-formulas Volume of a cone V = ⅓πr²h math-geometry-formulas Lateral surface area of a cone πrl, where l is the slant height math-geometry-formulas Total surface area of a cone SA = πr² + πrl math-geometry-formulas Volume of a pyramid V = ⅓ × base area × height math-geometry-formulas Volume of a sphere V = 4/3 πr³ math-geometry-formulas Surface area of a sphere SA = 4πr² math-geometry-formulas Volume of a hemisphere V = 2/3 πr³ math-geometry-formulas Slant height of a cone l = √(r² + h²) math-geometry-formulas Diagonal of a rectangle d = √(l² + w²) math-geometry-formulas Diagonal of a square d = s√2 math-geometry-formulas Space diagonal of a rectangular prism d = √(l² + w² + h²) math-geometry-formulas Pythagorean theorem a² + b² = c² math-geometry-formulas Distance formula d = √[(x₂−x₁)² + (y₂−y₁)²] math-geometry-formulas Midpoint formula ((x₁+x₂)/2, (y₁+y₂)/2) math-geometry-formulas Slope formula m = (y₂−y₁)/(x₂−x₁) math-geometry-formulas Sum of interior angles of an n-gon (n − 2) × 180° math-geometry-formulas Each interior angle of a regular n-gon (n − 2) × 180° / n math-geometry-formulas Sum of exterior angles of any polygon 360° math-geometry-formulas Number of diagonals of an n-gon n(n − 3)/2 math-geometry-formulas Equation of a circle (x − h)² + (y − k)² = r² math-geometry-formulas Surface area of a triangular prism 2 × base area + perimeter of base × height math-geometry-formulas Volume of a torus V = 2π²Rr² math-geometry-formulas Perimeter of an equilateral triangle P = 3s math-geometry-formulas Area of an annulus (ring) A = π(R² − r²) math-geometry-formulas Area scale factor If lengths scale by k, area scales by k² math-geometry-formulas Volume scale factor If lengths scale by k, volume scales by k³ math-geometry-formulas Triangle angle sum theorem The three angles of a triangle add up to 180° math-geometry-theorems Exterior angle theorem An exterior angle of a triangle equals the sum of the two remote interior angles math-geometry-theorems Isosceles triangle theorem Angles opposite equal sides are equal math-geometry-theorems Converse of the isosceles triangle theorem Sides opposite equal angles are equal math-geometry-theorems Triangle inequality theorem The sum of any two sides is greater than the third side math-geometry-theorems Pythagorean theorem In a right triangle, a² + b² = c² math-geometry-theorems Converse of the Pythagorean theorem If a² + b² = c², the triangle is a right triangle math-geometry-theorems 45-45-90 triangle Sides are in the ratio 1 : 1 : √2 math-geometry-theorems 30-60-90 triangle Sides are in the ratio 1 : √3 : 2 math-geometry-theorems Midsegment theorem A segment joining midpoints of two sides is parallel to the third side and half as long math-geometry-theorems Triangle angle bisector theorem An angle bisector splits the opposite side in the ratio of the adjacent sides math-geometry-theorems SSS congruence Three pairs of equal sides make triangles congruent math-geometry-theorems SAS congruence Two sides and the angle between them equal makes triangles congruent math-geometry-theorems ASA congruence Two angles and the side between them equal makes triangles congruent math-geometry-theorems AAS congruence Two angles and a non-included side equal makes triangles congruent math-geometry-theorems HL congruence Right triangles with equal hypotenuse and one equal leg are congruent math-geometry-theorems CPCTC Corresponding parts of congruent triangles are congruent math-geometry-theorems AA similarity Two triangles with two equal angles are similar math-geometry-theorems SAS similarity Two proportional sides with equal included angle make triangles similar math-geometry-theorems SSS similarity Three proportional sides make triangles similar math-geometry-theorems Vertical angles theorem Vertical angles are equal math-geometry-theorems Linear pair postulate Two angles that form a straight line add up to 180° math-geometry-theorems Corresponding angles postulate Parallel lines cut by a transversal make equal corresponding angles math-geometry-theorems Alternate interior angles theorem Parallel lines cut by a transversal make equal alternate interior angles math-geometry-theorems Same-side interior angles theorem Parallel lines cut by a transversal make supplementary same-side interior angles math-geometry-theorems Perpendicular bisector theorem A point on the perpendicular bisector is equally far from the segment's endpoints math-geometry-theorems Angle bisector theorem (points) A point on an angle bisector is equally far from the angle's sides math-geometry-theorems Centroid The point where a triangle's three medians meet math-geometry-theorems Circumcenter The point where perpendicular bisectors meet; the center of the circumscribed circle math-geometry-theorems Incenter The point where angle bisectors meet; the center of the inscribed circle math-geometry-theorems Orthocenter The point where a triangle's altitudes meet math-geometry-theorems Centroid ratio The centroid divides each median in a 2 : 1 ratio, longer part toward the vertex math-geometry-theorems Inscribed angle theorem An inscribed angle is half its intercepted arc math-geometry-theorems Central angle theorem A central angle equals its intercepted arc math-geometry-theorems Thales' theorem An angle inscribed in a semicircle is a right angle math-geometry-theorems Tangent-radius theorem A tangent is perpendicular to the radius at the point of tangency math-geometry-theorems Two tangents theorem Tangent segments from one outside point to a circle are equal math-geometry-theorems Chord-chord theorem If two chords cross, the products of their pieces are equal math-geometry-theorems Opposite angles in a cyclic quadrilateral They add up to 180° math-geometry-theorems Parallelogram theorem: sides Opposite sides are equal math-geometry-theorems Parallelogram theorem: angles Opposite angles are equal; consecutive angles add up to 180° math-geometry-theorems Parallelogram theorem: diagonals Diagonals bisect each other math-geometry-theorems Rectangle diagonals Diagonals of a rectangle are equal math-geometry-theorems Rhombus diagonals Diagonals are perpendicular and bisect the angles math-geometry-theorems Kite diagonals Diagonals are perpendicular math-geometry-theorems Isosceles trapezoid base angles Base angles of an isosceles trapezoid are equal math-geometry-theorems Side-splitter theorem A line parallel to one side of a triangle divides the other two sides proportionally math-geometry-theorems Geometric mean (altitude) theorem The altitude to the hypotenuse of a right triangle is the geometric mean of the two segments it makes math-geometry-theorems Law of sines a/sin A = b/sin B = c/sin C math-geometry-theorems Law of cosines c² = a² + b² − 2ab cos C math-geometry-theorems Euler line The centroid, circumcenter and orthocenter of a triangle lie on one line math-geometry-theorems Ceva's theorem Cevians through a point in a triangle satisfy (AF/FB)(BD/DC)(CE/EA) = 1 math-geometry-theorems Pick's theorem Area of a lattice polygon = i + b/2 − 1, with i inside points and b boundary points math-geometry-theorems Euler's polyhedron formula V − E + F = 2 math-geometry-theorems ∫ 0 dx C math-integral-rules ∫ k dx kx + C math-integral-rules ∫ xⁿ dx (n ≠ −1) xⁿ⁺¹/(n + 1) + C math-integral-rules ∫ 1/x dx ln|x| + C math-integral-rules ∫ eˣ dx eˣ + C math-integral-rules ∫ aˣ dx aˣ/ln a + C math-integral-rules ∫ sin x dx −cos x + C math-integral-rules ∫ cos x dx sin x + C math-integral-rules ∫ sec² x dx tan x + C math-integral-rules ∫ csc² x dx −cot x + C math-integral-rules ∫ sec x tan x dx sec x + C math-integral-rules ∫ csc x cot x dx −csc x + C math-integral-rules ∫ tan x dx −ln|cos x| + C math-integral-rules ∫ cot x dx ln|sin x| + C math-integral-rules ∫ sec x dx ln|sec x + tan x| + C math-integral-rules ∫ ln x dx x ln x − x + C math-integral-rules ∫ 1/(1 + x²) dx arctan x + C math-integral-rules ∫ 1/√(1 − x²) dx arcsin x + C math-integral-rules ∫ x² dx x³/3 + C math-integral-rules ∫ x³ dx x⁴/4 + C math-integral-rules ∫ √x dx (2/3) x^(3/2) + C math-integral-rules ∫ 1/x² dx −1/x + C math-integral-rules ∫ e^(2x) dx e^(2x)/2 + C math-integral-rules ∫ cos(3x) dx sin(3x)/3 + C math-integral-rules ∫ 2x dx x² + C math-integral-rules ∫ (3x² + 2) dx x³ + 2x + C math-integral-rules Constant of integration + C, since many functions share the same derivative math-integral-rules Linearity of integrals ∫ (af + bg) dx = a∫f dx + b∫g dx math-integral-rules Fundamental theorem of calculus, part 1 d/dx ∫ₐˣ f(t) dt = f(x) math-integral-rules Fundamental theorem of calculus, part 2 ∫ₐᵇ f(x) dx = F(b) − F(a), where F' = f math-integral-rules Definite integral The signed area between a curve and the x-axis over an interval math-integral-rules Indefinite integral The family of all antiderivatives of a function math-integral-rules Integration by parts ∫ u dv = uv − ∫ v du math-integral-rules Substitution (u-substitution) Let u = g(x), then replace g'(x) dx with du math-integral-rules Partial fractions Split a rational function into simpler fractions before integrating math-integral-rules Riemann sum A sum of rectangle areas that approximates a definite integral math-integral-rules Trapezoidal rule Approximating area using trapezoids instead of rectangles math-integral-rules Average value of a function (1/(b − a)) ∫ₐᵇ f(x) dx math-integral-rules Area between curves ∫ (top − bottom) dx math-integral-rules Disk method V = π ∫ [R(x)]² dx math-integral-rules Washer method V = π ∫ ([R(x)]² − [r(x)]²) dx math-integral-rules Shell method V = 2π ∫ (radius)(height) dx math-integral-rules Arc length L = ∫ √(1 + [f'(x)]²) dx math-integral-rules ∫₀¹ x dx 1/2 math-integral-rules ∫₀¹ x² dx 1/3 math-integral-rules ∫₀^π sin x dx 2 math-integral-rules ∫₁^e (1/x) dx 1 math-integral-rules ∫₀^1 eˣ dx e − 1 math-integral-rules Improper integral An integral with an infinite limit or an unbounded function, defined with a limit math-integral-rules Position from velocity s(t) = ∫ v(t) dt math-integral-rules Velocity from acceleration v(t) = ∫ a(t) dt math-integral-rules Even function integral over [−a, a] 2 ∫₀ᵃ f(x) dx math-integral-rules Odd function integral over [−a, a] 0 math-integral-rules Swapping limits of integration ∫ₐᵇ f = −∫ᵦᵃ f math-integral-rules Scalar A single number, as opposed to a vector math-linear-algebra Vector An ordered list of numbers; has magnitude and direction math-linear-algebra Magnitude (norm) of a vector Its length, √(x² + y² + ...) math-linear-algebra Unit vector A vector of length 1 math-linear-algebra Zero vector The vector whose entries are all 0 math-linear-algebra Vector addition Add matching entries math-linear-algebra Scalar multiplication Multiply every entry of a vector or matrix by the same number math-linear-algebra Dot product Sum of products of matching entries; a · b = |a||b| cos θ math-linear-algebra Perpendicular vectors Their dot product is 0 (orthogonal) math-linear-algebra Cross product A vector perpendicular to two 3D vectors; length |a||b| sin θ math-linear-algebra Linear combination A sum of vectors each multiplied by a scalar math-linear-algebra Span The set of all linear combinations of some vectors math-linear-algebra Linear independence No vector in the set is a linear combination of the others math-linear-algebra Linear dependence At least one vector can be built from the others math-linear-algebra Basis A linearly independent set of vectors that spans a space math-linear-algebra Dimension The number of vectors in a basis math-linear-algebra Standard basis of R² (1, 0) and (0, 1) math-linear-algebra Vector space A set where vectors can be added and scaled by numbers following set rules math-linear-algebra Subspace A vector space inside a larger one math-linear-algebra Matrix A rectangular array of numbers in rows and columns math-linear-algebra Order (size) of a matrix Rows × columns, such as 2 × 3 math-linear-algebra Square matrix A matrix with the same number of rows and columns math-linear-algebra Identity matrix A square matrix with 1s on the main diagonal and 0s elsewhere; AI = A math-linear-algebra Zero matrix A matrix with all entries 0 math-linear-algebra Diagonal matrix A square matrix with zeros off the main diagonal math-linear-algebra Symmetric matrix A matrix equal to its transpose math-linear-algebra Transpose Swap rows and columns; written Aᵀ math-linear-algebra Matrix addition Add matching entries; matrices need the same size math-linear-algebra Matrix multiplication rule An m × n matrix times an n × p matrix gives an m × p matrix math-linear-algebra Is matrix multiplication commutative? No. AB is usually not equal to BA math-linear-algebra Inverse matrix A⁻¹ such that A A⁻¹ = I math-linear-algebra Invertible (nonsingular) matrix A square matrix with an inverse; its determinant is not 0 math-linear-algebra Singular matrix A matrix with no inverse; determinant 0 math-linear-algebra Determinant of a 2×2 matrix [a b; c d] ad − bc math-linear-algebra Determinant meaning Scale factor of area or volume under the matrix's transformation math-linear-algebra Trace of a matrix The sum of the diagonal entries math-linear-algebra Rank The number of linearly independent rows (or columns) math-linear-algebra Null space All vectors x with Ax = 0 math-linear-algebra Column space The span of the columns of a matrix math-linear-algebra Row space The span of the rows of a matrix math-linear-algebra Linear transformation A function T where T(u + v) = T(u) + T(v) and T(cu) = cT(u) math-linear-algebra System of linear equations as a matrix Ax = b math-linear-algebra Augmented matrix The coefficient matrix with the constants column attached math-linear-algebra Gaussian elimination Using row operations to turn a matrix into row-echelon form math-linear-algebra Row operations Swap rows, multiply a row by a nonzero number, add a multiple of one row to another math-linear-algebra Row-echelon form A staircase form with leading entries moving to the right math-linear-algebra Reduced row-echelon form Row-echelon form with leading 1s and zeros above and below them math-linear-algebra Pivot A leading nonzero entry in a row used in elimination math-linear-algebra Eigenvalue A number λ such that Av = λv for some nonzero vector v math-linear-algebra Eigenvector A nonzero vector that only gets scaled by a matrix math-linear-algebra Characteristic polynomial det(A − λI); its roots are the eigenvalues math-linear-algebra Orthogonal matrix A square matrix whose columns are perpendicular unit vectors; Qᵀ = Q⁻¹ math-linear-algebra Projection The shadow of one vector onto another math-linear-algebra Rotation matrix in 2D [cos θ, −sin θ; sin θ, cos θ] math-linear-algebra Cramer's rule Solve a system using ratios of determinants math-linear-algebra Orthonormal basis A basis of perpendicular unit vectors math-linear-algebra Gram-Schmidt process A method to turn a basis into an orthonormal one math-linear-algebra Diagonalization Writing A = PDP⁻¹ with D diagonal math-linear-algebra Singular value decomposition (SVD) Factoring a matrix into rotations and stretches: A = UΣVᵀ math-linear-algebra Definition of a logarithm log_b(x) = y means bʸ = x math-log-rules Product rule log_b(xy) = log_b x + log_b y math-log-rules Quotient rule log_b(x/y) = log_b x − log_b y math-log-rules Power rule log_b(xⁿ) = n log_b x math-log-rules Change of base log_b x = log x / log b (any base) math-log-rules Log of 1 log_b 1 = 0 math-log-rules Log of the base log_b b = 1 math-log-rules Inverse property 1 b^(log_b x) = x math-log-rules Inverse property 2 log_b(bˣ) = x math-log-rules Common logarithm Base 10, written log x math-log-rules Natural logarithm Base e, written ln x math-log-rules Binary logarithm Base 2, written log₂ x math-log-rules Domain of log x x > 0 math-log-rules log of a negative number Not defined among the real numbers math-log-rules log 100 2 math-log-rules log 1,000 3 math-log-rules log 0.1 −1 math-log-rules log 0.01 −2 math-log-rules log 10 1 math-log-rules log₂ 8 3 math-log-rules log₂ 64 6 math-log-rules log₂ 1024 10 math-log-rules log₃ 81 4 math-log-rules log₅ 125 3 math-log-rules log₄ 2 1/2 math-log-rules log₂ (1/8) −3 math-log-rules ln e 1 math-log-rules ln 1 0 math-log-rules ln(e²) 2 math-log-rules e^(ln 5) 5 math-log-rules Expand log(ab/c) log a + log b − log c math-log-rules Condense 2 log x + log y log(x²y) math-log-rules Solve 2ˣ = 16 x = 4 math-log-rules Solve 3ˣ = 5 x = log 5 / log 3 (about 1.465) math-log-rules Solve log₂ x = 5 x = 32 math-log-rules Is log(x + y) = log x + log y? No. The product rule needs log(xy) math-log-rules Is log(x)/log(y) the same as log(x/y)? No. They are different quantities math-log-rules Graph of y = log x: vertical asymptote x = 0 math-log-rules Graph of y = log x: x-intercept x = 1 math-log-rules Value of e (to 5 decimals) 2.71828 math-log-rules Half-life formula idea Amount = start × (1/2)^(t / half-life) math-log-rules Decibel scale uses A logarithm of a power ratio (10 log₁₀) math-log-rules Richter scale uses A base-10 logarithm of shaking amplitude math-log-rules pH formula pH = −log₁₀[H⁺] math-log-rules Euclid Wrote the Elements, the foundation of geometry (about 300 BCE) math-mathematicians Pythagoras Greek thinker linked to the Pythagorean theorem (6th century BCE) math-mathematicians Archimedes Greek scholar who found area and volume of the sphere and estimated pi math-mathematicians Eratosthenes Greek scholar who estimated Earth's size and made a sieve for primes math-mathematicians Thales Early Greek geometer; the inscribed angle in a semicircle theorem math-mathematicians Hypatia Mathematician and teacher in Alexandria, about 400 CE math-mathematicians Diophantus Greek writer on equations, often called a father of algebra math-mathematicians Aryabhata Indian mathematician and astronomer (born 476 CE); worked on pi and sines math-mathematicians Brahmagupta Indian mathematician who gave rules for zero and negative numbers (628 CE) math-mathematicians Al-Khwarizmi Persian scholar whose book on equations gave us the word algebra; 'algorithm' comes from his name math-mathematicians Omar Khayyam Persian poet and mathematician who studied cubic equations math-mathematicians Fibonacci (Leonardo of Pisa) Brought Hindu-Arabic numerals to Europe; the Fibonacci sequence math-mathematicians Zu Chongzhi Chinese mathematician who found pi to about 7 digits (5th century) math-mathematicians Liu Hui Chinese mathematician who estimated pi with polygons (3rd century) math-mathematicians Madhava Indian mathematician who found infinite series for sine, cosine and pi (14th century) math-mathematicians Ramanujan Indian mathematician known for remarkable formulas and number theory (1887-1920) math-mathematicians René Descartes Created the coordinate plane linking algebra and geometry math-mathematicians Pierre de Fermat Number theory; Fermat's Last Theorem math-mathematicians Blaise Pascal Pascal's triangle and early probability math-mathematicians Isaac Newton Co-inventor of calculus; laws of motion and gravity math-mathematicians Gottfried Leibniz Co-inventor of calculus; the integral and derivative notation we use math-mathematicians Leonhard Euler Prolific Swiss mathematician; e, i, graph theory, Euler's formula math-mathematicians Carl Friedrich Gauss German 'prince of mathematicians'; number theory, the bell curve, 1 to 100 sum story math-mathematicians Joseph-Louis Lagrange Mechanics and number theory math-mathematicians Pierre-Simon Laplace Probability and celestial mechanics math-mathematicians Sophie Germain French mathematician who worked on primes and elasticity math-mathematicians Évariste Galois Showed which polynomial equations can be solved by formulas; founded group theory math-mathematicians Niels Henrik Abel Norwegian mathematician; proved the quintic has no general formula math-mathematicians Bernhard Riemann German mathematician; Riemann hypothesis, geometry of curved spaces math-mathematicians Georg Cantor Founded set theory; showed that some infinities are bigger than others math-mathematicians David Hilbert German mathematician who posed 23 famous problems in 1900 math-mathematicians Henri Poincaré French mathematician; topology and chaos math-mathematicians Emmy Noether German mathematician; abstract algebra and Noether's theorem math-mathematicians Ada Lovelace Wrote the first published computer program (1843) for Babbage's engine math-mathematicians Charles Babbage Designed early mechanical computers math-mathematicians Florence Nightingale Nurse who used statistical charts to improve health care math-mathematicians Mary Somerville Scottish science writer and mathematician (1780-1872) math-mathematicians Maria Gaetana Agnesi Italian mathematician who wrote an early calculus textbook math-mathematicians Kurt Gödel Showed math systems have true statements that cannot be proven (1931) math-mathematicians Alan Turing Founded theoretical computer science; broke wartime codes math-mathematicians John von Neumann Game theory and computer architecture math-mathematicians Claude Shannon Founded information theory math-mathematicians Katherine Johnson NASA mathematician who calculated orbits and flight paths math-mathematicians Dorothy Vaughan NASA mathematician and computer programmer math-mathematicians Mary Jackson NASA mathematician and the agency's first Black female engineer math-mathematicians Grace Hopper Computer scientist and Navy admiral; early programming languages math-mathematicians Maryam Mirzakhani Iranian mathematician; first woman to win the Fields Medal (2014) math-mathematicians Terence Tao Australian-American mathematician; Fields Medal in 2006 math-mathematicians Andrew Wiles Proved Fermat's Last Theorem (1994-95) math-mathematicians Paul Erdős Hungarian mathematician who wrote with hundreds of collaborators math-mathematicians Benoit Mandelbrot Fractal geometry math-mathematicians John Napier Scottish inventor of logarithms (1614) math-mathematicians Robert Recorde Introduced the equals sign in 1557 math-mathematicians William Oughtred Invented the slide rule; used × for multiplication math-mathematicians George Boole Boolean algebra, the basis of computer logic math-mathematicians Carl Jacobi German mathematician; elliptic functions and determinants math-mathematicians Augustin-Louis Cauchy French mathematician who made calculus rigorous math-mathematicians Johannes Kepler Astronomer and mathematician; laws of planetary motion math-mathematicians Galileo Galilei Italian scientist who used math to describe motion math-mathematicians Thomas Bayes English statistician; Bayes' theorem math-mathematicians Karl Pearson Founded modern statistics; the correlation coefficient math-mathematicians G. H. Hardy English mathematician who recognized Ramanujan's genius math-mathematicians Hipparchus Greek astronomer who began trigonometry tables math-mathematicians Kilo- means 1,000 times the base unit math-metric-units Hecto- means 100 times the base unit math-metric-units Deka- means 10 times the base unit math-metric-units Deci- means One tenth of the base unit math-metric-units Centi- means One hundredth of the base unit math-metric-units Milli- means One thousandth of the base unit math-metric-units Micro- means One millionth of the base unit math-metric-units Nano- means One billionth of the base unit math-metric-units Mega- means One million times the base unit math-metric-units Giga- means One billion times the base unit math-metric-units Tera- means One trillion times the base unit math-metric-units Base unit of length in the metric system Meter (m) math-metric-units Base unit of mass in the metric system Kilogram (kg) (Gram is the small everyday unit) math-metric-units Base unit of volume often used for liquids Liter (L) math-metric-units Millimeters in 1 centimeter 10 math-metric-units Centimeters in 1 meter 100 math-metric-units Millimeters in 1 meter 1,000 math-metric-units Meters in 1 kilometer 1,000 math-metric-units Centimeters in 1 kilometer 100,000 math-metric-units Grams in 1 kilogram 1,000 math-metric-units Milligrams in 1 gram 1,000 math-metric-units Kilograms in 1 metric ton 1,000 math-metric-units Milliliters in 1 liter 1,000 math-metric-units Liters in 1 kiloliter 1,000 math-metric-units Cubic centimeters in 1 milliliter 1 math-metric-units Cubic centimeters in 1 liter 1,000 math-metric-units Mass of 1 liter of water About 1 kilogram math-metric-units Mass of 1 milliliter of water About 1 gram math-metric-units Convert 3 m to cm 300 cm math-metric-units Convert 250 cm to m 2.5 m math-metric-units Convert 4.5 km to m 4,500 m math-metric-units Convert 7,000 g to kg 7 kg math-metric-units Convert 2.5 L to mL 2,500 mL math-metric-units Convert 600 mm to cm 60 cm math-metric-units Convert 0.8 kg to g 800 g math-metric-units Convert 35 mg to g 0.035 g math-metric-units Which is larger: 1 cm or 1 mm? 1 cm (10 mm) math-metric-units Which is longer: 1 km or 1,200 m? 1,200 m math-metric-units About how thick is a dime? About 1 millimeter math-metric-units About how wide is a fingernail? About 1 centimeter math-metric-units About how tall is a door? About 2 meters math-metric-units About how long is a 15-minute walk? About 1 kilometer math-metric-units About how much does a paper clip weigh? About 1 gram math-metric-units About how much does a big textbook weigh? About 1 kilogram math-metric-units About how much does a spoonful hold? About 5 milliliters (a teaspoon) math-metric-units About how much does a soda bottle hold? About 2 liters (large) or 0.5 liter (small) math-metric-units Symbol for centimeter cm math-metric-units Symbol for kilogram kg math-metric-units Symbol for milliliter mL math-metric-units Symbol for micrometer μm math-metric-units Which direction when going from a large unit to a small unit? Multiply (there are more small units) math-metric-units Hectare A metric unit of land area, 10,000 square meters math-metric-units Square meters in 1 hectare 10,000 math-metric-units Celsius freezing point of water 0 degrees C math-metric-units Celsius boiling point of water (sea level) 100 degrees C math-metric-units Normal human body temperature in Celsius About 37 degrees C math-metric-units Room temperature in Celsius About 20 to 22 degrees C math-metric-units Kelvin value of 0 degrees C 273.15 K math-metric-units Absolute zero in Celsius −273.15 degrees C math-metric-units Mass of 1 cubic meter of water About 1,000 kilograms (1 metric ton) math-metric-units Value of a penny 1 cent math-money Value of a nickel 5 cents math-money Value of a dime 10 cents math-money Value of a quarter 25 cents math-money Value of a half dollar 50 cents math-money Value of a dollar coin 100 cents, or $1 math-money Cents in a dollar 100 math-money Pennies in a nickel 5 math-money Nickels in a dime 2 math-money Value of 2 dimes and a nickel 25 cents, the same as a quarter math-money Quarters in a dollar 4 math-money Dimes in a dollar 10 math-money Nickels in a dollar 20 math-money Pennies in a dollar 100 math-money Quarters in $5 20 math-money Total of 2 quarters and 1 dime 60 cents math-money Total of 3 dimes and 2 nickels 40 cents math-money Total of 1 quarter, 1 dime, 1 nickel 40 cents math-money Total of 4 dimes and 3 pennies 43 cents math-money Fewest coins for 41 cents 1 quarter, 1 dime, 1 nickel, 1 penny math-money Fewest coins for 30 cents 1 quarter and 1 nickel math-money Fewest coins for 65 cents 2 quarters, 1 dime, 1 nickel math-money Common bills in US money $1, $2, $5, $10, $20, $50, $100 math-money Change from $5 for a $3.50 purchase $1.50 math-money Change from $10 for a $6.25 purchase $3.75 math-money Change from $20 for a $13.80 purchase $6.20 math-money Change from a dollar for 35 cents 65 cents math-money Counting up to make change Start at the price and count up to the amount paid math-money Write 7 dollars and 5 cents with a dollar sign $7.05 math-money Write 275 cents in dollars $2.75 math-money Write $0.08 in cents 8 cents math-money Cost of 3 items at $1.25 each $3.75 math-money Cost of 4 items at $2.50 each $10.00 math-money Unit price The cost for one unit, such as one ounce or one item math-money Better buy: 6 for $3 or 10 for $4.50 10 for $4.50 (45 cents each vs 50 cents each) math-money Sales tax A percent of the price added at the register math-money Total with 10% tax on $20 $22 math-money Total with 5% tax on $40 $42 math-money Tip of 20% on $30 $6 math-money Tip of 15% on $40 $6 math-money Quick 10% of a price Move the decimal one place left math-money 20% off a $50 item Pay $40 math-money 25% off a $60 item Pay $45 math-money 50% off a $18 item Pay $9 math-money Discount An amount taken off the price math-money Profit Money earned minus money spent math-money Revenue Money coming in from sales math-money Budget A plan for how to spend and save money math-money Income Money you receive math-money Expense Money you spend math-money Savings Money set aside for later math-money Interest Money paid for using money, or earned on savings math-money Simple interest on $100 at 5% for 2 years $10 math-money Compound interest Interest earned on both the original amount and past interest math-money Principal The starting amount of a loan or deposit math-money Allowance saved: $4 a week for 6 weeks $24 math-money Need vs. want A need is required to live; a want is nice to have math-money Wage Pay for work, often per hour math-money Pay for 8 hours at $12 per hour $96 math-money Salary Fixed pay per year, not by the hour math-money Exchange rate How much of one currency equals another math-money Currency The money used in a country math-money Check vs. cash A check is a written order for the bank to pay; cash is physical money math-money Divisible by 2 if The last digit is even (0, 2, 4, 6, 8) math-number-theory Divisible by 3 if The sum of the digits is divisible by 3 math-number-theory Divisible by 4 if The last two digits form a number divisible by 4 math-number-theory Divisible by 5 if The last digit is 0 or 5 math-number-theory Divisible by 6 if It is divisible by both 2 and 3 math-number-theory Divisible by 8 if The last three digits form a number divisible by 8 math-number-theory Divisible by 9 if The sum of the digits is divisible by 9 math-number-theory Divisible by 10 if The last digit is 0 math-number-theory Divisible by 11 if The alternating sum of the digits is divisible by 11 math-number-theory Is 2,346 divisible by 3? Yes, digit sum 15 math-number-theory Is 4,125 divisible by 9? No, digit sum 12 math-number-theory Is 1,716 divisible by 4? Yes, 16 is divisible by 4 math-number-theory Is 121 divisible by 11? Yes: 1 − 2 + 1 = 0 math-number-theory Factors of 12 1, 2, 3, 4, 6, 12 math-number-theory Factors of 24 1, 2, 3, 4, 6, 8, 12, 24 math-number-theory Factors of 36 1, 2, 3, 4, 6, 9, 12, 18, 36 math-number-theory Factors of 48 1, 2, 3, 4, 6, 8, 12, 16, 24, 48 math-number-theory Factors of 60 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60 math-number-theory Factors of 100 1, 2, 4, 5, 10, 20, 25, 50, 100 math-number-theory Prime factorization of 12 2² × 3 math-number-theory Prime factorization of 36 2² × 3² math-number-theory Prime factorization of 60 2² × 3 × 5 math-number-theory Prime factorization of 100 2² × 5² math-number-theory Prime factorization of 72 2³ × 3² math-number-theory Prime factorization of 84 2² × 3 × 7 math-number-theory Prime factorization of 90 2 × 3² × 5 math-number-theory Prime factorization of 120 2³ × 3 × 5 math-number-theory Prime factorization of 144 2⁴ × 3² math-number-theory GCF of 12 and 18 6 math-number-theory GCF of 24 and 36 12 math-number-theory GCF of 15 and 25 5 math-number-theory GCF of 48 and 72 24 math-number-theory GCF of 17 and 31 1 (both prime) math-number-theory LCM of 4 and 6 12 math-number-theory LCM of 6 and 8 24 math-number-theory LCM of 3 and 5 15 math-number-theory LCM of 9 and 12 36 math-number-theory LCM of 10 and 15 30 math-number-theory Relationship of GCF and LCM GCF(a, b) × LCM(a, b) = a × b math-number-theory Coprime (relatively prime) Two numbers whose only common factor is 1 math-number-theory Euclidean algorithm Find the GCF by repeatedly replacing the larger number with the remainder math-number-theory Sieve of Eratosthenes A method that crosses out multiples to find primes math-number-theory Fundamental theorem of arithmetic Every integer above 1 has exactly one prime factorization math-number-theory Largest prime factor idea Test divisors only up to the square root math-number-theory Number of factors of a prime 2 math-number-theory Is there a largest prime? No. Euclid proved there are infinitely many primes math-number-theory Is 97 prime? Yes math-number-theory Is 101 prime? Yes math-number-theory Is 111 prime? No: 3 × 37 math-number-theory Is 143 prime? No: 11 × 13 math-number-theory Is 221 prime? No: 13 × 17 math-number-theory Remainder of 17 ÷ 5 2 math-number-theory 17 mod 5 2 math-number-theory Modular arithmetic Arithmetic where numbers wrap around after reaching a set value math-number-theory Clock arithmetic example 5 hours after 10 o'clock is 3 o'clock (15 mod 12) math-number-theory Even + even Even math-number-theory Odd + odd Even math-number-theory Odd + even Odd math-number-theory Odd × odd Odd math-number-theory Even × any integer Even math-number-theory Sum of the first n odd numbers n² math-number-theory Sum of the first n whole numbers n(n + 1)/2 math-number-theory Sum of 1 to 100 5,050 math-number-theory Goldbach's conjecture Every even number above 2 is the sum of two primes (unproven) math-number-theory Fermat's Last Theorem No positive integers satisfy aⁿ + bⁿ = cⁿ for n above 2; proved in 1994-95 math-number-theory Pythagorean triple Three whole numbers a, b, c with a² + b² = c² math-number-theory Common Pythagorean triples 3-4-5, 5-12-13, 8-15-17, 7-24-25 math-number-theory Natural (counting) numbers 1, 2, 3, 4, ... math-number-types Whole numbers 0, 1, 2, 3, ... math-number-types Integers ..., −3, −2, −1, 0, 1, 2, 3, ... math-number-types Rational numbers Numbers that can be written as a fraction of two integers math-number-types Irrational numbers Numbers that cannot be written as a fraction math-number-types Real numbers All rational and irrational numbers math-number-types Imaginary numbers Multiples of i, where i² = −1 math-number-types Complex numbers Numbers of the form a + bi math-number-types Even numbers Whole numbers divisible by 2 math-number-types Odd numbers Whole numbers not divisible by 2 math-number-types Prime numbers Numbers above 1 with exactly two factors math-number-types Composite numbers Numbers above 1 that are not prime math-number-types Square numbers 1, 4, 9, 16, 25, ... math-number-types Cube numbers 1, 8, 27, 64, 125, ... math-number-types Triangular numbers 1, 3, 6, 10, 15, 21, ... math-number-types Fibonacci numbers 1, 1, 2, 3, 5, 8, 13, 21, ... math-number-types Perfect numbers Numbers equal to the sum of their proper factors: 6, 28, 496 math-number-types Is 6 perfect? Yes: 1 + 2 + 3 = 6 math-number-types Is 28 perfect? Yes: 1 + 2 + 4 + 7 + 14 = 28 math-number-types Palindromic number A number that reads the same backward and forward (121) math-number-types Twin primes Primes that differ by 2 (11 and 13) math-number-types Mersenne prime A prime of the form 2ⁿ − 1 (7, 31, 127) math-number-types Pi is rational or irrational? Irrational math-number-types √2 is rational or irrational? Irrational math-number-types √9 is rational or irrational? Rational (it equals 3) math-number-types 0.5 is rational or irrational? Rational math-number-types 0.333... is rational or irrational? Rational (1/3) math-number-types Is e rational or irrational? Irrational math-number-types Is zero positive or negative? Neither math-number-types Is zero even? Yes math-number-types Is 1 prime? No math-number-types Is 0 an integer? Yes math-number-types Is −4 a whole number? No math-number-types Is −4 an integer? Yes math-number-types Is 3/4 an integer? No, it is rational math-number-types Is every integer a rational number? Yes (n = n/1) math-number-types Is every rational number an integer? No math-number-types Is every real number rational? No math-number-types Ordinal numbers First, second, third: tell position math-number-types Cardinal numbers One, two, three: tell how many math-number-types Nominal numbers Numbers used as names or labels, like a jersey number math-number-types Opposite of 7 −7 math-number-types Reciprocal of 7 1/7 math-number-types Absolute value of −7 7 math-number-types Negative of a negative A positive: −(−7) = 7 math-number-types Googol 10^100 math-number-types Googolplex 10 to the power of a googol math-number-types Infinity Not a number; the idea of going on without end math-number-types Golden ratio About 1.618, an irrational number: (1 + √5)/2 math-number-types What does PEMDAS stand for? Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (Please Excuse My Dear Aunt Sally) math-order-of-operations What does BODMAS stand for? Brackets, Orders, Division and Multiplication, Addition and Subtraction math-order-of-operations Which comes first, parentheses or exponents? Parentheses math-order-of-operations Which is done first, multiplication or division? Whichever comes first, going left to right math-order-of-operations Which is done first, addition or subtraction? Whichever comes first, going left to right math-order-of-operations Is multiplication always done before division? No. They have equal rank; work left to right math-order-of-operations 2 + 3 × 4 14 math-order-of-operations (2 + 3) × 4 20 math-order-of-operations 10 − 2 × 3 4 math-order-of-operations (10 − 2) × 3 24 math-order-of-operations 8 ÷ 2 × 4 16 math-order-of-operations 8 ÷ (2 × 4) 1 math-order-of-operations 12 − 4 + 2 10 math-order-of-operations 12 − (4 + 2) 6 math-order-of-operations 3² + 4 13 math-order-of-operations 3 × 2² 12 math-order-of-operations (3 × 2)² 36 math-order-of-operations 5 + 2³ 13 math-order-of-operations 20 ÷ 5 + 3 7 math-order-of-operations 20 ÷ (5 + 3) 2.5 math-order-of-operations 6 + 4 ÷ 2 8 math-order-of-operations (6 + 4) ÷ 2 5 math-order-of-operations 2 × (3 + 4) − 5 9 math-order-of-operations 18 ÷ 3 × 2 12 math-order-of-operations 7 − 3 × 2 + 1 2 math-order-of-operations 4 + 6 × (5 − 3) 16 math-order-of-operations 100 − 5² 75 math-order-of-operations 3 + 2 × 5² 53 math-order-of-operations (8 − 2)² ÷ 4 9 math-order-of-operations 2 + 3 × 4 − 5 9 math-order-of-operations 9 − 3 ÷ 3 + 1 9 math-order-of-operations 5 × 4 ÷ 2 + 1 11 math-order-of-operations −3² −9 (Square first, then negate) math-order-of-operations (−3)² 9 math-order-of-operations 6 ÷ 2(1 + 2) 9 by strict left-to-right; the notation is ambiguous, so write it clearly math-order-of-operations Where do roots fit in PEMDAS? With exponents math-order-of-operations Where does the fraction bar fit? It groups the top and bottom like parentheses math-order-of-operations Evaluate (4 + 2) / (5 − 2) 2 math-order-of-operations Evaluate 2 + |3 − 5| 4 (Absolute value bars group like parentheses) math-order-of-operations √16 + 3 × 2 10 math-order-of-operations Evaluate 5 + 3(2 + 1)² 32 math-order-of-operations 1³ 1 math-perfect-cubes 2³ 8 math-perfect-cubes 3³ 27 math-perfect-cubes 4³ 64 math-perfect-cubes 5³ 125 math-perfect-cubes 6³ 216 math-perfect-cubes 7³ 343 math-perfect-cubes 8³ 512 math-perfect-cubes 9³ 729 math-perfect-cubes 10³ 1000 math-perfect-cubes 11³ 1331 math-perfect-cubes 12³ 1728 math-perfect-cubes 13³ 2197 math-perfect-cubes 14³ 2744 math-perfect-cubes 15³ 3375 math-perfect-cubes 16³ 4096 math-perfect-cubes 17³ 4913 math-perfect-cubes 18³ 5832 math-perfect-cubes 19³ 6859 math-perfect-cubes 20³ 8000 math-perfect-cubes 1² 1 math-perfect-squares 2² 4 math-perfect-squares 3² 9 math-perfect-squares 4² 16 math-perfect-squares 5² 25 math-perfect-squares 6² 36 math-perfect-squares 7² 49 math-perfect-squares 8² 64 math-perfect-squares 9² 81 math-perfect-squares 10² 100 math-perfect-squares 11² 121 math-perfect-squares 12² 144 math-perfect-squares 13² 169 math-perfect-squares 14² 196 math-perfect-squares 15² 225 math-perfect-squares 16² 256 math-perfect-squares 17² 289 math-perfect-squares 18² 324 math-perfect-squares 19² 361 math-perfect-squares 20² 400 math-perfect-squares 21² 441 math-perfect-squares 22² 484 math-perfect-squares 23² 529 math-perfect-squares 24² 576 math-perfect-squares 25² 625 math-perfect-squares 26² 676 math-perfect-squares 27² 729 math-perfect-squares 28² 784 math-perfect-squares 29² 841 math-perfect-squares 30² 900 math-perfect-squares 31² 961 math-perfect-squares 32² 1024 math-perfect-squares 33² 1089 math-perfect-squares 34² 1156 math-perfect-squares 35² 1225 math-perfect-squares 36² 1296 math-perfect-squares 37² 1369 math-perfect-squares 38² 1444 math-perfect-squares 39² 1521 math-perfect-squares 40² 1600 math-perfect-squares How many ones in a ten? 10 math-place-value How many tens in a hundred? 10 math-place-value How many hundreds in a thousand? 10 math-place-value How many thousands in a million? 1,000 math-place-value How many millions in a billion? 1,000 math-place-value How many billions in a trillion? 1,000 math-place-value Which place is the first to the left of the ones? Tens math-place-value Place values from the right in 5-digit numbers Ones, tens, hundreds, thousands, ten thousands math-place-value What is the value of the 4 in 4,567? 4,000 math-place-value What is the value of the 5 in 4,567? 500 math-place-value What is the value of the 6 in 4,567? 60 math-place-value What is the value of the 7 in 4,567? 7 math-place-value Write 3,000 + 400 + 20 + 1 3,421 math-place-value Write 5 hundreds + 0 tens + 8 ones 508 math-place-value Expanded form of 7,306 7,000 + 300 + 6 math-place-value Expanded form of 24,018 20,000 + 4,000 + 10 + 8 math-place-value Which place is the first to the right of the decimal point? Tenths math-place-value Second place after the decimal point Hundredths math-place-value Third place after the decimal point Thousandths math-place-value Fourth place after the decimal point Ten-thousandths math-place-value Fifth place after the decimal point Hundred-thousandths math-place-value Sixth place after the decimal point Millionths math-place-value What is the value of the 3 in 0.37? 3 tenths, or 0.3 math-place-value What is the value of the 7 in 0.37? 7 hundredths, or 0.07 math-place-value What is the value of the 2 in 5.012? 2 thousandths, or 0.002 math-place-value How many hundredths in a tenth? 10 math-place-value Name for 1,000,000 One million math-place-value Name for 1,000,000,000 One billion math-place-value Name for 1,000,000,000,000 One trillion math-place-value Name for 10^15 One quadrillion math-place-value Name for 10^18 One quintillion math-place-value How many zeros in one million? 6 math-place-value How many zeros in one billion? 9 math-place-value How many zeros in one trillion? 12 math-place-value What is a googol? 1 followed by 100 zeros (10^100) math-place-value Round 3,468 to the nearest hundred 3,500 math-place-value Round 7,249 to the nearest thousand 7,000 math-place-value Round 85 to the nearest ten 90 (Round half up) math-place-value Round 3.14159 to the nearest hundredth 3.14 math-place-value Which is greater, 0.5 or 0.45? 0.5 (0.50 versus 0.45) math-place-value What does multiplying by 10 do to place value? Moves every digit one place to the left math-place-value What does dividing by 10 do to place value? Moves every digit one place to the right math-place-value What does multiplying by 100 do to a decimal point? Moves it two places to the right math-place-value Standard form of seven million, four hundred thousand 7,400,000 math-place-value How is 10,000 read? Ten thousand math-place-value How is 100,000 read? One hundred thousand math-place-value Digit in the ten-thousands place of 82,915 8 math-place-value Digit in the millions place of 6,248,000 6 math-place-value 2⁰ 1 math-powers 2¹ 2 math-powers 2² 4 math-powers 2³ 8 math-powers 2⁴ 16 math-powers 2⁵ 32 math-powers 2⁶ 64 math-powers 2⁷ 128 math-powers 2⁸ 256 math-powers 2⁹ 512 math-powers 2¹⁰ 1024 math-powers 2¹¹ 2048 math-powers 2¹² 4096 math-powers 2¹³ 8192 math-powers 2¹⁴ 16384 math-powers 2¹⁵ 32768 math-powers 2¹⁶ 65536 math-powers 2¹⁷ 131072 math-powers 2¹⁸ 262144 math-powers 2¹⁹ 524288 math-powers 2²⁰ 1048576 math-powers 3² 9 math-powers 3³ 27 math-powers 3⁴ 81 math-powers 3⁵ 243 math-powers 3⁶ 729 math-powers 3⁷ 2187 math-powers 3⁸ 6561 math-powers 3⁹ 19683 math-powers 3¹⁰ 59049 math-powers 5² 25 math-powers 5³ 125 math-powers 5⁴ 625 math-powers 5⁵ 3125 math-powers 5⁶ 15625 math-powers 5⁷ 78125 math-powers 5⁸ 390625 math-powers 6² 36 math-powers 6³ 216 math-powers 6⁴ 1296 math-powers 6⁵ 7776 math-powers 6⁶ 46656 math-powers 7² 49 math-powers 7³ 343 math-powers 7⁴ 2401 math-powers 7⁵ 16807 math-powers 7⁶ 117649 math-powers 10² 100 math-powers 10³ 1000 math-powers 10⁴ 10000 math-powers 10⁵ 100000 math-powers 10⁶ 1000000 math-powers What is the 1st prime number? 2 math-prime-numbers What is the 2nd prime number? 3 math-prime-numbers What is the 3rd prime number? 5 math-prime-numbers What is the 4th prime number? 7 math-prime-numbers What is the 5th prime number? 11 math-prime-numbers What is the 6th prime number? 13 math-prime-numbers What is the 7th prime number? 17 math-prime-numbers What is the 8th prime number? 19 math-prime-numbers What is the 9th prime number? 23 math-prime-numbers What is the 10th prime number? 29 math-prime-numbers What is the 11th prime number? 31 math-prime-numbers What is the 12th prime number? 37 math-prime-numbers What is the 13th prime number? 41 math-prime-numbers What is the 14th prime number? 43 math-prime-numbers What is the 15th prime number? 47 math-prime-numbers What is the 16th prime number? 53 math-prime-numbers What is the 17th prime number? 59 math-prime-numbers What is the 18th prime number? 61 math-prime-numbers What is the 19th prime number? 67 math-prime-numbers What is the 20th prime number? 71 math-prime-numbers What is the 21st prime number? 73 math-prime-numbers What is the 22nd prime number? 79 math-prime-numbers What is the 23rd prime number? 83 math-prime-numbers What is the 24th prime number? 89 math-prime-numbers What is the 25th prime number? 97 math-prime-numbers What is the 26th prime number? 101 math-prime-numbers What is the 27th prime number? 103 math-prime-numbers What is the 28th prime number? 107 math-prime-numbers What is the 29th prime number? 109 math-prime-numbers What is the 30th prime number? 113 math-prime-numbers What is the 31st prime number? 127 math-prime-numbers What is the 32nd prime number? 131 math-prime-numbers What is the 33rd prime number? 137 math-prime-numbers What is the 34th prime number? 139 math-prime-numbers What is the 35th prime number? 149 math-prime-numbers What is the 36th prime number? 151 math-prime-numbers What is the 37th prime number? 157 math-prime-numbers What is the 38th prime number? 163 math-prime-numbers What is the 39th prime number? 167 math-prime-numbers What is the 40th prime number? 173 math-prime-numbers What is the 41st prime number? 179 math-prime-numbers What is the 42nd prime number? 181 math-prime-numbers What is the 43rd prime number? 191 math-prime-numbers What is the 44th prime number? 193 math-prime-numbers What is the 45th prime number? 197 math-prime-numbers What is the 46th prime number? 199 math-prime-numbers What is the 47th prime number? 211 math-prime-numbers What is the 48th prime number? 223 math-prime-numbers What is the 49th prime number? 227 math-prime-numbers What is the 50th prime number? 229 math-prime-numbers What is the 51st prime number? 233 math-prime-numbers What is the 52nd prime number? 239 math-prime-numbers What is the 53rd prime number? 241 math-prime-numbers What is the 54th prime number? 251 math-prime-numbers What is the 55th prime number? 257 math-prime-numbers What is the 56th prime number? 263 math-prime-numbers What is the 57th prime number? 269 math-prime-numbers What is the 58th prime number? 271 math-prime-numbers What is the 59th prime number? 277 math-prime-numbers What is the 60th prime number? 281 math-prime-numbers What is the only even prime number? 2 math-prime-numbers Is 1 a prime number? No. A prime has exactly two factors, and 1 has only one. math-prime-numbers Is 91 prime? No: 91 = 7 × 13 (It looks prime but is not) math-prime-numbers Is 51 prime? No: 51 = 3 × 17 math-prime-numbers Is 57 prime? No: 57 = 3 × 19 math-prime-numbers Is 87 prime? No: 87 = 3 × 29 math-prime-numbers Probability A number from 0 to 1 telling how likely an event is math-probability-terms Impossible event Probability 0 math-probability-terms Certain event Probability 1 math-probability-terms Outcome One possible result of an experiment math-probability-terms Event A set of outcomes math-probability-terms Sample space The set of all possible outcomes math-probability-terms Experiment (trial) An action with an uncertain result, like rolling a die math-probability-terms Theoretical probability Favorable outcomes divided by all equally likely outcomes math-probability-terms Experimental probability Number of times it happened divided by number of trials math-probability-terms Complement of an event All outcomes where the event does not happen; P(not A) = 1 − P(A) math-probability-terms Independent events One event does not change the probability of the other math-probability-terms Dependent events One event changes the probability of the other math-probability-terms Mutually exclusive events Events that cannot happen at the same time math-probability-terms Multiplication rule (independent) P(A and B) = P(A) × P(B) math-probability-terms Addition rule P(A or B) = P(A) + P(B) − P(A and B) math-probability-terms Conditional probability P(A | B) = P(A and B) / P(B) math-probability-terms Bayes' theorem P(A | B) = P(B | A) P(A) / P(B) math-probability-terms Law of large numbers With many trials, experimental probability gets close to theoretical probability math-probability-terms Expected value The long-run average: sum of each value times its probability math-probability-terms Random variable A variable whose value comes from a random process math-probability-terms Probability of heads on a fair coin 1/2 math-probability-terms Probability of a 4 on a fair die 1/6 math-probability-terms Probability of an even number on a die 1/2 math-probability-terms Probability of a heart from a standard deck 13/52 = 1/4 math-probability-terms Probability of an ace from a standard deck 4/52 = 1/13 math-probability-terms Cards in a standard deck 52 math-probability-terms Probability of two heads in two fair flips 1/4 math-probability-terms Probability of a sum of 7 with two dice 6/36 = 1/6 math-probability-terms Number of outcomes when rolling two dice 36 math-probability-terms Number of outcomes in 3 coin flips 8 math-probability-terms Fundamental counting principle Multiply the number of choices at each step math-probability-terms Outfits from 3 shirts and 4 pants 12 math-probability-terms Factorial n! n × (n − 1) × ... × 1 math-probability-terms Permutation An ordered arrangement math-probability-terms Permutation formula P(n, r) = n! / (n − r)! math-probability-terms Combination A selection where order does not matter math-probability-terms Combination formula C(n, r) = n! / (r!(n − r)!) math-probability-terms Ways to arrange 5 books on a shelf 5! = 120 math-probability-terms P(5, 2) 20 math-probability-terms C(5, 2) 10 math-probability-terms C(6, 3) 20 math-probability-terms C(10, 2) 45 math-probability-terms C(n, 0) and C(n, n) Both equal 1 math-probability-terms Pascal's triangle A triangle where each number is the sum of the two above it; rows give combinations math-probability-terms Binomial distribution The distribution of the number of successes in a fixed number of independent yes/no trials math-probability-terms With replacement Each item is put back before the next draw math-probability-terms Without replacement Items are not put back; later probabilities change math-probability-terms Tree diagram A branching picture that lists outcomes of a multi-step event math-probability-terms Venn diagram Overlapping circles showing how sets share members math-probability-terms Odds Ratio of favorable outcomes to unfavorable outcomes math-probability-terms Fair game A game where each player has the same expected value math-probability-terms Simulation Using random numbers or objects to imitate a real chance process math-probability-terms What do the seven letters I V X L C D M stand for? 1, 5, 10, 50, 100, 500, 1000 math-roman-numerals 1 I math-roman-numerals 2 II math-roman-numerals 3 III math-roman-numerals 4 IV math-roman-numerals 5 V math-roman-numerals 6 VI math-roman-numerals 7 VII math-roman-numerals 8 VIII math-roman-numerals 9 IX math-roman-numerals 10 X math-roman-numerals 11 XI math-roman-numerals 12 XII math-roman-numerals 13 XIII math-roman-numerals 14 XIV math-roman-numerals 15 XV math-roman-numerals 16 XVI math-roman-numerals 17 XVII math-roman-numerals 18 XVIII math-roman-numerals 19 XIX math-roman-numerals 20 XX math-roman-numerals 21 XXI math-roman-numerals 22 XXII math-roman-numerals 23 XXIII math-roman-numerals 24 XXIV math-roman-numerals 25 XXV math-roman-numerals 26 XXVI math-roman-numerals 27 XXVII math-roman-numerals 28 XXVIII math-roman-numerals 29 XXIX math-roman-numerals 30 XXX math-roman-numerals 35 XXXV math-roman-numerals 40 XL math-roman-numerals 44 XLIV math-roman-numerals 49 XLIX math-roman-numerals 50 L math-roman-numerals 60 LX math-roman-numerals 70 LXX math-roman-numerals 80 LXXX math-roman-numerals 90 XC math-roman-numerals 99 XCIX math-roman-numerals 100 C math-roman-numerals 105 CV math-roman-numerals 144 CXLIV math-roman-numerals 200 CC math-roman-numerals 400 CD math-roman-numerals 444 CDXLIV math-roman-numerals 500 D math-roman-numerals 900 CM math-roman-numerals 1000 M math-roman-numerals 1492 MCDXCII math-roman-numerals 1776 MDCCLXXVI math-roman-numerals 1999 MCMXCIX math-roman-numerals 2000 MM math-roman-numerals 2024 MMXXIV math-roman-numerals 2026 MMXXVI math-roman-numerals 3000 MMM math-roman-numerals 3999 MMMCMXCIX math-roman-numerals Which Roman numerals can be repeated up to three times? I, X, C and M math-roman-numerals Sequence An ordered list of numbers following a rule math-sequences-series Term of a sequence One number in the list math-sequences-series Arithmetic sequence Each term is found by adding the same number math-sequences-series Common difference The number added each time in an arithmetic sequence math-sequences-series Geometric sequence Each term is found by multiplying by the same number math-sequences-series Common ratio The number multiplied each time in a geometric sequence math-sequences-series Explicit formula, arithmetic aₙ = a₁ + (n − 1)d math-sequences-series Explicit formula, geometric aₙ = a₁ · rⁿ⁻¹ math-sequences-series Recursive formula Gives each term from the term before it math-sequences-series Sum of an arithmetic series Sₙ = n(a₁ + aₙ)/2 math-sequences-series Sum of a finite geometric series Sₙ = a₁(1 − rⁿ)/(1 − r) math-sequences-series Sum of an infinite geometric series S = a₁/(1 − r), when |r| < 1 math-sequences-series Next term: 2, 4, 6, 8, ... 10 (add 2) math-sequences-series Next term: 5, 10, 15, 20, ... 25 (add 5) math-sequences-series Next term: 3, 6, 12, 24, ... 48 (multiply by 2) math-sequences-series Next term: 1, 4, 9, 16, ... 25 (squares) math-sequences-series Next term: 1, 3, 6, 10, ... 15 (triangular numbers, add 5) math-sequences-series Next term: 100, 90, 80, ... 70 (subtract 10) math-sequences-series Next term: 81, 27, 9, 3, ... 1 (divide by 3) math-sequences-series Next term: 1, 1, 2, 3, 5, 8, ... 13 (Fibonacci) math-sequences-series Next term: 2, 6, 18, 54, ... 162 (multiply by 3) math-sequences-series Next term: 1, 2, 4, 7, 11, ... 16 (differences grow by 1) math-sequences-series Next term: 1, 8, 27, 64, ... 125 (cubes) math-sequences-series Next term: 2, 3, 5, 7, 11, ... 13 (primes) math-sequences-series 10th term of 3, 7, 11, ... 39 math-sequences-series 20th term of 5, 8, 11, ... 62 math-sequences-series 8th term of 2, 4, 8, ... 256 math-sequences-series Common difference of 7, 12, 17, ... 5 math-sequences-series Common ratio of 5, 15, 45, ... 3 math-sequences-series Common ratio of 16, 8, 4, ... 1/2 math-sequences-series Sum of 1 + 2 + ... + 10 55 math-sequences-series Sum of 1 + 2 + ... + 100 5,050 math-sequences-series Sum of the first 5 odd numbers 25 math-sequences-series Sum of the first 10 even numbers 110 math-sequences-series Sum of 1 + 1/2 + 1/4 + 1/8 + ... forever 2 math-sequences-series Fibonacci rule Each term is the sum of the two before it math-sequences-series Fibonacci numbers up to 144 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144 math-sequences-series Ratio of consecutive Fibonacci numbers Approaches the golden ratio, about 1.618 math-sequences-series Triangular numbers formula n(n + 1)/2 math-sequences-series Square numbers formula n² math-sequences-series Pentagonal number idea n(3n − 1)/2: 1, 5, 12, 22, ... math-sequences-series Pascal's triangle rows 1; 1 1; 1 2 1; 1 3 3 1; 1 4 6 4 1 math-sequences-series Row sums of Pascal's triangle Powers of 2 math-sequences-series Sigma notation Σ means add up the terms for the given range math-sequences-series Σ from k=1 to 4 of k 10 math-sequences-series Σ from k=1 to 3 of k² 14 math-sequences-series Σ from k=1 to 5 of 2 10 math-sequences-series Pattern Something that repeats or changes by a rule math-sequences-series Growth pattern A pattern in which a number or shape grows by a rule each step math-sequences-series Limit of a sequence The value the terms get close to math-sequences-series Converge The terms or sum settle toward a number math-sequences-series Diverge The terms or sum do not settle toward a number math-sequences-series Harmonic series 1 + 1/2 + 1/3 + 1/4 + ... ; it grows without bound math-sequences-series Arithmetic mean of two numbers Their average math-sequences-series Geometric mean of two numbers The square root of their product math-sequences-series Geometric mean of 4 and 9 6 math-sequences-series Compound interest as a sequence A geometric sequence, multiplying by (1 + r) each period math-sequences-series Set A collection of distinct objects math-sets-logic Element (member) An object that is in a set math-sets-logic Empty set The set with no elements, written ∅ or { } math-sets-logic Subset A set whose elements are all in another set math-sets-logic Proper subset A subset that is not the whole set math-sets-logic Union (A ∪ B) Elements in A or B or both math-sets-logic Intersection (A ∩ B) Elements in both A and B math-sets-logic Complement (A′) Elements in the universal set that are not in A math-sets-logic Universal set The set of everything under consideration math-sets-logic Disjoint sets Sets with no elements in common math-sets-logic Cardinality The number of elements in a set, written |A| math-sets-logic Roster form Listing the elements: {1, 2, 3} math-sets-logic Set-builder notation {x | x > 2} means all x such that x is greater than 2 math-sets-logic Venn diagram Overlapping circles showing how sets relate math-sets-logic Cardinality of a union |A ∪ B| = |A| + |B| − |A ∩ B| math-sets-logic If A = {1,2,3} and B = {3,4}, A ∪ B {1, 2, 3, 4} math-sets-logic If A = {1,2,3} and B = {3,4}, A ∩ B {3} math-sets-logic Number of subsets of a set with n elements 2ⁿ math-sets-logic Number of subsets of {a, b, c} 8 math-sets-logic Is ∅ a subset of every set? Yes math-sets-logic Natural numbers symbol ℕ math-sets-logic Integers symbol ℤ math-sets-logic Rational numbers symbol ℚ math-sets-logic Real numbers symbol ℝ math-sets-logic Statement (proposition) A sentence that is either true or false math-sets-logic Negation The statement with the opposite truth value (not P) math-sets-logic Conjunction P and Q; true only when both are true math-sets-logic Disjunction P or Q; true when at least one is true math-sets-logic Conditional If P then Q; false only when P is true and Q is false math-sets-logic Hypothesis The 'if' part of a conditional math-sets-logic Conclusion The 'then' part of a conditional math-sets-logic Converse Swap the parts: if Q then P math-sets-logic Inverse Negate both parts: if not P then not Q math-sets-logic Contrapositive Swap and negate: if not Q then not P; same truth as the original math-sets-logic Biconditional P if and only if Q; true when both have the same truth value math-sets-logic Truth table A table listing the truth value of a statement for every case math-sets-logic Tautology A statement that is always true math-sets-logic Contradiction A statement that is always false math-sets-logic De Morgan's law 1 not (P and Q) is the same as (not P) or (not Q) math-sets-logic De Morgan's law 2 not (P or Q) is the same as (not P) and (not Q) math-sets-logic Counterexample An example showing a statement is false math-sets-logic Direct proof Proving a statement by a chain of logical steps from facts math-sets-logic Proof by contradiction Assume the opposite and show it leads to something impossible math-sets-logic Proof by induction Show it for a first case, then show each case leads to the next math-sets-logic Inductive reasoning Reaching a general rule from patterns math-sets-logic Deductive reasoning Reaching a conclusion that must follow from given facts math-sets-logic Axiom A basic statement accepted without proof math-sets-logic Lemma A small proven result used to prove a bigger one math-sets-logic Corollary A result that follows easily from a theorem math-sets-logic Q.E.D. Latin 'quod erat demonstrandum': which was to be shown math-sets-logic Necessary condition Something that must be true for another thing to be true math-sets-logic Sufficient condition Something that, if true, guarantees another thing math-sets-logic Quantifier ∀ For all math-sets-logic Quantifier ∃ There exists math-sets-logic Boolean value True or false, often written 1 or 0 math-sets-logic AND gate output 1 only if both inputs are 1 math-sets-logic OR gate output 1 if at least one input is 1 math-sets-logic NOT gate output The opposite of the input math-sets-logic Binary number system Base 2, using only digits 0 and 1 math-sets-logic Binary 101 in decimal 5 math-sets-logic Decimal 10 in binary 1010 math-sets-logic Binary 1111 in decimal 15 math-sets-logic √1 1 math-square-roots √4 2 math-square-roots √9 3 math-square-roots √16 4 math-square-roots √25 5 math-square-roots √36 6 math-square-roots √49 7 math-square-roots √64 8 math-square-roots √81 9 math-square-roots √100 10 math-square-roots √121 11 math-square-roots √144 12 math-square-roots √169 13 math-square-roots √196 14 math-square-roots √225 15 math-square-roots √256 16 math-square-roots √289 17 math-square-roots √324 18 math-square-roots √361 19 math-square-roots √400 20 math-square-roots √441 21 math-square-roots √484 22 math-square-roots √529 23 math-square-roots √576 24 math-square-roots √625 25 math-square-roots √676 26 math-square-roots √729 27 math-square-roots √784 28 math-square-roots √841 29 math-square-roots √900 30 math-square-roots √961 31 math-square-roots √1024 32 math-square-roots √1089 33 math-square-roots √1156 34 math-square-roots √1225 35 math-square-roots √1296 36 math-square-roots √1369 37 math-square-roots √1444 38 math-square-roots √1521 39 math-square-roots √1600 40 math-square-roots Data Facts or measurements that have been collected math-statistics-terms Population The whole group you want to learn about math-statistics-terms Sample A smaller group taken from the population math-statistics-terms Random sample A sample where every member of the population has an equal chance of being chosen math-statistics-terms Bias A systematic error that tilts results in one direction math-statistics-terms Survey A way of collecting data by asking people questions math-statistics-terms Census A survey of every member of the population math-statistics-terms Variable A characteristic that can take different values math-statistics-terms Categorical data Data sorted into groups or names (Eye color) math-statistics-terms Numerical (quantitative) data Data that are numbers you can measure or count math-statistics-terms Discrete data Data that can only take separate values, like counts math-statistics-terms Continuous data Data that can take any value in a range, like height math-statistics-terms Mean Sum of all values divided by how many values math-statistics-terms Median Middle value of ordered data math-statistics-terms Mode Most frequently occurring value math-statistics-terms Range Maximum minus minimum math-statistics-terms Outlier A value far away from the rest of the data math-statistics-terms Quartile One of three values that split ordered data into four equal parts math-statistics-terms Q1 (first quartile) The median of the lower half of the data math-statistics-terms Q3 (third quartile) The median of the upper half of the data math-statistics-terms Interquartile range (IQR) Q3 − Q1 math-statistics-terms Five-number summary Minimum, Q1, median, Q3, maximum math-statistics-terms Outlier rule (1.5 × IQR) Values below Q1 − 1.5·IQR or above Q3 + 1.5·IQR are flagged as outliers math-statistics-terms Percentile The value below which a given percent of data fall math-statistics-terms Variance The average squared distance from the mean math-statistics-terms Standard deviation The square root of variance; typical distance from the mean math-statistics-terms Z-score How many standard deviations a value is from the mean: (x − mean)/s math-statistics-terms Frequency How many times a value occurs math-statistics-terms Relative frequency Frequency divided by the total count math-statistics-terms Cumulative frequency A running total of frequencies math-statistics-terms Histogram A bar graph for numerical data grouped into intervals math-statistics-terms Box plot (box-and-whisker) A graph of the five-number summary math-statistics-terms Dot plot A graph with a dot above a number line for each data value math-statistics-terms Stem-and-leaf plot A table that splits each value into a stem and a leaf math-statistics-terms Scatter plot A graph of ordered pairs showing how two variables relate math-statistics-terms Bar graph A graph using separate bars to compare categories math-statistics-terms Pie chart A circle graph that shows parts of a whole math-statistics-terms Line graph A graph that connects points to show change over time math-statistics-terms Symmetric distribution The left and right sides mirror each other math-statistics-terms Skewed right A longer tail on the right; mean is usually greater than median math-statistics-terms Skewed left A longer tail on the left; mean is usually less than median math-statistics-terms Normal distribution A symmetric bell-shaped distribution math-statistics-terms Empirical rule About 68%, 95% and 99.7% of normal data fall within 1, 2 and 3 standard deviations of the mean math-statistics-terms Correlation A measure of the strength and direction of a linear relationship math-statistics-terms Correlation coefficient (r) A number from −1 to 1 measuring linear correlation math-statistics-terms Positive correlation As one variable increases, so does the other math-statistics-terms Negative correlation As one variable increases, the other decreases math-statistics-terms Correlation vs. causation Two things moving together does not prove one causes the other math-statistics-terms Line of best fit A line drawn to match the trend in a scatter plot math-statistics-terms Regression A method for fitting a model to describe how one variable depends on another math-statistics-terms Residual Actual value minus predicted value math-statistics-terms Extrapolation Predicting beyond the range of the data math-statistics-terms Interpolation Predicting between known data points math-statistics-terms Hypothesis test A procedure for deciding whether data support a claim math-statistics-terms Null hypothesis The default claim that nothing special is going on math-statistics-terms P-value The probability of results at least as extreme as observed if the null hypothesis is true math-statistics-terms Confidence interval A range of values likely to contain the true population value math-statistics-terms Margin of error The amount added and subtracted from an estimate to form an interval math-statistics-terms Sampling distribution The distribution of a statistic over many samples math-statistics-terms Central limit theorem Sample means tend toward a normal shape as sample size grows math-statistics-terms Control group The group in an experiment that does not get the treatment math-statistics-terms Random assignment Using chance to place subjects in experiment groups math-statistics-terms Observational study A study that observes without assigning treatments math-statistics-terms Experiment A study where researchers assign treatments math-statistics-terms + Plus: add math-symbols − Minus: subtract math-symbols × Times: multiply math-symbols ÷ Divided by: divide math-symbols = Equals: is the same value as math-symbols ≠ Is not equal to math-symbols < Is less than math-symbols > Is greater than math-symbols ≤ Is less than or equal to math-symbols ≥ Is greater than or equal to math-symbols ≈ Is approximately equal to math-symbols ≡ Is identical to or congruent (mod) to math-symbols ± Plus or minus math-symbols ∓ Minus or plus math-symbols √ Square root (radical) math-symbols ∛ Cube root math-symbols π Pi: about 3.14159, circumference over diameter math-symbols ∞ Infinity: unbounded, without end math-symbols % Percent: out of 100 math-symbols ‰ Per mille: out of 1,000 math-symbols ° Degree: a unit of angle or temperature math-symbols ′ Prime: minutes, feet, or a related variable math-symbols ″ Double prime: seconds, inches math-symbols |x| Absolute value of x math-symbols ! Factorial math-symbols Σ Sigma: sum of a series math-symbols Π (capital pi) Product of a series math-symbols Δ Delta: change in math-symbols ∂ Partial derivative math-symbols ∫ Integral math-symbols ∮ Contour or closed-loop integral math-symbols lim Limit math-symbols θ Theta: commonly an angle math-symbols λ Lambda: wavelength, eigenvalue math-symbols μ Mu: mean of a population math-symbols σ Sigma (lowercase): standard deviation math-symbols ε (epsilon) A very small positive number in limits math-symbols φ Phi: the golden ratio (about 1.618) or an angle math-symbols e Euler's number, about 2.71828 math-symbols i Imaginary unit, where i² = −1 math-symbols ℝ The set of real numbers math-symbols ℚ The set of rational numbers math-symbols ℤ The set of integers math-symbols ℕ The set of natural numbers math-symbols ℂ The set of complex numbers math-symbols ∈ Is an element of math-symbols ∉ Is not an element of math-symbols ⊂ Is a subset of math-symbols ∪ Union: in either set math-symbols ∩ Intersection: in both sets math-symbols ∅ Empty set: no elements math-symbols ∀ For all math-symbols ∃ There exists math-symbols ∴ Therefore math-symbols ∵ Because math-symbols ⇒ Implies math-symbols ⇔ If and only if math-symbols ∥ Is parallel to math-symbols ⊥ Is perpendicular to math-symbols ∠ Angle math-symbols △ Triangle math-symbols ≅ Is congruent to math-symbols ∼ Is similar to math-symbols ∝ Is proportional to math-symbols f(x) Function f of x math-symbols f⁻¹ Inverse function of f math-symbols n! n factorial: product of 1 to n math-symbols a:b Ratio of a to b math-symbols ⌊x⌋ Floor: greatest integer not above x math-symbols ⌈x⌉ Ceiling: least integer not below x math-symbols mod Modulo: the remainder after division math-symbols · Dot: multiplication math-symbols → (arrow) Approaches, or maps to math-symbols Seconds in a minute 60 math-time Minutes in an hour 60 math-time Seconds in an hour 3,600 math-time Hours in a day 24 math-time Minutes in a day 1,440 math-time Seconds in a day 86,400 math-time Days in a week 7 math-time Days in a year 365 math-time Days in a leap year 366 math-time Weeks in a year 52 (plus 1 or 2 days) math-time Months in a year 12 math-time Years in a decade 10 math-time Years in a century 100 math-time Years in a millennium 1,000 math-time Months with 31 days January, March, May, July, August, October, December math-time Months with 30 days April, June, September, November math-time Days in February 28, or 29 in a leap year math-time Knuckle trick for month lengths Knuckles are 31-day months, valleys between are shorter months math-time Leap year rule Every 4 years, except century years not divisible by 400 math-time Is 2000 a leap year? Yes math-time Is 1900 a leap year? No math-time Is 2028 a leap year? Yes math-time Is 2026 a leap year? No math-time Order of the days of the week Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday math-time Order of the months January through December math-time Seasons in order (northern hemisphere) Winter, spring, summer, fall math-time Quarter past 3 3:15 math-time Half past 7 7:30 math-time Quarter to 9 8:45 math-time Ten minutes to 6 5:50 math-time Meaning of a.m. From midnight to just before noon math-time Meaning of p.m. From noon to just before midnight math-time 12:00 a.m. is Midnight math-time 12:00 p.m. is Noon math-time 3:00 p.m. in 24-hour time 15:00 math-time 18:30 in 12-hour time 6:30 p.m. math-time 00:00 in 12-hour time 12:00 a.m. (midnight) math-time Elapsed time from 2:15 to 4:45 2 hours 30 minutes math-time Elapsed time from 9:40 a.m. to 11:10 a.m. 1 hour 30 minutes math-time Time 45 minutes after 3:30 4:15 math-time Time 2 hours before 1:00 p.m. 11:00 a.m. math-time Minutes in 2 and a half hours 150 math-time Hours in 3 days 72 math-time Hours in a week 168 math-time Minutes in a quarter hour 15 math-time Convert 90 minutes to hours 1.5 hours math-time Convert 2 hours to seconds 7,200 math-time Convert 3 weeks to days 21 math-time Convert 18 months to years 1.5 years math-time Fortnight Two weeks (14 days) math-time Biweekly Every two weeks math-time Semester Half of a school year math-time Which hand moves faster, hour or minute? Minute hand math-time How far does the minute hand move in 5 minutes? From one number to the next (30 degrees) math-time Degrees the minute hand turns in an hour 360 math-time Time zone A region that shares the same standard time math-time Number of time zones in the contiguous US Four: Eastern, Central, Mountain, Pacific math-time Greenwich Mean Time (GMT) Time at the 0 degree longitude line through Greenwich, England math-time Calendar year starts and ends January 1 to December 31 math-time BCE and CE Before Common Era and Common Era, used to number years math-time Digital vs. analog clock Digital shows numbers; analog uses hands on a dial math-time SOH-CAH-TOA: sine sin = opposite / hypotenuse math-trig-identities SOH-CAH-TOA: cosine cos = adjacent / hypotenuse math-trig-identities SOH-CAH-TOA: tangent tan = opposite / adjacent math-trig-identities Cosecant csc θ = 1 / sin θ math-trig-identities Secant sec θ = 1 / cos θ math-trig-identities Cotangent cot θ = 1 / tan θ = cos θ / sin θ math-trig-identities Tangent in terms of sine and cosine tan θ = sin θ / cos θ math-trig-identities Pythagorean identity sin² θ + cos² θ = 1 math-trig-identities Pythagorean identity with tangent 1 + tan² θ = sec² θ math-trig-identities Pythagorean identity with cotangent 1 + cot² θ = csc² θ math-trig-identities Even-odd: sin(−θ) −sin θ (sine is odd) math-trig-identities Even-odd: cos(−θ) cos θ (cosine is even) math-trig-identities Even-odd: tan(−θ) −tan θ (tangent is odd) math-trig-identities Cofunction: sin(90° − θ) cos θ math-trig-identities Cofunction: cos(90° − θ) sin θ math-trig-identities Cofunction: tan(90° − θ) cot θ math-trig-identities Supplementary: sin(180° − θ) sin θ math-trig-identities Supplementary: cos(180° − θ) −cos θ math-trig-identities Shift: sin(θ + 180°) −sin θ math-trig-identities Shift: cos(θ + 180°) −cos θ math-trig-identities Shift: tan(θ + 180°) tan θ math-trig-identities Sum formula for sine sin(A + B) = sin A cos B + cos A sin B math-trig-identities Difference formula for sine sin(A − B) = sin A cos B − cos A sin B math-trig-identities Sum formula for cosine cos(A + B) = cos A cos B − sin A sin B math-trig-identities Difference formula for cosine cos(A − B) = cos A cos B + sin A sin B math-trig-identities Sum formula for tangent tan(A + B) = (tan A + tan B) / (1 − tan A tan B) math-trig-identities Difference formula for tangent tan(A − B) = (tan A − tan B) / (1 + tan A tan B) math-trig-identities Double-angle: sine sin 2θ = 2 sin θ cos θ math-trig-identities Double-angle: cosine cos 2θ = cos² θ − sin² θ math-trig-identities Double-angle: cosine (other form 1) cos 2θ = 1 − 2 sin² θ math-trig-identities Double-angle: cosine (other form 2) cos 2θ = 2 cos² θ − 1 math-trig-identities Double-angle: tangent tan 2θ = 2 tan θ / (1 − tan² θ) math-trig-identities Power reducing: sin² θ (1 − cos 2θ) / 2 math-trig-identities Power reducing: cos² θ (1 + cos 2θ) / 2 math-trig-identities Half-angle: sin(θ/2) ±√((1 − cos θ) / 2) math-trig-identities Half-angle: cos(θ/2) ±√((1 + cos θ) / 2) math-trig-identities Product to sum: sin A cos B ½[sin(A + B) + sin(A − B)] math-trig-identities Product to sum: cos A cos B ½[cos(A − B) + cos(A + B)] math-trig-identities Law of sines a / sin A = b / sin B = c / sin C math-trig-identities Law of cosines c² = a² + b² − 2ab cos C math-trig-identities Area of a triangle with trig ½ ab sin C math-trig-identities Arc length formula (radians) s = rθ math-trig-identities Sector area formula (radians) A = ½ r² θ math-trig-identities Degrees to radians Multiply by π/180 math-trig-identities Radians to degrees Multiply by 180/π math-trig-identities Amplitude of y = A sin(Bx) |A| math-trig-identities Period of y = A sin(Bx) 2π / |B| math-trig-identities Phase shift of y = sin(x − C) C units to the right math-trig-identities Inverse sine range −π/2 to π/2 math-trig-identities Inverse cosine range 0 to π math-trig-identities Inverse tangent range −π/2 to π/2 (not including the ends) math-trig-identities Euler's formula e^(iθ) = cos θ + i sin θ math-trig-identities Euler's identity e^(iπ) + 1 = 0 math-trig-identities De Moivre's theorem (cos θ + i sin θ)ⁿ = cos nθ + i sin nθ math-trig-identities sin 0° 0 math-trig-values sin 30° 1/2 math-trig-values sin 45° √2/2 math-trig-values sin 60° √3/2 math-trig-values sin 90° 1 math-trig-values sin 120° √3/2 math-trig-values sin 135° √2/2 math-trig-values sin 150° 1/2 math-trig-values sin 180° 0 math-trig-values sin 210° −1/2 math-trig-values sin 225° −√2/2 math-trig-values sin 240° −√3/2 math-trig-values sin 270° −1 math-trig-values sin 300° −√3/2 math-trig-values sin 315° −√2/2 math-trig-values sin 330° −1/2 math-trig-values sin 360° 0 math-trig-values cos 0° 1 math-trig-values cos 30° √3/2 math-trig-values cos 45° √2/2 math-trig-values cos 60° 1/2 math-trig-values cos 90° 0 math-trig-values cos 120° −1/2 math-trig-values cos 135° −√2/2 math-trig-values cos 150° −√3/2 math-trig-values cos 180° −1 math-trig-values cos 210° −√3/2 math-trig-values cos 225° −√2/2 math-trig-values cos 240° −1/2 math-trig-values cos 270° 0 math-trig-values cos 300° 1/2 math-trig-values cos 315° √2/2 math-trig-values cos 330° √3/2 math-trig-values cos 360° 1 math-trig-values tan 0° 0 math-trig-values tan 30° √3/3 (1/√3) math-trig-values tan 45° 1 math-trig-values tan 60° √3 math-trig-values tan 90° Undefined math-trig-values tan 120° −√3 math-trig-values tan 135° −1 math-trig-values tan 150° −√3/3 math-trig-values tan 180° 0 math-trig-values tan 225° 1 math-trig-values tan 270° Undefined math-trig-values sin(π/6) 1/2 math-trig-values sin(π/4) √2/2 math-trig-values sin(π/3) √3/2 math-trig-values sin(π/2) 1 math-trig-values cos(π/6) √3/2 math-trig-values cos(π/4) √2/2 math-trig-values cos(π/3) 1/2 math-trig-values cos(π) −1 math-trig-values sin(π) 0 math-trig-values cos(2π/3) −1/2 math-trig-values sin(5π/6) 1/2 math-trig-values sin(3π/2) −1 math-trig-values sin(7π/6) −1/2 math-trig-values cos(5π/4) −√2/2 math-trig-values cos(7π/4) √2/2 math-trig-values tan(π/4) 1 math-trig-values tan(π/3) √3 math-trig-values tan(π/6) √3/3 math-trig-values csc 30° 2 math-trig-values sec 60° 2 math-trig-values cot 45° 1 math-trig-values sec 0° 1 math-trig-values Where is sine positive? Quadrants I and II math-trig-values Where is cosine positive? Quadrants I and IV math-trig-values Where is tangent positive? Quadrants I and III math-trig-values Memory rule for signs (All Students Take Calculus) All positive in I, Sine in II, Tangent in III, Cosine in IV math-trig-values The unit circle point at 90° (0, 1) math-trig-values The unit circle point at 180° (−1, 0) math-trig-values The unit circle point at 45° (√2/2, √2/2) math-trig-values Range of sine and cosine −1 to 1 math-trig-values Period of sine and cosine 360° or 2π math-trig-values Period of tangent 180° or π math-trig-values Inches in 1 foot 12 math-us-units Feet in 1 yard 3 math-us-units Inches in 1 yard 36 math-us-units Feet in 1 mile 5,280 math-us-units Yards in 1 mile 1,760 math-us-units Ounces in 1 pound 16 math-us-units Pounds in 1 ton 2,000 math-us-units Teaspoons in 1 tablespoon 3 math-us-units Tablespoons in 1 fluid ounce 2 math-us-units Tablespoons in 1 cup 16 math-us-units Fluid ounces in 1 cup 8 math-us-units Cups in 1 pint 2 math-us-units Pints in 1 quart 2 math-us-units Cups in 1 quart 4 math-us-units Quarts in 1 gallon 4 math-us-units Pints in 1 gallon 8 math-us-units Cups in 1 gallon 16 math-us-units Fluid ounces in 1 gallon 128 math-us-units Square feet in 1 square yard 9 math-us-units Square inches in 1 square foot 144 math-us-units Square feet in 1 acre 43,560 math-us-units Cubic feet in 1 cubic yard 27 math-us-units Cubic inches in 1 cubic foot 1,728 math-us-units Convert 5 ft to inches 60 inches math-us-units Convert 2 yd to feet 6 feet math-us-units Convert 48 inches to feet 4 feet math-us-units Convert 3 lb to oz 48 oz math-us-units Convert 32 oz to lb 2 lb math-us-units Convert 3 gal to qt 12 qt math-us-units Convert 6 cups to pints 3 pints math-us-units Convert 10 qt to gal 2.5 gal math-us-units Convert 1 mile to yards 1,760 yards math-us-units Convert 4,000 lb to tons 2 tons math-us-units Inches in 1 centimeter (about) 0.39 inches math-us-units Centimeters in 1 inch 2.54 exactly math-us-units Meters in 1 yard 0.9144 exactly math-us-units Kilometers in 1 mile (about) 1.61 kilometers math-us-units Miles in 1 kilometer (about) 0.62 miles math-us-units Pounds in 1 kilogram (about) 2.2 pounds math-us-units Grams in 1 ounce (about) 28.35 grams math-us-units Liters in 1 gallon (US, about) 3.785 liters math-us-units Liters in 1 quart (about) 0.95 liters math-us-units Feet in 1 meter (about) 3.28 feet math-us-units Fahrenheit freezing point of water 32 degrees F math-us-units Fahrenheit boiling point of water (sea level) 212 degrees F math-us-units Normal body temperature in Fahrenheit About 98.6 degrees F math-us-units Formula: Fahrenheit to Celsius C = (F − 32) × 5/9 math-us-units Formula: Celsius to Fahrenheit F = C × 9/5 + 32 math-us-units Convert 212 degrees F to Celsius 100 degrees C math-us-units Convert 0 degrees C to Fahrenheit 32 degrees F math-us-units Convert 20 degrees C to Fahrenheit 68 degrees F math-us-units Convert 100 degrees F to Celsius About 37.8 degrees C math-us-units At what temperature are Celsius and Fahrenheit equal? −40 degrees math-us-units Which is longer, a foot or a meter? A meter (about 3.28 feet) math-us-units Which is heavier, a pound or a kilogram? A kilogram math-us-units Which is bigger, a quart or a liter? A liter is slightly bigger math-us-units Which is bigger, a mile or a kilometer? A mile math-us-units What is a baker's dozen? 13 math-us-units What is a dozen? 12 math-us-units What is a gross? 144 (12 dozen) math-us-units What is a score? 20 math-us-units What is a ream of paper? 500 sheets math-us-units What is a nautical mile? About 1.15 miles (1,852 meters) math-us-units What is a knot? A speed of one nautical mile per hour math-us-units What is a fathom? 6 feet, used to measure water depth math-us-units What is a furlong? One eighth of a mile (220 yards) math-us-units What is a hand (horse height)? 4 inches math-us-units Factor A number that multiplies with another to make a product (3 and 4 are factors of 12) math-vocab-3-5 Product The answer to a multiplication problem math-vocab-3-5 Quotient The answer to a division problem math-vocab-3-5 Dividend The number being divided (In 12 ÷ 3, it is 12) math-vocab-3-5 Divisor The number you divide by (In 12 ÷ 3, it is 3) math-vocab-3-5 Remainder What is left over after dividing math-vocab-3-5 Multiple What you get when you multiply a number by 1, 2, 3, ... (Multiples of 5: 5, 10, 15) math-vocab-3-5 Prime number A number greater than 1 with exactly two factors, 1 and itself math-vocab-3-5 Composite number A number with more than two factors math-vocab-3-5 Array Objects arranged in equal rows and columns math-vocab-3-5 Equation A math sentence with an equals sign math-vocab-3-5 Expression Numbers and operation signs with no equals sign (3 × 4 + 2) math-vocab-3-5 Variable A letter that stands for an unknown number math-vocab-3-5 Numerator The top number in a fraction; how many parts we have math-vocab-3-5 Denominator The bottom number in a fraction; how many equal parts in the whole math-vocab-3-5 Equivalent fractions Fractions that name the same amount (1/2 and 2/4) math-vocab-3-5 Mixed number A whole number and a fraction together (2 1/3) math-vocab-3-5 Improper fraction A fraction whose numerator is greater than or equal to its denominator (7/4) math-vocab-3-5 Unit fraction A fraction with 1 on top (1/5) math-vocab-3-5 Simplest form A fraction that cannot be made smaller; the top and bottom share no factor but 1 math-vocab-3-5 Decimal A number written with a decimal point to show parts of a whole (0.75) math-vocab-3-5 Decimal point The dot that separates ones from tenths math-vocab-3-5 Tenths The first place right of the decimal point math-vocab-3-5 Hundredths The second place right of the decimal point math-vocab-3-5 Round Replace a number with a nearby number that is easier to use math-vocab-3-5 Estimate Find an answer that is close but not exact math-vocab-3-5 Regroup Trade between place values, such as 10 ones for 1 ten math-vocab-3-5 Place value The value of a digit based on where it sits in a number math-vocab-3-5 Perimeter The distance around a shape math-vocab-3-5 Area The amount of flat space inside a shape, in square units math-vocab-3-5 Volume The amount of space inside a solid, in cubic units math-vocab-3-5 Capacity How much a container can hold math-vocab-3-5 Angle Formed where two rays meet at a point math-vocab-3-5 Right angle An angle of exactly 90 degrees, like a square corner math-vocab-3-5 Acute angle An angle less than 90 degrees math-vocab-3-5 Obtuse angle An angle greater than 90 degrees and less than 180 degrees math-vocab-3-5 Straight angle An angle of exactly 180 degrees math-vocab-3-5 Parallel lines Lines that stay the same distance apart and never meet math-vocab-3-5 Perpendicular lines Lines that meet at a right angle math-vocab-3-5 Intersecting lines Lines that cross each other math-vocab-3-5 Line segment Part of a line with two endpoints math-vocab-3-5 Ray Part of a line with one endpoint that goes on forever in one direction math-vocab-3-5 Point An exact location with no size math-vocab-3-5 Polygon A closed flat shape made of straight sides math-vocab-3-5 Quadrilateral Any polygon with 4 sides math-vocab-3-5 Pentagon A polygon with 5 sides math-vocab-3-5 Hexagon A polygon with 6 sides math-vocab-3-5 Octagon A polygon with 8 sides math-vocab-3-5 Symmetry When a shape can be folded so both halves match exactly math-vocab-3-5 Line of symmetry The fold line that makes two matching halves math-vocab-3-5 Congruent Exactly the same size and shape math-vocab-3-5 Face A flat surface of a solid shape math-vocab-3-5 Edge Where two faces of a solid meet math-vocab-3-5 Vertex A corner point of a shape or solid; plural is vertices math-vocab-3-5 Elapsed time The amount of time that passes from start to end math-vocab-3-5 Data Facts or numbers collected to learn something math-vocab-3-5 Bar graph A graph that uses bars to compare amounts math-vocab-3-5 Line plot A number line with marks above it to show how often values occur math-vocab-3-5 Pictograph A graph that uses pictures to show data math-vocab-3-5 Survey Asking a group of people questions to collect data math-vocab-3-5 Mean The average: add all values and divide by how many there are math-vocab-3-5 Median The middle value when the data are in order math-vocab-3-5 Mode The value that appears most often math-vocab-3-5 Range The highest value minus the lowest value math-vocab-3-5 Commutative property You can swap the order of numbers when adding or multiplying (3 × 4 = 4 × 3) math-vocab-3-5 Associative property You can regroup numbers when adding or multiplying and get the same answer math-vocab-3-5 Distributive property Multiplying a sum equals multiplying each part and adding (3 × (4 + 2) = 3 × 4 + 3 × 2) math-vocab-3-5 Identity property Adding 0 or multiplying by 1 leaves a number unchanged math-vocab-3-5 Inverse operations Operations that undo each other (Addition and subtraction) math-vocab-3-5 Ordered pair Two numbers in parentheses that name a point on a grid ((3, 5)) math-vocab-3-5 Coordinate plane A flat grid made by a horizontal and a vertical number line math-vocab-3-5 Digit One of the symbols 0 to 9 math-vocab-3-5 Compatible numbers Numbers that are easy to compute with in your head math-vocab-3-5 Unit A fixed amount used to measure, like an inch or a gram math-vocab-3-5 Weight How heavy something is math-vocab-3-5 Mass The amount of matter in an object math-vocab-3-5 Fraction A number that names equal parts of a whole math-vocab-3-5 Ratio A comparison of two quantities by division (3 to 5, or 3:5) math-vocab-6-8 Rate A ratio that compares two different kinds of units (60 miles per hour) math-vocab-6-8 Unit rate A rate with 1 as the second quantity ($2 per pound) math-vocab-6-8 Proportion An equation that says two ratios are equal math-vocab-6-8 Percent A ratio out of 100 math-vocab-6-8 Integer A whole number or its opposite: ..., −2, −1, 0, 1, 2, ... math-vocab-6-8 Rational number A number that can be written as a fraction of two integers math-vocab-6-8 Irrational number A number that cannot be written as a fraction; its decimal never ends or repeats (π, √2) math-vocab-6-8 Real numbers All rational and irrational numbers together math-vocab-6-8 Absolute value A number's distance from zero on the number line math-vocab-6-8 Opposite numbers Two numbers the same distance from zero on opposite sides (5 and −5) math-vocab-6-8 Negative number A number less than zero math-vocab-6-8 Greatest common factor (GCF) The largest factor two numbers share math-vocab-6-8 Least common multiple (LCM) The smallest multiple two numbers share math-vocab-6-8 Prime factorization Writing a number as a product of primes (12 = 2 × 2 × 3) math-vocab-6-8 Exponent A small raised number that tells how many times to use the base as a factor math-vocab-6-8 Base The number that is multiplied by itself in a power math-vocab-6-8 Power A number written with a base and an exponent math-vocab-6-8 Square root A number that, multiplied by itself, gives the original number math-vocab-6-8 Scientific notation Writing a number as a number from 1 to 10 times a power of 10 (3.2 × 10^4) math-vocab-6-8 Term A number, variable, or product in an expression math-vocab-6-8 Coefficient The number multiplied by a variable (In 7x, the coefficient is 7) math-vocab-6-8 Constant A term that does not change; a plain number math-vocab-6-8 Like terms Terms with the same variable part (3x and 5x) math-vocab-6-8 Inequality A statement that two values are not equal (x > 4) math-vocab-6-8 Solution A value that makes an equation or inequality true math-vocab-6-8 Slope How steep a line is; rise over run math-vocab-6-8 Y-intercept The point where a graph crosses the y-axis math-vocab-6-8 Function A rule that gives exactly one output for each input math-vocab-6-8 Input The value you put into a function math-vocab-6-8 Output The value that comes out of a function math-vocab-6-8 Linear equation An equation whose graph is a straight line math-vocab-6-8 Quadrant One of the four regions of a coordinate plane math-vocab-6-8 Origin The point (0, 0) where the axes cross math-vocab-6-8 Pythagorean theorem In a right triangle, a² + b² = c² math-vocab-6-8 Hypotenuse The longest side of a right triangle, opposite the right angle math-vocab-6-8 Legs The two shorter sides of a right triangle math-vocab-6-8 Circumference The distance around a circle math-vocab-6-8 Radius The distance from the center of a circle to its edge math-vocab-6-8 Diameter The distance across a circle through its center; twice the radius math-vocab-6-8 Pi (π) The ratio of a circle's circumference to its diameter; about 3.14159 math-vocab-6-8 Surface area The total area of all the faces of a solid math-vocab-6-8 Prism A solid with two identical parallel bases and rectangular sides math-vocab-6-8 Pyramid A solid with one base and triangular faces that meet at a point math-vocab-6-8 Net A flat pattern that folds up into a solid math-vocab-6-8 Similar figures Shapes with the same angles and proportional sides math-vocab-6-8 Scale factor The number you multiply lengths by to enlarge or shrink a figure math-vocab-6-8 Transformation A move that changes a figure's position or size math-vocab-6-8 Translation Slide a figure without turning or flipping it math-vocab-6-8 Reflection Flip a figure over a line math-vocab-6-8 Rotation Turn a figure around a point math-vocab-6-8 Dilation Enlarge or shrink a figure from a center point math-vocab-6-8 Complementary angles Two angles that add up to 90 degrees math-vocab-6-8 Supplementary angles Two angles that add up to 180 degrees math-vocab-6-8 Vertical angles Opposite angles formed by two crossing lines; they are equal math-vocab-6-8 Adjacent angles Two angles that share a side and a vertex math-vocab-6-8 Transversal A line that crosses two or more other lines math-vocab-6-8 Mean The average of a set of numbers math-vocab-6-8 Median The middle value of ordered data math-vocab-6-8 Mode The most frequent value math-vocab-6-8 Range Greatest value minus least value math-vocab-6-8 Outlier A value far from the rest of the data math-vocab-6-8 Quartile One of the three values that split ordered data into four equal parts math-vocab-6-8 Interquartile range (IQR) The third quartile minus the first quartile math-vocab-6-8 Box plot A graph that shows the median, quartiles and extremes of data math-vocab-6-8 Histogram A bar graph with no gaps that shows how data are spread across intervals math-vocab-6-8 Probability A number from 0 to 1 that tells how likely an event is math-vocab-6-8 Sample space The set of all possible outcomes math-vocab-6-8 Compound event An event made of two or more simple events math-vocab-6-8 Variable A letter that stands for a number that can change math-vocab-6-8 Evaluate Find the value of an expression by putting in numbers math-vocab-6-8 Simplify Rewrite an expression in its simplest form math-vocab-6-8 Inverse operation An operation that undoes another math-vocab-6-8 Reciprocal The number that multiplies with another to make 1 (The reciprocal of 3/4 is 4/3) math-vocab-6-8 Terminating decimal A decimal that ends (0.25) math-vocab-6-8 Repeating decimal A decimal with a digit or group of digits that repeats forever (0.333...) math-vocab-6-8 Markup An amount added to a cost to set the selling price math-vocab-6-8 Discount An amount subtracted from a price math-vocab-6-8 Simple interest Interest earned only on the starting amount: I = Prt math-vocab-6-8 Domain The set of all possible inputs of a function math-vocab-9-12 Range (of a function) The set of all possible outputs of a function math-vocab-9-12 Relation A set of ordered pairs math-vocab-9-12 Inverse function A function that undoes another function (f⁻¹(x)) math-vocab-9-12 Composition of functions Using the output of one function as the input of another (f(g(x))) math-vocab-9-12 Polynomial A sum of terms, each a number times a variable to a whole-number power math-vocab-9-12 Degree (of a polynomial) The highest exponent of the variable math-vocab-9-12 Leading coefficient The coefficient of the term with the highest power math-vocab-9-12 Monomial A polynomial with one term math-vocab-9-12 Binomial A polynomial with two terms math-vocab-9-12 Trinomial A polynomial with three terms math-vocab-9-12 Root (zero) An input that makes a function equal zero math-vocab-9-12 Quadratic A polynomial of degree 2 (ax² + bx + c) math-vocab-9-12 Parabola The U-shaped graph of a quadratic function math-vocab-9-12 Vertex (of a parabola) The highest or lowest point of the parabola math-vocab-9-12 Axis of symmetry The line that splits a parabola into mirror halves math-vocab-9-12 Discriminant b² − 4ac; tells how many real roots a quadratic has math-vocab-9-12 Imaginary unit (i) The number whose square is −1 math-vocab-9-12 Complex number A number of the form a + bi math-vocab-9-12 Conjugate The binomial with the opposite sign in the middle (a + bi and a − bi) math-vocab-9-12 Asymptote A line that a graph gets closer and closer to math-vocab-9-12 Exponential function A function where the variable is in the exponent (y = 2^x) math-vocab-9-12 Logarithm The exponent you need on a base to get a number math-vocab-9-12 Natural logarithm A logarithm with base e, written ln math-vocab-9-12 Euler's number (e) An irrational constant, about 2.71828 math-vocab-9-12 Sequence A list of numbers in order math-vocab-9-12 Arithmetic sequence A sequence with a constant difference between terms math-vocab-9-12 Geometric sequence A sequence with a constant ratio between terms math-vocab-9-12 Series The sum of the terms of a sequence math-vocab-9-12 Sigma (Σ) The symbol meaning add up a list of terms math-vocab-9-12 Matrix A rectangular array of numbers in rows and columns math-vocab-9-12 System of equations Two or more equations with the same variables math-vocab-9-12 Linear programming Finding the best value of a quantity under linear limits math-vocab-9-12 Theorem A statement that has been proven math-vocab-9-12 Postulate (axiom) A statement accepted as true without proof math-vocab-9-12 Proof A logical argument that shows a statement is true math-vocab-9-12 Conjecture A guess based on a pattern that has not been proven math-vocab-9-12 Counterexample An example that shows a statement is false math-vocab-9-12 Converse A conditional statement with its parts swapped (If Q, then P) math-vocab-9-12 Contrapositive The converse of the inverse; it always has the same truth as the original statement math-vocab-9-12 Inductive reasoning Drawing a general rule from patterns math-vocab-9-12 Deductive reasoning Drawing a conclusion from facts that are known to be true math-vocab-9-12 Congruent triangles Triangles with all matching sides and angles equal math-vocab-9-12 Locus The set of all points that follow a given rule math-vocab-9-12 Inscribed Drawn inside another figure and touching it math-vocab-9-12 Circumscribed Drawn around another figure and touching it math-vocab-9-12 Tangent (to a circle) A line that touches a circle at exactly one point math-vocab-9-12 Secant A line that cuts through a circle at two points math-vocab-9-12 Chord A segment with both endpoints on a circle math-vocab-9-12 Arc A part of a circle's edge math-vocab-9-12 Sector A pie-slice region of a circle math-vocab-9-12 Radian An angle measure; the angle whose arc equals the radius math-vocab-9-12 Sine In a right triangle, opposite over hypotenuse math-vocab-9-12 Cosine In a right triangle, adjacent over hypotenuse math-vocab-9-12 Tangent (trigonometry) In a right triangle, opposite over adjacent math-vocab-9-12 Unit circle A circle of radius 1 centered at the origin math-vocab-9-12 Amplitude Half the distance between the highest and lowest values of a wave math-vocab-9-12 Period The length of one full cycle of a repeating graph math-vocab-9-12 Limit The value a function gets close to as the input approaches a number math-vocab-9-12 Continuous Having a graph with no breaks, jumps or holes math-vocab-9-12 Derivative The rate at which a function changes; the slope of its tangent math-vocab-9-12 Integral The accumulated total under a curve; area under a graph math-vocab-9-12 Vector A quantity with both size and direction math-vocab-9-12 Scalar A quantity with size only math-vocab-9-12 Permutation An arrangement of items in which order matters math-vocab-9-12 Combination A selection of items in which order does not matter math-vocab-9-12 Standard deviation A measure of how spread out data are around the mean math-vocab-9-12 Normal distribution A bell-shaped, symmetric distribution math-vocab-9-12 Correlation A measure of how two variables move together math-vocab-9-12 Regression line The line of best fit through a scatter plot math-vocab-9-12 Residual The difference between an actual value and the predicted value math-vocab-9-12 Conic section A curve from slicing a cone: circle, ellipse, parabola, hyperbola math-vocab-9-12 Ellipse An oval shape; the set of points whose distances to two foci add to a constant math-vocab-9-12 Hyperbola A curve with two branches; the set of points whose distances to two foci differ by a constant math-vocab-9-12 Polar coordinates Describing a point by distance from the origin and an angle math-vocab-9-12 Rational function A function that is a ratio of two polynomials math-vocab-9-12 Extraneous solution A solution that comes from the working but does not fit the original equation math-vocab-9-12 Add Put numbers together to find how many in all math-vocab-k-2 Plus sign (+) The sign that means add math-vocab-k-2 Subtract Take away to find how many are left math-vocab-k-2 Minus sign (−) The sign that means subtract math-vocab-k-2 Equals (=) Means the same amount as math-vocab-k-2 Sum The answer when you add (In 3 + 4 = 7, the sum is 7) math-vocab-k-2 Difference The answer when you subtract (In 9 − 4 = 5, the difference is 5) math-vocab-k-2 Number A word or symbol that tells how many math-vocab-k-2 Digit One of the ten symbols 0 to 9 used to write numbers math-vocab-k-2 Zero The number that means none math-vocab-k-2 Even number A number you can split into two equal groups with none left over (2, 4, 6, 8, 10) math-vocab-k-2 Odd number A number that leaves one left over when you make pairs (1, 3, 5, 7, 9) math-vocab-k-2 Greater than (>) Bigger than (The open side faces the bigger number) math-vocab-k-2 Less than (<) Smaller than math-vocab-k-2 Compare Look at two things to see which is more, less, or the same math-vocab-k-2 More A larger amount math-vocab-k-2 Fewer A smaller number of things you can count math-vocab-k-2 Same Not different; equal math-vocab-k-2 Order Put numbers or things in a line by size or by rule math-vocab-k-2 Count Say numbers in order to find how many math-vocab-k-2 Skip count Count by jumping over numbers (2, 4, 6, 8) math-vocab-k-2 Number line A line with numbers placed in order, used to count and add math-vocab-k-2 Ones The place on the right in a number (In 36, the 6 is in the ones place) math-vocab-k-2 Tens The place next to the ones; each ten is a group of 10 (In 36, the 3 is in the tens place) math-vocab-k-2 Hundreds The place that counts groups of 100 math-vocab-k-2 Half One of two equal parts math-vocab-k-2 Fourth (quarter) One of four equal parts math-vocab-k-2 Whole All of something, with nothing missing math-vocab-k-2 Equal parts Parts that are exactly the same size math-vocab-k-2 Circle A round shape with no corners math-vocab-k-2 Triangle A shape with 3 sides and 3 corners math-vocab-k-2 Square A shape with 4 equal sides and 4 square corners math-vocab-k-2 Rectangle A shape with 4 sides and 4 square corners math-vocab-k-2 Side A straight edge of a flat shape math-vocab-k-2 Corner (vertex) The point where two sides meet math-vocab-k-2 Cube A solid shape like a dice; 6 square faces math-vocab-k-2 Sphere A solid shape shaped like a ball math-vocab-k-2 Cone A solid shape like an ice cream cone math-vocab-k-2 Cylinder A solid shape like a can math-vocab-k-2 Pattern Something that repeats in the same way (red, blue, red, blue) math-vocab-k-2 Measure Find out how long, tall, heavy or full something is math-vocab-k-2 Length How long something is math-vocab-k-2 Height How tall something is math-vocab-k-2 Longer Having more length math-vocab-k-2 Shorter Having less length math-vocab-k-2 Heavier Weighing more math-vocab-k-2 Lighter Weighing less math-vocab-k-2 Inch A small unit for length; about the width of a thumb math-vocab-k-2 Foot A unit of length equal to 12 inches math-vocab-k-2 Penny A coin worth 1 cent math-vocab-k-2 Nickel A coin worth 5 cents math-vocab-k-2 Dime A coin worth 10 cents math-vocab-k-2 Quarter (coin) A coin worth 25 cents math-vocab-k-2 Hour hand The short hand on a clock math-vocab-k-2 Minute hand The long hand on a clock math-vocab-k-2 Graph A picture that shows information so it is easy to compare math-vocab-k-2 Tally mark A mark used to keep count; four lines and a slash make five math-vocab-k-2 Before Comes earlier in order (4 comes before 5) math-vocab-k-2 After Comes later in order (6 comes after 5) math-vocab-k-2 Between In the middle of two things (5 is between 4 and 6) math-vocab-k-2 Double Two times a number; add it to itself (Double 4 is 8) math-vocab-k-2 Number sentence A math idea written with numbers and signs (2 + 3 = 5) math-vocab-k-2 Fact family Add and subtract facts that use the same three numbers (2, 3, 5) math-vocab-k-2 Group A set of things that belong together math-vocab-k-2 Row Things lined up side to side math-vocab-k-2 Column Things lined up top to bottom math-vocab-k-2 In all Add math-word-problem-keywords Altogether Add math-word-problem-keywords Total Add math-word-problem-keywords Sum Add math-word-problem-keywords Combined Add math-word-problem-keywords Plus Add math-word-problem-keywords More than Add (but 'A is 3 more than B' means A = B + 3) math-word-problem-keywords Increased by Add math-word-problem-keywords Gained Add math-word-problem-keywords Both Add the two amounts math-word-problem-keywords How many are left Subtract math-word-problem-keywords Remain Subtract math-word-problem-keywords Difference Subtract math-word-problem-keywords Fewer than Subtract math-word-problem-keywords Less than Subtract (order is flipped: '5 less than 9' is 9 − 5) math-word-problem-keywords Decreased by Subtract math-word-problem-keywords Take away Subtract math-word-problem-keywords How many more Subtract to compare two amounts math-word-problem-keywords Lost Subtract math-word-problem-keywords Gave away Subtract math-word-problem-keywords Minus Subtract math-word-problem-keywords Times Multiply math-word-problem-keywords Product Multiply math-word-problem-keywords Of (with a fraction or percent) Multiply (1/2 of 10 is 1/2 × 10) math-word-problem-keywords Each (with a total needed) Multiply math-word-problem-keywords Every Multiply or divide, depending on the question math-word-problem-keywords Twice Multiply by 2 math-word-problem-keywords Double Multiply by 2 math-word-problem-keywords Triple Multiply by 3 math-word-problem-keywords Per Multiply or divide; it marks a rate (Miles per hour) math-word-problem-keywords At this rate Multiply a rate by an amount math-word-problem-keywords Groups of Multiply math-word-problem-keywords Rows of Multiply math-word-problem-keywords Quotient Divide math-word-problem-keywords Divided equally Divide math-word-problem-keywords Shared equally Divide math-word-problem-keywords Split among Divide math-word-problem-keywords Half of Divide by 2 math-word-problem-keywords Average Add the values, then divide by how many math-word-problem-keywords Per person Divide the total by the number of people math-word-problem-keywords How many in each group Divide math-word-problem-keywords How many groups Divide math-word-problem-keywords Ratio of Compare using division or a fraction math-word-problem-keywords Percent of Convert to a decimal and multiply math-word-problem-keywords Is (as an equals sign) = math-word-problem-keywords What is The unknown value, a variable math-word-problem-keywords At most ≤ math-word-problem-keywords At least ≥ math-word-problem-keywords No more than ≤ math-word-problem-keywords No less than ≥ math-word-problem-keywords More than (inequality) > math-word-problem-keywords Fewer than (inequality) < math-word-problem-keywords Is the same as = math-word-problem-keywords Consecutive numbers Numbers in order with no gaps, such as n, n + 1, n + 2 math-word-problem-keywords Consecutive even numbers n, n + 2, n + 4 math-word-problem-keywords Perimeter Add the lengths of all the sides math-word-problem-keywords Area Multiply (for a rectangle, length times width) math-word-problem-keywords Step 1 of solving a word problem Read it and decide what the question asks math-word-problem-keywords Step 2 of solving a word problem Pick out the numbers and units you need math-word-problem-keywords Step 3 of solving a word problem Choose the operation and write an equation math-word-problem-keywords Step 4 of solving a word problem Solve it, then check that the answer makes sense and has the right units math-word-problem-keywords Extra information A number in a problem you do not need math-word-problem-keywords Multi-step problem A problem needing two or more operations math-word-problem-keywords