Number Sense Tricks Bryant Heath 18 October 2007∗ ∗Revised and Edited : 12 April 2018 1 Contents 1 Numerical Tricks 6 1.1 Introduction: FOILing/LIOFing When Multiplying . 6 1.2 Multiplying: The Basics . 8 1.2.1 Multiplying by 11 Trick . 8 1.2.2 Multiplying by 101 Trick . 10 1.2.3 Multiplying by 25 Trick . 11 1.2.4 Multiplying by 75 trick . 12 1.2.5 Multiplying by Any Fraction of 100, 1000, etc... 13 1.2.6 Double and Half Trick . 15 1.2.7 Multiplying Two Numbers Near 100 . 16 1.2.8 Squares Ending in 5 Trick . 19 1.2.9 Squares from 41-59 . 19 1.2.10 Multiplying Two Numbers Equidistant from a Third Number . 20 1.2.11 Multiplying Reverses . 22 1.3 Standard Multiplication Tricks . 22 1.3.1 Extending Foiling . 22 1.3.2 Factoring of Numerical Problems . 24 1.3.3 Sum of Consecutive Squares . 27 1.3.4 Sum of Squares: Factoring Method . 27 1.3.5 Sum of Squares: Special Case . 28 1.3.6 Difference of Squares . 29 1.3.7 Multiplying Two Numbers Ending in 5 . 30 1.3.8 Multiplying Mixed Numbers . 31 a 1.3.9 a × Trick.......................................... 33 b 1.3.10 Combination of Tricks . 34 1.4 Dividing Tricks . 35 1.4.1 Finding a Remainder when Dividing by 4, 8, etc... 36 1.4.2 Finding a Remainder when Dividing by 3, 9, etc... 36 1.4.3 Finding a Remainder when Dividing by 11 . 37 1.4.4 Finding Remainders of Other Integers . 37 1.4.5 Remainders of Expressions . 38 1.4.6 Dividing by 9 Trick . 40 a b 1.4.7 Converting 40 and 80 , etc... to Decimals . 41 1.5 Adding and Subtracting Tricks . 42 1.5.1 Subtracting Reverses . 42 1.5.2 Switching Numbers and Negating on Subtraction . 43 a a 1.5.3 + + ··· ............................. 44 b · (b + 1) (b + 1) · (b + 2) a b 1.5.4 + Trick.......................................... 44 b a a na − 1 1.5.5 − .......................................... 45 b nb + 1 2 Memorizations 47 2.1 Important Numbers . 47 2.1.1 Squares . 47 2.1.2 Cubes . 49 2.1.3 Powers of 2; 3; 5........................................ 51 2.1.4 Important Fractions . 53 2.1.5 Special Integers . 56 2.1.6 Roman Numerals . 59 2.1.7 Platonic Solids and Euler's Formula . 60 2.1.8 π and e Approximations . 61 2 2.1.9 Distance and Velocity Conversions . 62 2.1.10 Conversion between Distance ! Area, Volume . 63 2.1.11 Fluid and Weight Conversions . 64 2.1.12 Celsius to Fahrenheit Conversions . 65 2.2 Formulas . 65 2.2.1 Sum of Series . 65 2.2.2 Fibonacci Numbers . 69 2.2.3 Integral Divisors . 72 2.2.4 Number of Diagonals of a Polygon . 75 2.2.5 Exterior/Interior Angles . 75 2.2.6 Triangular, Pentagonal, etc... Numbers . 76 2.2.7 Finding Sides of a Triangle . 77 2.2.8 Equilateral Triangle Formulas . 80 2.2.9 Formulas of Solids . 80 2.2.10 Combinations and Permutations . 81 2.2.11 Trigonometric Values . 83 2.2.12 Trigonometric Formulas . 86 2.2.13 Graphs of Sines/Cosines . 87 2.2.14 Vertex of a Parabola . 88 2.2.15 Discriminant and Roots . 89 3 Miscellaneous Topics 90 3.1 Random Assortment of Problems . 90 3.1.1 GCD and LCM . 90 3.1.2 Perfect, Abundant, and Deficient Numbers . 92 3.1.3 Sum and Product of Coefficients in Binomial Expansion . 92 3.1.4 Sum/Product of the Roots . 94 3.1.5 Finding Units Digit of xn .................................. 95 3.1.6 Exponent Rules . 97 3.1.7 Log Rules . 98 3.1.8 Square Root Problems . 101 3.1.9 Finding Approximations of Square Roots . 101 3.1.10 Complex Numbers . 103 3.1.11 Function Inverses . 105 3.1.12 Patterns . 106 3.1.13 Probability and Odds . 107 3.1.14 Sets . 108 3.2 Changing Bases . 109 3.2.1 Converting Integers . 109 3.2.2 Converting Decimals . 112 3.2.3 Performing Operations . 113 3.2.4 Changing Between Bases: Special Case . 115 3.2.5 Changing Bases: Sum of Powers . 117 3.2.6 Changing Bases: Miscellaneous Topics . 117 3.3 Repeating Decimals . 117 3.3.1 In the form: :aaaaa : : : ....................................118 3.3.2 In the form: :ababa : : : ....................................118 3.3.3 In the form: :abbbb : : : ....................................119 3.3.4 In the form: :abcbcbc : : : ...................................119 3.4 Modular Arithmetic . 120 3.5 Fun with Factorials! . 121 3.5.1 1 · 1! + 2 · 2! + ··· + n · n! ..................................121 a! ± b! 3.5.2 .............................................122 c! 3.5.3 Wilson's Theorem . 122 3 3.6 Basic Calculus . ..