AAS theorem, 159
Abel, Niels, 47
absolute value, 302
absolutely convergent, 302
algebra, 20
distributive law, 29–30
fundamental theorem of, 46, 141, 234
golden rule of, 27
graphs and, 41–47
rules of, 26–30
solving for x, 36–40
solving for y, 48–49
alternate interior angles, 157
alternating series, 295
altitude of triangle, 172
“American Pi” (Lesser), 202
angles
alternate interior, 157
bisector, 160
corresponding, 156–157
perpendicular, 155
right, 155
sums of, 156 (fig.)
sums of, of pentagon, 167 (fig.)
vertical, 156
anti-derivatives, 296
approximations, 128
arc sine function, 214
Archimedes, 196
area, 169–172
of circles, 182–193
of ellipses, 190 (fig.)
of square, 173–174
arithmetic, fundamental theorem of, 141
ASA theorem, 159
axioms, 125
corresponding angle, 157
of Euclid, 153–154
SAS, 159
base ten, 61–62
basepoints, 277
bell curve, 194
formula, 247 (fig.)
Benjamin, Arthur, 322
binary arithmetic, 297–298
binary representation, 139
Binet’s formula, 232
Fibonacci numbers and, 116
for Lucas numbers, 118
binomial coefficients, 76–77
Pascal’s identity and, 87
birthdays, 304–305
“Blowin’ in the Wind” (Dylan), 133
Brown, Dan, 119
Brown, Ethan, 201
cadences, 102–103
Caesar, Julius, 64
calculus, 162
differential, 257
integral, 257
magic of, 255–263
calendar
calculating, 63–70
Gregorian, 64
history of, 64
Julian, 64
leap years, 64
mnemonics for, 65–66
month codes, 66–67
weekday names, 66
year codes, 67–68
Camp, Dane, 133
Cantor, Georg, 289
cards, decks of, 73
See also poker hands
Carroll, Lewis, 199
casting out nines, 53–59
division and, 57–58
multiplication and, 56–57
central angle theorem, 186
Cervantes, Miguel de, 64–65
chain rule, 271–272
check digit, 62
checkerboards
dominos covering, 122–123
trominos covering, 137
circles, 155
area of, 182–193
circumference of, 182–193
diameter of, 182–183
magic of, 181–182
major arc of, 186
minor arc of, 186
trigonometry and, 210–219
unit, 188, 210, 237, 238 (fig.)
circumference, 182–193
definition of, 183
circumscribed polygons, 196
clinometer, 208
clocks, 60 (fig.)
coefficients, 32–33
College Mathematics Journal, 119
combinations, 76
commutative property, 9
complements, 16–17
completing the square, 39–40
complex numbers, 233–240
complex plane, 235
unit circle in, 238 (fig.)
complex roots, 46
compound interest, 242
concavity, 275
conditionally convergent, 301
cone, 192
congruences, 60–61
congruent objects, 158
congruent triangles, 158 (fig.)
constant terms, 33
constructive proofs, 130
contradiction, 126
convergence, 290–291
absolute, 302
conditional, 301
converse statements, 126
corollaries, 129
corresponding angles, 156
axiom, 157
cosecant function, 208
cosine function, 206, 212 (fig.)
graphs of, 228 (fig.)
cotangent function, 208
countability, 288
counterexamples, 126
counting, 29
Fibonacci numbers, 102–110
critical points, 264
cryptography, 145–146
cubes
sums of, 11–12
sums of, identity, 136
cubics, 44
cylinder
surface area of, 191
volume of, 191
da Vinci, Leonardo, 119
The Da Vinci Code (Brown, D.), 119
degree of polynomials, 44
delegations, 139
denominators, 23
derivatives, 228
product rule for, 266–268
Descartes, René, 41
diameter, 182–183
differences of squares formula, 14, 33
proof of, 34
differential calculus, 257
differentiation, 261
rules, 265–274
digital roots, 54
Dirichlet, Peter, 147
distinct powers, 94
distributive law
definition of, 29
FOIL and, 31
illustration of, 29–30
negative numbers and, 30
divergence, 291
dividing by zero,
division
mental, 20–24
number of digits, 22
domino, 107
on checkerboard, 122–123
Dunham, Douglas, 154
Dylan, Bob, 133
e (number), 240–250
The Elements (Euclid), 153
eleven, 60–63
ellipses, 189
area of, 190 (fig.)
foci of, 190
encryption, 145–146
endpoints, average of, 164 (fig.)
equations
beautiful, 232
linear, 36–37
solving, 36–40
equilateral triangle, 159, 161, 205
Escher, M. C., 154
estimation
mental, 20–24
of number of digits, 21–22
axioms of, 153–154
Euclidean geometry, 154
Euler-Mascheroni constant, 292
Euler’s equation, 251–253
even numbers theorem, 125–126
exclamation points, 71–72
existence proof, 129
exponential decay, 243
exponential formula, 242
exponential functions, 243 (fig.), 265
exponentiation, law of, 129
exponents, law of, 47
factorials, 71–72
fast mental calculations, 12–24
Fermat, Pierre de, 41
on prime numbers, 145
Feynman, Richard, 294
Fibonacci numbers, 2, 178, 232
addition of squares of, 108
applications of, 99
Binet’s formula and, 116
consecutive, 111
counting, 102–110
distant neighbors, 109–110
first thirteen, 99 (table)
greatest common divisors of, 112
limericks and, 120
Lucas numbers and, 117 (table)
magic of, 97–102, 113 (table)
multiplication of, 109 (table), 110 (table)
in nature, 99
neighbors, 109
Pascal’s triangle and, 95, 103–104
patterns, 110–120
poetry of, 120 (table)
prime, 112–113
rabbits, 97–99
relatively prime, 111
sequences that add to n, 103 (table)
shifting, 103
squares of, 106 (table)
sum of even, 100–101
sum of odd, 102
as tilings, 105
Fibonacci sequence, 97
financial transactions, 145–146
flushes, 79–80
foci of ellipses, 190
applications of, 32
distributive law and, 31
football fields, 151–152
formulas
bell curve, 247 (fig.)
differences of squares, 14, 33, 34
exponential, 242
Hero’s, 219
quadratic, 38
Wallis’s, 197
four of a kind, 81
fractions, addition of, 127–128
full house, 80
fundamental theorem of algebra, 46, 234
fundamental theorem of arithmetic, 141
Gadbois, Steve, 83
number patterns, 5–6
geometric series, 280–290
finite, 283
infinite, 284–286
geometrical proofs, 136
geometry
classics, 153–169
of complex numbers, 235–240
Euclidean, 154
hyperbolic, 154
plane, 154
proof of, 252
Goldbach’s conjecture, 147
cultural perceptions of, 119
naming of, 118
properties of, 119
golden rectangle, 119
golden rule of algebra, 27
Google, 244
graphs
algebra and, 41–47
lines in, 41
ordered pairs and, 41
parabolas, 43–44
parallel, 42
slope in, 41
x-axis of, 41
y-axis of, 41–42
y-intercept in, 41
great circles, 154
greatest common divisors, 112
Gregorian calendar, 64
Gregory XIII (Pope), 64
Guinness Book of World Records, 198
Hardy, G. H., 121
harmonic series, 290–294
Hartl, Michael, 201
Harvey Mudd College, 3
Hemachandra, 102
Hero’s formula, 219
hexadecimal system, 61–62
hexagons, 195–196
hockey stick identity, 89–90, 90 (fig.)
hockey teams, 89–90
hyperbolic geometry, 154
hypotenuse, 173
i (number), 232
ice cream flavors, 76–77
identity
hockey stick, 89–90, 90 (fig.)
sum of cubes, 136
trigonometric, 219–225
if-then theorem, 125
imaginary axis, 235
induction
Fibonacci numbers and, 109, 134
proof by, 131–140
strong, 140
index, 311
infinite integers, 298
infinite series, 45
geometric, 284–286
impossible, 294–302
infinity, 279–280
inscribed hexagons, 195–196
integers, 125
composite, 141
infinite, 298
integral calculus, 257
integration, 296
interest, compound, 242
Internet, 145–146
The Interview (film), 28
inverse trigonometric functions, 214, 215 (fig.)
irrational numbers, 127–131
ISBN, 62–63
isosceles right triangle, 204
isosceles triangles, 159
“Jabberwocky” (Carroll), 199
Jackson, Andrew, 247
joker, 83
Julian calendar, 64
junk hands (poker), 82
Kilmer, Joyce, 253
Lambert, Johann Heinrich, 197
laws
of exponentiation, 129
of exponents, 47
of sines, 218–219
leap years, 64
legs, 173
lemmas, 268
length, products and, 238
Leonardo of Pisa, 97
Liber Abaci (Leonardo of Pisa), 97
license plates, 75
limericks, 120
limits, 260
products of, 262
of quotients, 262
sums and, 262
linear approximation, 276
linear equations, 36–37
lines
in graphs, 41
midpoint of, 164 (fig.)
parallel, 156
secant, 258
tangent, 257, 259 (fig.)
logarithms, 243–244
lotteries, 76–84
lowest terms, 128
Lu, Chao, 198
Lucas, Édouard, 117
Lucas numbers
Binet’s formula for, 118
Fibonacci numbers and, 117 (table)
magic
of algebra, 25–26
of calculus, 255–263
of circles, 181–182
of Fibonacci numbers, 97–102, 113 (table)
of infinity, 279–280
mathematics and, 1
of nine, 51–52
numbers and, 51–53
squares, 302–305
major arc, 186
major system, 199
Markowski, George, 119
Marloshkovips, Tony, 199
Mars, 143
matching problem, 248
mathematics
laws of, 121
magic and, 1
permanence and, 121
max-min problems, 264–265
mediant, 115
memory palace, 202
mental calculations
addition, 15
division, 20–24
estimation, 20–24
fast, 12–24
multiplication, 17–20
subtraction, 16–17
Mertens constant, 292
midpoint, 161
of line, 164 (fig.)
triangle, 165
minor arc, 186
“Misconceptions About the Golden Ratio” (Markowski), 119
modular arithmetic, 60–63
month codes, 66–67
mountain heights, 203–204, 209, 218
multiples of nine, 52–53
multiplication
addition method, 18
casting out nines and, 56–57
of close numbers, 34–35
commutative property of, 9
of Fibonacci numbers, 109 (table), 110 (table)
mental, 17–20
of negative numbers, 30
of numbers close to 100, 14–15
subtraction method in, 18
table, 21 (table), 23–24
See also products
my favorite number, 2520
why?, 141
negative numbers, 37
distributive law and, 30
multiplication of, 30
Newton, Isaac, 47
n-gon, 166
nine (number)
casting out, 53–59
magic of, 51–52
multiples of, 52–53
Not a Wake (Keith), 199
number patterns, 5–12
beauty of, 12
factorial, 73 (table)
Fibonacci, 110–120
Gauss and, 5–6
numbers
composite, 112
even, 125–126
interesting, 147
irrational, 127–131
magic and, 51–53
odd, 7, 10–11, 92–93, 126, 131–132
Pascal’s triangle with, 85
perfect, 146
rational, 127–131
real, 37
relatively prime, 111
study of, 5
sum of first n, 7
sum of first n even, 7
sum of first n odd, 7
triangular, 6–7
obtuse triangle, 159
odd numbers
in Pascal’s triangle, 92–93
sum of first n, 7
sums of, 131–132
theorem about, 126
in triangles, 10–11
one (number), 140–141
one pair, 81
1.61, 113–115
optimization, 255
theorem, 264–265
ordered pairs, 41
origin, 210
oversubtracting, 16
Pacioli, Luca, 119
Palais, Bob, 201
parabolas, 43
properties of, 44
vertex of, 44
parallel, 42
parallel lines, 156
parallelograms, 150
partial sums, 282
Pascal’s identity, 86
binomial coefficients and, 87
Pascal’s triangle, 84–95
addition of numbers in, 86–87
Fibonacci numbers and, 95, 103–104
odd numbers in, 92–93
patterns in, 92
as right triangle, 89
symmetry in, 85
pentagons, 166
sums of angles of, 167 (fig.)
perfect numbers, 146
perfect squares, 8
perimeters, 169–172
of inscribed hexagons, 195–196
of rectangles, 152
period, 229
permutations, 76
perpendicular angles, 155
perpendicular bisectors, 161
phonetic codes, 199
celebrating, 198–202
digits of, 195–197
memorizing, 198–202
surprising appearance of, 193–195
transcendental, 197
Pi-Day, 198
plane geometry, 154
plus or minus symbol, 38
Poe, Edgar Allan, 198–199
poetry, 120 (table)
poker hands, 76–78
flushes, 79–80
four of a kind, 81
full house, 80
jokers in, 83
junk, 82
one pair, 81
straight, 80
straight flush, 80
three of a kind, 81
two pair, 81
value of, 83–84
wild cards, 83
polar form, 238
polygons, 166
circumscribed, 196
triangulation of, 168
polyhedrons, 232
polynomials
cubic, 44
degree of, 44
quadratic, 44
quartic, 44
quintic, 44
roots of, 45
Taylor, 276
population dynamics, 97–99
postulates. See axioms
prime numbers
Fermat on, 145
Fibonacci, 112–113
infinitely many, 144–145
proofs and, 140–147
relatively, 111
two as, 140–141
Wilson’s theorem on, 144
product rule
for derivatives, 266–268
proofs, 266
products
of complex numbers, 238–240
length and, 238
of limits, 262
of numbers that add to 20, 12 (table)
rule of, 73–76
proofs
by algebra, 139
axioms in, 125
constructive, 130
by contradiction, 126
converse statements in, 126
counterexamples in, 126
existence, 129
geometrical, 136
of God’s equation, 252
by induction, 131–140
involving trominos, 137–138
irrational numbers, 127–131
of negatives, 121–122
product rule, 266
of Pythagorean theorem, 173–174
of quotient rule, 271
rational numbers, 127–131
strong induction, 140
of sum of cubes identity, 136
of unique factorization theorem, 142
value of, 121–127
pseudoprimes, 145
public key cryptography, 145–146
Pythagorean theorem, 173–177
in area of square calculation, 173–174
proofs of, 173–174
Pythagorean triple, 205
quadrants, 214
quadratic equations, 37
completing squares, 39–40
quadratic formula, 38
quadratic functions, 43
quadratic polynomials, 44
quadrilaterals, 166
definition, 149
midpoints of, 150 (fig.)
quartics, 44
quintics, 44
quotients
of limits, 262
rule, 270–271
rabbits, 97–99
radians, 226–230
Ramanujan, Srinivasa, 189
rational numbers
averages of, 128
proofs, 127–131
“The Raven” (Poe), 198–199
real line, 232
real numbers, 37
reciprocals, 23
rectangles, 7 (fig.)
fences, 150–151
perimeters of, 152
relatively prime numbers, 111
right angles, 155
right triangles
isosceles, 204
Pascal’s triangle as, 89
roots
complex, 46
digital, 54
of polynomials, 45
square, 37
rule of products, 73–76
rule of sum, 73–76
SAS axiom, 159
secant function, 208
secant line, 258
slope of, 260 (fig.)
Secrets of Mental Math (Benjamin and Shermer), 62
semi-perimeter, 219
Shakespeare, William, 64–65
Sierpinski triangle, 93
sine function, 206, 212 (fig.)
graphs of, 228 (fig.)
sines, law of, 218–219
slant height, 192
slope, 258
in graphs, 41
of secant line, 260 (fig.)
solving for x, 36–40
solving for y, 48–49
sphere, 191
square root, 37
squares, 8 (fig.)
addition of, of Fibonacci numbers, 108
addition of consecutive, 106
area of, 173–174
of Fibonacci numbers, 106 (table)
magic, 302–305
perfect, 8
quick calculations of, 13–14
SSS theorem, 161
Stirling’s approximation, 193–194, 232
straight, 80
straight flush, 80
strong induction, 140
subtraction
complements in, 16
mental, 16–17
method in multiplication, 18
sums
of angles, 156 (fig.)
of angles of pentagon, 167 (fig.)
of cubes, 11–12
of cubes identity, 136
of even-positioned Fibonacci numbers, 100–101
of first 100 numbers, 5–6
of first n even numbers, 7
of first n numbers, 7
of first n odd numbers, 7
limits and, 262
of numbers in multiplication table, 21, 23–24
of odd numbers, 131–132
of odd-positioned Fibonacci numbers, 102
partial, 282
rule of, 73–76
See also addition
supplementary angles, 155–156, 213
surface area
of cone, 192
of cylinder, 191
of sphere, 191
syllables, 102–103
symmetry in Pascal’s triangle, 85
tangent function, 206
tangent line, 257
approximation of, 259 (fig.)
tau, 201
Taylor, Brook, 275
Taylor polynomials, 276
Taylor series, 274–278
Tetris, 124
The, 1–310
theorems
AAS, 159
ASA, 159
central angle, 186
even numbers, 125–126
if-then, 125
odd numbers, 126
optimization, 264–265
Pythagorean, 173–177
30-60-90 triangle, 205
three of a kind, 81
tilings
breakable, 106
Fibonacci numbers as, 105
identities, 107
unbreakable, 106
transcendental, 197
tree heights, 208–209
altitude of, 172
congruent, 158 (fig.)
isosceles right, 204
midpoint theorem, 165
obtuse, 159
Sierpinski, 93–94
30–60–90, 205
See also Pascal’s triangle
triangular numbers, 6–7
triangulation, 168
trigonometric graphs, 226–230
trigonometric identities, 219–225
circles and, 210–219
trominos, 137
“The Twelve Days of Christmas,” 90–91
twin primes, 147
two pair, 81
unbreakable tilings, 106
uncountable sets, 289
unique factorization theorem, 141
proof of, 142
unit, 141
unit circle, 188, 210, 237, 238 (fig.)
variables, 26
vertex, 44
vertical angle theorem, 156
volume, 191
von Lindemann, Ferdinand, 197
Wallis’s formula, 197
water fountain, 44 (fig.)
weekday names, 66
wild cards, 83
Wilson’s theorem, 144
x, solving for, 36–40
x-axis, 41
y, solving for, 48–49
y-axis, 41–42
year codes, 67–68
y-intercept, 41
zip codes, 75