Index

A

About Time: Einstein’s Unfinished Revolution (Davies), 49

Agnesi, Maria, 32

American usage. See also British usage

of billion, 11

of trillion, 11

Appel, Kenneth, 59

The Art of Mathematics (King), 78

a^x + b^x = c^x, nontrivial solution to. See Fermat’s Last Theorem

B

Balance scale, 35–36

Barrow, John, 45

Beiler, Albert H., 5

Billion, in scientific articles, 10–11

British vs. American usage, 11

Birthday, probability, 9–10

Boltzman, Ludwig, 41

Brainteasers, 82–86

answers to, 87–101

British usage. See also American usage

of billion, 11

of trillion, 11

C

Cantor, Georg, 19

Cantor’s Diagonal Proof, 19–20

Carson, Johnny, 9

Casti, John L., 20

Cauchy, Augustin-Louis, 58

Chess, 26

Coins, tossing, 41–42, 44, 47–48

Complex numbers, xiv, 18

Composite numbers, xii–xiii

Compounding, 22–23

The Constants of Nature: The Numbers That Encode the Deepest Secrets of the Universe (Barrow), 45

Convergent series, 41

Countable infinities, 19

Counterintuitive reality, 27–28

Cubes, 65

D

Dangers of induction. See Induction, dangers of

Dantzig, Tobias, 73

Davies, Paul, 49

Degrees of freedom, 8

Derbyshire, John, 42

Diagonal number, 20. See also Cantor’s Diagonal Proof

Dice, 21

Divergent series, 41

Doxiadis, Apostolos, 38

Dunham, William, 67

E

Empty set, 47

Erdös, Paul, 76

Error in probability, 64–65

Euclid, xii, 33

Euclid (Lindsay), 80

Euclid’s proof of an infinity of primes, 59

Euler, Leonhard, 7, 35, 43, 60, 77

Euler’s identity, 77

Even, defined, 75

Even number

Goldbach’s conjecture, 35

infinity of, 17–18

sum of, 64

sum of two primes and, 35

F

Factorial, 2. See also Probability

Fermat, Pierre de, 3–5

Fermat’s Enigma (Singh), 57

Fermat’s Last Theorem, 3–5, 27

Field(s), 7, 70. See also Ring(s)

Frye, Roger, 7

Fundamental Theorem of Arithmetic, 73

G

Game, Set, and Math: Enigmas and Conundrums (Stewart), 4

Game probability. See also Probability

chess, 26

coins, tossing, 41–42, 44, 47–48

dice, 21

nim, 29–32

tic-tac-toe, 27

World Series, 51–55

Gardner, Martin, 10, 28, 48–49, 65

Gauss, Carl, 21–22

Gender, probability. See Sexes, probability

Germaine, Sophie, 32

Gödel, Kurt, 1

Gödel’s theorem, 1

Goldbach, Christian, 35

Goldbach’s conjecture, 35, 38–39

Graham’s Number, 35

Gravity, 70

The Guinness Book of World Records, 35

H

Haken, Wolfgang, 59

Hardy, G. H., 45

Harmonic series, 28, 41

Havil, Julian, 71

Heisenberg’s uncertainty principle, 1–2

Hypatia, 32

I

Imaginary numbers, xiv, 16–17

Induction, dangers of, 5–7

Infinite line probability, 8–9

Infinity(ies), 18–19

countable, 19

of even numbers, 17–18

of integers, 17–18

same size, 18–19

Integers, xii. See also Whole numbers

infinity of, 17–18

square root of, 73

Intuition vs. logic, 67

Irrational numbers, 16, 22

concept, xiii

rings and fields, 7

K

Kahneman, Daniel, 38

King, Jerry P., 78

Kovalevsky, Sonya, 32

Kronecker, Leopold, xii

L

Lady Luck: The Theory of Probability (Weaver), 9

Lagrange, Joseph Louis, 22

Largest number (10^{70,000,000,000,000}), 45

Largest prime, 33–34

Last Year at Marienbad, 29

Limit, to series, 28

Lindsay, Vachel, 80

Linguistic vagueness, 14

Literacy test, 65

Living and nonliving, 14

Lloyd, Seth, 26

Logic, intuition vs., 67

Logical paradox, 46

Losing. See Winning and losing

M

Marienbad game. See Nim, game of

Mathematical Circus (Gardner), 65

Mathematical Intelligencer, 77

The Mathematical Universe: An Alphabetical Journey through the Great Proofs, Problems, and Personalities (Dunham), 67

Mathematical truths, xi, 1, 13

vs. scientific truth, 5, 75–76

Mathematicians, xv

women, 32

Measure of ignorance, 58

Medical test results, probability, 45–46

N

Natural number(s), xii, 16

sum of square numbers, 22

uninteresting, 37

Negative whole numbers, xiii, 16

Nim, game of, 29–32

how to play, 29–30

winning strategy, 31–32

Noether, Emmy, 32

Nonintuitive truths, 71

Nonplussed! Mathematical Proof of Implausible Ideas (Havil), 71

Nontrivial solution, to equation (a^x + b^x = c^x). See Fermat’s Last Theorem

Nothingness, 36–37

Number. See specific number

Number (Dantzig), 73

O

Odd number, 39

sequence sum, 78

sum of, 64

Ogilvy, C. Stanley, 76

One-inch n -cube, 42–43

P

Paulos, John Allen, 27

The Penguin Dictionary of Curious and Interesting Numbers (Wells), 37

Perfect number(s). See also Sublime number

defined, 43

largest, 43

perfect, 43

smallest, 43

Physical universe, 36–37

Pi (π), 22, 67–68, 77

Poincaré, Henri, 41

Power, zero, 37

Prime numbers, xii

Goldbach’s conjecture, 35

largest, 33–34

sum of two, 35

3n + 1 form, 42

Probability, 2–3

birthday, 9–10

coins, tossing, 41–42, 44, 47–48

hitting 1-in-n event in n attempts, 43–45

infinite line, 8–9

measure of ignorance, 58

medical test results, 45–46

most common error in, 64–65

sexes, 11–12

winning and losing, 51–55

Proof, 13–15, 58. See also Truths

Pythagorean theorem, 3

Q

Quadrillion, 11

Questions, unanswered, 25

Quine, W. V., 4, 37

R

Ramanujan, Srinivasa, 45

Random numbers, 48–49

Rare numbers, 72. See also Perfect number(s)

Rational numbers, 16

concept, xiii

nature of, 73

rings and fields, 7, 70

Real numbers, xiv

Recreations in the Theory of Numbers (Beiler), 5

Rees, Martin, 70

Riemann’s Prime Number Hypothesis, 25

Right triangles, 3. See also Pythagorean theorem

Ring(s), 7, 70. See also Field(s)

Russell, Bertrand, 46

S

Sautoy, Marcus du, 58

Scale, 35–36

Scientific American (Stewart), 9

Scientific notation, 10–11

Scientific truth, vs. mathematical truths, 5, 75–76

The Secret Lives of Numbers (Szpiro), 41, 63

Series

convergent, 41

divergent, 41

limit to, 28

Set, 46–47

empty, 47

Sexes, probability, 11–12

Singh, Simon, 57

Small-world situation, 68–70

Social scientists, 10

Spatial dimensions, 70

Square root, xiii

Squares

consecutive, 76

pair of, 76

Stewart, Ian, 4

Strogatz, Steven, 69–70

Sublime number, 43. See also Perfect number(s)

Switching sides, 64

Szpiro, George G., 41, 63

T

Tautology, 13–14

Taxonomic problem, 14–15

Theorem. See specific theorem

Tic-tac-toe, 27

Tomorrow’s Math: Unsolved Problems for the Amateur (Ogilvy), 76

Transcendental number, 76–77

Trial-and-error methods, 2

Trillion, in scientific articles, 10–11

British vs. American usage, 11

Truths, xi, 13–14. See also Tautology

mathematical vs. scientific, 5, 75–76

nonintuitive, 71

Tversky, Amos, 38

U

Uncertainty principle. See Heisenberg’s uncertainty principle

Uncle Petros and Goldbach’s Conjecture (Doxiadis), 38–39

Uninteresting (natural) number, 37

Unpredictability, 2

W

Watts, Duncan, 69–70

Weaver, Warren, 9

Wells, David, 37

Wheels, Life, and Other Mathematical Amusements (Gardner), 10

Whole numbers, xii, 62–63. See also Integers

negative, xiii, 16

Wiles, Sir Andrew John, 4

Wilson Index, 73

Winning and losing, 51–55. See also Game probability

Women mathematicians, 32

World Series, 51–55

X

X-axis, 7

Z

Zeno’s paradox, 17

Zero (0), xiii, 15, 16

number raised to power, 37