INDEX

addition

associative law of, 187

commutative law of, 187

of complex numbers, 122–124, 123 (fig.), 124 (fig.)

of vectors, 125

aesthetic theory, 145

algebra

Euler on, 59, 68

imaginary numbers and, 52

variables in, 172–173

alternating current, 139n

Amalric, Marie, 156

angle, 81, 85

cosine of, 89–90, 106–107

radian in, 91, 107–108, 130–131

sine of, 89–90, 106–107

See also hypotenuse; triangle

angle sweeper

coordinates in, 85–86

cosine, sine functions in, 86–90, 88 (fig.)

radians in, 93–94

right triangle in, 85

trig function and, 86–87

in unit circle, 86

vector and, 132–134, 132 (fig.), 133 (fig.)

Anhalt Dessau, Princess of, 59–60

Apostol, Tom, 201n

arc length, 93

Archimedes, 43, 63

Argand, Jean-Robert, 115–117, 116 (fig.)

Aristotle, 38–40, 159

arithmetic

brain and, 155

distributive law in, 111n

grounding metaphor for, 152–153

–1 and, 126

arrow, 119 (fig.)

art

Euler’s formula and, 141–142

infinity in, 163

by McElheny, 162–163

associative law

of addition, 187

of multiplication, 187

Bair, Jaques, 210n

Baker, Nicholson, 3–4

Basel problem, 44, 44n

Basic Metaphor of Infinity (BMI), 160–161

beauty

brain and, 146

of Euler’s formula, 1–4, 14–15, 141–146, 148–149

Hardy on, 144

limbic system in, 143–144

mathematical, 142–144

pleasure and, 142

subjectivity in, 143

sublime and, 145–146

Bell, Eric Temple, 58

Bellos, Alex, 73

Benjamin, Arthur, 203n, 207n

Berkeley, George, 39, 172

Bernoulli, Johann, 56, 63, 69

Blanton, John D., 209n

BMI. See Basic Metaphor of Infinity

Bolyai, János, 65–66

Bombelli, Rafael, 53, 122–123

Bouwsma, Oets Kolk “O.K.,” 4

Boyer, Carl, 166

brain, 146, 155–156

Bruce, Ian, 209n

calculus

differential, 28

infinitesimals in, 169

infinity in, 25–26, 39

on rate of change, 24–28

Calculus volumes I and II (Apostol), 201n

Calinger, Ronald, 70

Cantor, Georg, 39–40

Cardano, Gerolamo, 49, 52–53

Catherine II, 62

Chudnovsky, Gregory and David, 45–46

circle

right triangle in, 85

triangle in, 83–86, 88 (fig.)

trig functions and, 83–84

See also unit circle

circumference, 41–42, 45, 93

commutative law

of addition, 187

of multiplication, 187

complex analysis, 141

complex number

addition of, 122–124, 123 (fig.), 124 (fig.)

defining, 187

multiplication of, 126–128

real number and, 117–118

in science, 122

speed, direction in, 118–119, 119 (fig.)

as vector, 120–122, 125, 135–136

in vector rotation, 135–136

complex plane

arrow in, 119 (fig.)

e in, 135

Gauss on, 117, 138

compound interest, 18–20

conceptual metaphor, in mathematics, 153–154, 161

constant, 187

coordinates, 117

in angle sweeper, 85–86

on plane, 81–82, 82 (fig.)

cosine, 80n

of angle, 89–90, 106–107

function, 77, 79–81, 84–90, 88 (fig.), 188

pi, 96–97

Cotes, Roger, 202n

cubic equation, 52–53, 188

de Moivre, Abraham

Euler on, 105–106

formula of, 103–106, 173–175

imaginary, real numbers in, 106

De Morgan, Augustus, 35

decimal, 42n

Declaration of Independence, 9, 11

Dedekind, Richard, 32n

Dehaene, Stanislas, 156

Descartes, René, 81

Devlin, Keith, 14, 149

Diderot, Denis, 69–71

differential calculus, 28

digits, of pi, 45–46

direction, 118–119, 119 (fig.)

distributive law

in arithmetic, 111n

defining, 188

for i, 111–112

divergent series, 37–38

division

infinitesimals in, 171

by zero, 26–27

Dudley, Underwood, vi

Dunham, William, 12, 56, 168

in complex plane, 135

defining, 188

as Euler’s number, 16–17, 17n

in hat-check problem, 21–22

imaginary number and, 109, 130, 135

as infinity, 20–21

irrational numbers and, 30–31, 33

as limit, 20–21

variable exponent and, 22, 22n

Edison, Thomas, 139–140

Einstein, Albert, 121

electricity, 139–140, 154

Elements (Euclid), 59, 166

emotional intelligence, 143n

Enlightenment

Diderot in, 70

Euler in, 10–13

intellectuals in, 8–9

Newton in, 10, 12

“An Essential Tension in Mathematics Education” (Moreno-Armella), 210n

Euclid, 59

Euler, Leonhard

on algebra, 59, 68

on Basel problem, 44n

Bernoulli and, 56, 63, 69

on de Moivre formula, 105–106

death of, 7–8

derivation by, 173–182

Euler, Leonhard (continued)

Diderot and, 69–71

in Enlightenment, 10–13

Frederick and, 61–62, 68

Gauss on, 71

general formula of, 154

on i, 47

on imaginary numbers, 54

on infinitesimals, 168–172

Introductio by, 165–166

intuition of, 167–168

memory of, 57

Opera Omnia of, 12

pi and, 42–45

Princess of Anhalt Dessau and, 59–60

on real numbers, 106

in technology, 12–13

Thiébault on, 55

on transcendental functions, 37

on trigonometry, 103

Euler’s formula

art and, 141–142

beauty of, 1–4, 14–15, 141–146, 148–149

explication of, 161–162

exponents in, 76–77

geometric interpretation of, 134–135

as “God’s Equation,” 13–14, 14n

i in, 29–30, 48–49

infinite sum in, 108–112

infinity in, 29–30, 41, 112

integer in, 15, 44

–1 in, 15n

numbers in, 72–75

pi in, 112–113

proof of, 165

in science, 5

on time, 25

vectors and, 130

Euler’s number, 16–17, 17n

evolution, 155–156

exponential growth, 29

exponential models, 140

exponents

defining, 19, 188

in Euler’s formula, 76–77

fractional, 24n

imaginary, 109

non-integers as, 23–24

variable, 22, 22n

factorial

defining, 188–189

operator, 106

symbol, 106

female mathematicians, 67n

Fermat’s Last Theorem, 147n

Feynman, Richard, 2, 14

4-D number, 121–122

⁴i, 48–49, 48–51

fraction

infinite sum of, 99–102

irrational number as, 42

numerator in, 97–98, 110

pi as, 42

in Zeno’s paradox, 100 (fig.)

fractional exponent, 24n

Français, Jacques, 115

Frederick II (king), 59–62, 68, 70

function, 23

cosine as, 77, 79–81, 84–90, 88 (fig.), 188

in rate of change, 28–29

sine as, 77–81, 86, 88 (fig.), 89–90, 191

transcendental, 37

Galileo, 92

Gardner, Martin, 151

Gauss, Carl Friedrich

Bolyai and, 65–66

on complex plane, 117, 138

on Euler, 71

Germain and, 66–67

Stewart on, 67

General Electric, 139

geometric

calculation, 124–126, 130

interpretation, 134–137

intuition, 168n

geometry, 122–124

Germain, Marie-Sophie, 66–67

“God’s Equation,” 13–14, 14n

Grandi’s series, 37

Gregory, James, 35–37, 43–44

grounding metaphor, in mathematics, 152–153

Hamilton, William Rowan, 121

Hardy, G. H.

on beauty, 144

Platonism of, 150, 158

Ramanujan and, 157–158

Hatch, Robert A., 64

hat-check problem, 21–22

Hawking, Steven, 6

Hermite, Charles, 34

Hersh, Reuben, 151–152

Hertz, Heinrich, 162

Hidden Figures (movie), 13

Hilbert, David, 40

Hippasus, 32

Hooke, Robert, 63–64

hypotenuse

in circle, 85

defining, 189

of right triangle, 78–80

defining, 189

distributive law and, 111–112

Euler on, 47

in Euler’s formula, 29–30, 48–49

as imaginary number, 48, 50

to ith power, 183–184

multiplication of, 127–130, 127 (fig.)

as number, 51, 72

pi and, 48–49

as square root, 48, 183–184

imaginary exponent, 109

imaginary number

in algebra, 52

Argand on, 116–117, 116 (fig.)

in cubic equations, 52

in de Moivre formula, 106

defined, 189

e and, 109, 130, 135

Euler on, 54

i as, 48, 50

Leibniz on, 54, 106

imaginary number (continued)

on plane, 116–117, 116 (fig.)

real number, counterpart to, 48

trigonometry and, 103

zero and, 117–118

Imagining Numbers (Particularly the Square Root of Minus Fifteen) (Mazur), 150–151

impossible number, 54

Indian mathematics, 73–74, 74n

See also Ramanujan, Srinivasa

infinite

irrational numbers and, 31

sets, 40

infinite sum, 96–97

in Euler’s formula, 108–112

of fractions, 99–102

Leibniz on, 36–37, 43–44

pi and, 43

as transcendental function, 37

trig functions and, 105–106

trigonometry and, 36

infinitesimals

Berkeley, Newton on, 39

in calculus, 169

in division, 171

Euler on, 168–172

Leibniz on, 169

multiplication of, 171–172

infinity

actual, 159–160

in art, 163

in calculus, 25–26, 39

e as, 20–21

in Euler’s formula, 29–30, 41, 112

metaphor and, 159–160

as number, 38

Platonists on, 159

potential, 38–39, 159

Zeno’s paradox in, 99–100

See also potential infinities

Innumeracy: Mathematical Illiteracy and Its Consequences (Paulos), 154–155

integer

in Euler’s formula, 15, 44

in polynomial equation, 34

Introductio in Analysin Infinitorum (Euler), 165–166

intuition, mathematical

of Euler, 167–168

in geometry, 168n

Platonism and, 158–159

of Ramanujan, 162

irrational number, 41

defining, 189

e and, 30–31, 33

as fraction, 42

infinite and, 31

pi as, 42

Jefferson, Thomas, 9, 11

Jezler, Christoph, 68

Kac, Mark, 58

Kant, Immanuel, 9, 12

Kasner, Edward, 137

Kline, Morris, 92, 169–170

Klyve, Dominic, 202n

Lagrange, Joseph-Louis, 62

Lakoff, George, 152–154, 160–161, 168n

Lambert, Johann, 33

Laplace, Pierre-Simon, 71

Le Lionnais, François, 147

Leibniz, Gottfried Wilhelm

Gregory and, 36–37, 43–44

on imaginary numbers, 54, 106

on infinite sums, 36–37, 43–44

on infinitesimals, 169

Newton and, 28, 64–65

Letters of Euler on Different Subjects in Natural Philosophy Addressed to a German Princess (Euler), 59–60

limbic system, 143–144, 146

limit, 20–21, 170n

Liouville, Joseph, 34–35

Littlewood, J. E., 157

Locke, John, 10

Longinus, 145

Louis XIV (king), 104

The Magic of Math: Solving for x and Figuring Out Why (Benjamin), 203n, 207n

The Mathematical Experience (Hersh), 151

mathematics, 153–156