Index

Abel, Niels, 132, 235

Abu Kamil, 57

al-Karajå, 60

Al-Khwārizmī, 57, 253

algebraic irrationals, 180

Almagest, 65

analytic geometry, 77

Anderhub, J. H., 32

angle

trisection of, 201

Apéry, Roger, 138

aperiodic tiling, 258

apotome, 50

Arago, Dominique François Jean, 97

Archimedean spiral, 276

Archimedes, 49

Aristotle, 13

arithmetic axioms, 249

Arithmetica Integra, 74

Arithmetica infinitorum, 85

Arithmetical Theorem, 135

Aryabhata, 254

Axiom of Eudoxus, 40, 43, 49, 167, 251

Beatty, Samuel, 261

Bell numbers, 293

Berlin Academy, 104

Bernoulli, Jacob, 96

Besicovitch’s Constant, 233

Besicovitch, A. S., 233

Bézout’s Identity, 166

Bhaskara II, 54

Binet formula, 221

binomial, 50

Blake, William, 152

Bolzano, Bernard, 237

Borel, Emil, 230

bride’s chair, 45

Brouncker, Lord, 93

Burkert, Walter, 12

Caliph

al-Ma’mūn, 56

Harun al-Rashid, 56

Omar ibn Khattab, 56

Cantor, Georg, 204, 239

Cantor–Heine–Méray Model, 242

Cartwright, Dame Mary, 113

Catalan constant, 112

catenery, 253

Cauchy

distribution, 252

sequence, 242

Cauchy, Augustin-Louis, 237

Cavalieri Principle, 86

Chaitin, Gregory, 231

Champernowne’s

Constant, 232

Champernowne, David, 232

Chaucer, 66

Cicero, 65

commensurability, 13

commensurable in square, 48

common elements, 134

complementary sequences, 262

complete ordered field, 250

completeness axiom, 250

constructible numbers, 203

continued fractions, 92, 183

Conway Constant, 136

Conway, John Horton, 133

Copeland, A. H., 233

Copeland–Erds Constant, 233

coprime, 252

Cosmological Theorem, 135

countable set, 204

counter-earth, 14

Crelle, August, 236

Davis, Philip J., 277

De Divina Proportione, 74

De Morgan, Augustus, 47

de Stainville, M. J., 109

Dedekind

cut, 241, 244

Model, 244

Dedekind, Richard, 39, 239

Dehn, Max, 269

Delian Problem, 201

dense set, 167

Descartes, René, 77

Diehard test suite, 229

Diophantine approximation, 154

Dirichlet’s Diophantine

Approximation Theorem, 163, 269

Dirichlet’s Simultaneous

Approximation Theorem, 270

Dirichlet, Johann Peter Gustav

Lejeune, 162

Discourse on Method, 80

Dobinski’s formula, 293

Drawer Principle, 162

Dudeney, Henry, 268

dunce’s cap, 64

Duns Scotus, 64

duplication of the cube, 201

dynamical systems, 258

Dyson, Freeman, 184

Einstein’s field equations, 253

equimultiples, 41

equivalence relation, 289, 290

Eratosthenes, 49

Erds, Paul, 112, 233

Euclid, 38

Euclid’s Elements, 10, 11, 21, 39

Eudemian Summary, 10, 14

Eudemus of Rhodes, 14

Eudoxus of Cnidus, 39, 41, 49

Eudoxus’s method of exhaustion, 49, 238

Euler’s Constant, 263

Euler, Leonhard, 96, 182, 274

Euler–Mascheroni constant, 112

exotic elements, 134

factorial function, 274

Fermat’s Last Theorem, 199

Fermat, Pierre de, 77

Fibonacci of Pisa, 60, 62

Fibonacci Sequence, 166

field, 250

Floor function, 113, 146, 157, 261, 266

Flos, 63

Folium of Descartes, 84

Fourier, Joseph, 109

Fraenkel, Aviezri S., 267

Frederick II, 63

Frege, Gottlob, 251

Freiman’s constant, 175

Galois, Évariste, 132

Gamma Function, 274

Gelfond Schneider Theorem, 198

Gelfond, Aleksandr, 184, 199

geometric

curves, 80

mean, 50

Golden Ratio, 28, 74, 174, 220, 236, 255

Gray, Robert, 208

Great Year, 65

greatest common divisors, 166, 213

Hall’s Ray, 176

Hall, Marshall, 176

Hamilton, Sir William Rowan, 237

Hankel, Hermann, 238

Hardy, G. H., 32

harmonic

oscillator, 253

series, 138, 238

height of a polynomial, 205

Hermite Identity, 192

Hermite, Charles, 113, 190, 193

hexagram, 27

Hilbert, David, 11, 169, 198, 248

Hindu civilization, 53

Hippasus of Metapontum, 20

Hobbes, Thomas, 85

Horn Angle, 40

House of Wisdom, 56

Hurwitz, Adolf, 169

incommensurability, 48

indicator function, 257

irrational numbers, 29, 53, 63, 65, 74, 77, 97, 165, 167, 172, 174, 176, 199, 204, 209, 232, 233, 237, 242, 245, 247

addition of, 126

alternative definition of, 3

and rational approximation, 160

and the Littlewood Conjecture, 172

and the quintic equation, 132

and the Zeta function, 153

as non-surd algebraic, 132

birth of, 9

Cantor’s construction of, 209

Cauchy definition of, 237

denseness of, 167

elementary definition of, 2

generation of using polynomials, 131

nineteenth-century definition of, 239

non-existence of, 204

quadratic, 176

Stifel’s view of them, 74

the most irrational, 219

Weierstrass view of, 239

Iwamoto, Y., 119

Johannes of Palermo, 63

Jowett, Benjamin, 29

Kant, Immanuel, 18

Khinchin, Aleksandr, 210

Klein, Felix, 169

Knorr, Wilbur R., 33, 47

Kronecker, Leopold, 204

La Disme, 76

Lady Isobel’s Casket, 268

Lagrange Spectrum, 174

Lambek, Joachim (Jim), 265

Lambert, Johann Heinrich, 104

least common multiple, 145

Lehmer, Dick, 229

Leibniz, Gottfried Wilhelm, 182

Liber Abaci, 62, 74

Library of

Alexandria, 11, 56

Babel, 234

Life of Brian, The, 52

Lindemann, Ferdinand, 194

Lindemann–Weierstrass Theorem, 200

Liouville Approximation Theorem, 183

Liouville Number, 183, 186

Liouville, Joseph, 183

Littlewood, John Edensor, 172

logarithmic spiral, 253, 276

Look and Say Sequence, 133

Louis XIV, 286

Luther, Martin, 75

Madhava of Sangamagramma, 55

Markov

numbers, 175, 223

spectrum, 222

McCabe, Robert L., 33

mean, 20

proportion, 50

Mean Value Theorem, 184, 294

mechanical curves, 80

medial, 50

Méray, Charles, 239

method of normals, 80

Minkowski, Hermann, 169

Moser, Leo, 265

Murray, James, 23

National Institute of Standards and

Technology, 254

Test Suite, 229

Newcomb, Simon, 252

Niven polynomial, 116

Niven, Ivan, 116

normal, 230

distribution, 253

in base b, 230

numbers, 230

number of the beast, 75

Ohm, Martin, 236

Omar Khayyam, 61

Rubaiyat, 61

Online Encyclopedia of Integer Sequences, 209

order axioms, 250

ordered field, 250

Oresme, Nicole, 66, 138, 259

Pacioli, Luca, 74

Patruno, Gregg N., 127

Penrose Tiles, 258

pentagram, 25

Père Lachaise, 153, 286

permeability of free space, 254

Perron, Oskar, 175

Pigeonhole Principle, 162, 226

Plato, 12, 21, 29

Plato’s Dialogues, 29

Poincaré, Henri, 190, 239

Poinsot, Louis, 110

Poisson distribution, 253

Pope Leo X, 75

prime counting function, 147

Prime Number Theorem, 147

Proclus, 10, 15, 38

Profound Doctor, 66

Proteus, 90

Pythagoras

constant, 1

of Samos, 9, 10

Pythagoras’s Theorem, 18, 38, 45

quadratic

irrationals, 176, 186

surd, 212

quadratrix, 81

quadrature, 85

rational

line, 278

point, 278

roots theorem, 130

Rayleigh, Lord, 261

Riccati Equation, 102

Riccati, Count Jacopo Francesco, 102

Riemann Hypothesis, 199

Rolle’s Theorem, 294

Roth, Klaus, 181

Ruffini, Paolo, 132

Schneider, Theodor, 199

schnitt, 241

sexagesimal, 63

Siegel, Carl, 184, 199

Sierpiński, Waclaw, 231

similar figures, 36

simple

continued fractions, 211

pendulum, 252

simply normal in base b, 230

size of a polynomial, 208

Skolem, Thoralf, 268

Smith, Henry, 162

Spiral of Theodorus, 7, 35, 47, 272

squaring the circle, 15, 201

Śrīpati, 53

star polygon, 25

Steinhaus, Hugo, 259

Stevin, Simon, 76

Stifel, Michael, 74

Stupor Mundi, 63

Subtle Doctor, 64

Sum of Two Squares, 80

surd, 125

etymology of, 57

Swineshead, Richard, 66

Tannery, Paul, 248

Thales of Miletus, 9, 10, 12, 16

Theodorus Constant, 277

Thomae, Carl, 251, 257

Thomas Bradwardine, 66

Thue, Axel, 184

transcendental

functions, 81

numbers, 183, 185, 197, 199, 204, 210, 232

transuranic elements, 134

triangle inequality, 156

van der Poorten, Alfred, 139

van Pesch, J. G., 14

van Roomen, Adriaan, 70

Vesica Pisces, 28

Vièta, Francois, 70

Waclaw Sierpiński, 259

Wallis, John, 85, 93

Wantzel, Pierre, 202

Weierstrass, Karl, 200, 236

Weierstrass–Heine Model, 241

well-tempered musical scale, 254

Weyl, Hermann, 199

Wiles, Andrew, 199

Year of the Irrational, 239

zero

etymology of, 62

Zeta function, 137