A

al-Abbas (ibn Abd-al-Muttalib, uncle of Mohammed), 46

Abbasids, 46, 47, 48

Abbott, Edwin A., 145, 146, 340

Abel, Niels Henrik, 116, 117, 127–128, 129, 130, 131, 154, 217, 333, 334

Abraham, 19, 323

Abstraction

category theory, 304

literal symbolism, 2, 3

numbers, 1

Académie des Sciences, 117, 122, 155, 209, 210, 211, 227, 230, 332

Acnode, 260

Action at a distance, 336

Addition

complex numbers, 13

determinants, 171

fields, 197

integers, 282, 348

matrices 173

vectors, 136, 140, 141

Ahmes, 29–30

Airy, George, 182

Akkadian language and writings, 21, 22

d’Alembert, Jean Le Rond, 105

Alexandria, Egypt, 31, 33–34, 42, 44–45, 46, 48

l’Algebra, 82–83, 84

Algebra

as abstraction, 1–2, 231, 293–298

analysis distinguished from, 1

classification of topics, 299, 350

early textbooks, 35, 38, 41, 49, 66, 67, 71, 74–75, 82–83, 87, 88, 90–91, 93, 94

etymology, 43, 46

defined, 1

father of, 31, 41–42

first lecture, 72

future of, 319–320

geometric approach to, 32, 33, 88–89

Algebraic closure, 108, 196, 212

Algebraic geometry, 340.

See also Rings and ring theory

applications, 279

axiomatic approach, 231, 238, 293–298, 301, 305–306

conic equations, 241–248, 255, 260

determinants, 245, 250

eccentricity of ellipse, 244–245

Erlangen program, 275–276, 293, 347

fitting curves to points, 170–171

homogeneous coordinates, 153–154, 248, 251, 255, 259, 260, 345

hyperbolic, 256, 273

invariants, 243–244, 245, 250, 260–261

line at infinity, 247, 248

line geometry, 251–252, 257, 270, 292

matrices, 245, 250

notation, 242, 245–246, 344

Nullstellensatz (Zero Points Theorem), 262–264, 295, 345, 346

points at infinity, 247, 248, 344

projective, 247–248, 255, 256, 270, 273, 293

symmetry in, 248, 250, 251

three-dimensional, 247, 248

transformations, 250–251, 271–273

variety concept, 263–264

Algebraic number theory, 41, 262, 297

Birch and Swinnerton-Dyer conjecture, 289

defined, 288

p-adic numbers, 289–292, 306, 349

Algebraic numbers, 288

Algebraic Surfaces, 296

Algebraic topology

applications, 315

cat’s anus theorem, 348

Brouwer’s fixed-point theorem, 285–286, 348

dimensionality, 283, 284, 285

function theory, 267–268, 281, 284, 305, 344

fundamental groups, 282–283, 284, 302, 348

homeomorphism, 280, 347–348

hyperspheres, 283, 348

invariants, 284, 285

Jordan loops, 281–282

Lefschetz’s fixed-point theorem, 295

loop families, 282–283

mapping a space into itself, 285–286, 305

Möbius (Listing) strip, 250, 280

motivitic cohomology, 2, 350

Poincaré conjecture, 283, 349

point-set (analytical) approach, 285

projective plane, 249–250

of Riemann surfaces, 281, 348

of spheres, 281, 282

of toruses, 281–282

transformations, 250

Algebras

for bracketed triplets, 150–151

classification, 160

Clifford, 156

of complex numbers, 143–144, 149–150, 160, 173

defined, 143

division algebra, 196

Grassmann, 142, 154–155

Lie, 347

matrices, 173

n-dimensional, 156

noncommutative, 152, 159, 160, 335

octonions as, 152–153, 335

quaternions as, 151, 152

real numbers as, 160

zero vector factorization, 159–160

Algorithms, 28, 48–49, 319

Ali (ibn Abi Talib, son-in-law of Mohammed and fourth Caliph), 46, 325–326

American Mathematical Society, 298, 299

Analysis, 1, 104, 129, 267, 284, 285, 293, 305

Analysis situs, 283, 285

Analytical geometry, 91–92, 97, 153, 255

physical models, 253–254, 258–259

plane curves, 257–258

Analytical Society, 180

Anderson, Alexander, 89, 91

Appollonius, 344

Archimedes, 31, 33, 54–55

Areal coordinates, 345

Aristotle, 183, 307

Arithmetica, 35, 38, 41, 49, 224

Arithmetica Universalis, 99

Ars magna, 74–75, 78–79, 82

Artin, Emil, 301

Artin, Michael, 307, 310, 331, 338, 340, 346

Associative rule, 144, 212

Astrology, 73, 74, 329

Astronomical Society of London, 181

Astronomy, 23, 51, 164, 337

Atiyah, Michael, 310, 319

Augusta Victoria, Empress of Germany, 235

Ausdehnunsglehre, 154–156, 159

d’Avalos, Alfonso, 78

Ayres, Jr., Frank, 338

B

Babbage, Charles, 180

Babylonians

astronomy, 23

first empire, 21–22, 23

mathematics, 23, 25–29, 31, 39, 51, 55, 162

number system, 24–25, 28

problem texts, 25–27

second empire, 323

Baghdad, 46, 47, 48, 51, 53

Balzac, Honoré de, 207

Barycentric Calculus, 153

Barycentric coordinates, 345

Basis, in vector space, 140

Bell, E. T., 126, 206, 208, 209, 228, 333

Beltrami, Eugenio, 277, 346

Berlin Academy, 228

Bernoulli, Jacob, 169

Bernoulli, Johann, 98

Bernoulli numbers, 169, 337

Betti, Enrico, 277

Binomial theorem, 66, 328

Birch and Swinnerton-Dyer conjecture, 289

Birkhoff, Garrett, 299–301, 304

Birkhoff, George, 301, 350

Bishop, Errett, 287

Bismarck, Otto von, 235

Bolyai, János, 153, 256, 273

Bombelli, Rafael, 82–85, 89

Boole, George, 133, 181, 184–187, 338–339

Borrow, George, 334–335

Botta, Paul Émile, 22

Brahma-Gupta, 47

Bring, Erland, 333

Brioschi, Francesco, 277

British Association, 130

British Association for the Advancement of Science, 181

British mathematical culture, 97–99, 103–105, 130, 148, 149–153, 157–158, 174–191

Brougham, Henry, 339

Brouwer, L. E. J., 285, 287–288, 290

Brouwer’s fixed-point theorem, 285–286, 348

Bryn Mawr College, 239

A Budget of Paradoxes, 183

Burger, Dionys, 146

Byzantine empire, 45, 48, 53, 68

C

See Complex numbers.

Calabi, Eugenio, 317, 318

Calabi conjecture, 318

Calculus, 99, 104, 116, 159, 168, 169, 179–181, 185, 187, 254, 255, 257, 284

Caltech, 316

Cambridge Mathematical Journal, 185

Cambridge University, 99, 261, 333

Campbell, Thomas, 339

Cantor, Georg, 287

Cardano, Giambatista, 74

Cardano, Girolamo, 71, 72–75, 77–80, 83, 168

Carnap, Rudolf, 308

Cartan, Henri, 312

Cartesian coordinate system, 91, 158, 262

Castelnuovo, Guido, 277, 293, 295–296

Category theory, 304–308, 312, 315, 318, 351

Catherine de’ Medici, 85–86

Catholics, 85–86, 126

Cat’s anus theorem, 348

Cauchy, Augustin-Louis, 125–126, 127, 129, 130, 131, 132, 152, 155, 172, 174, 208, 209, 227, 228, 231, 293, 299, 332, 333

Cayley, Arthur, 153, 174–175, 181, 185, 187, 191, 213, 214, 217, 260, 261, 270, 317, 335

Cayley numbers, 335

Cayley tables, 189–191, 212, 213

Cayleyan, 260, 345

Cech, Eduard, 351

Characteristic function, 149

Charles I, King of Spain, 329

Charles V, Holy Roman Emperor, 74, 78, 85, 329

Charles IX, King of France, 85–86

Charles X, King of France, 207, 209

Chevalier, Auguste, 206, 211

China

Early Han dynasty, 162–163, 336

language, 337

mathematical culture, 47, 48, 162–164

Nestorians, 325

Qin dynasty, 162–163

Christianity, 43–44, 45, 69, 146, 325

Christina, Queen of Sweden, 94

Christ’s College, Cambridge, 180

Chuquet, Nicolas, 71–72

Cissoid, 260, 345

Class field theory, 299

Clay Mathematics Institute, 283, 349

Clifford, William Kingdon, 156

Clifford algebras, 156

Clock arithmetic, 289–290

Code breaking, 86–87, 315

Colbert, Jean-Baptiste, 332

Collège de France, 309, 313

Combinatorial math, 284, 285

Commutativity, 343

Abelian groups, 214, 216, 341

complex numbers, 152, 190, 198

in fields, 196

rings, 238, 262, 306

rule, 132, 144

Completed infinities, 287

Complex numbers ()

algebraic closure, 108

as an algebra, 143–144, 149–150, 160, 173

“big,” 107

commutativity, 152, 198

coordinates, 293, 346

cube roots of, 61–62

in cubic equations, 58, 61, 75, 77, 79

discovery and acceptance, 75, 79, 81, 82, 83, 84, 89, 94, 104, 110, 116, 148, 179

Gaussian integers, 226

matrix representation of, 173

modulus, 13, 107

multiplying, 13, 84, 149–151, 173

as points, 148

properties, 11–13, 108, 143, 196

quadruplets, 151

real numbers as, 106

on Riemann surfaces, 266

rings, 226, 233, 260, 264

roots of unity, 109–114

triplets, 150–151

as vector space, 143–144

Complex variable theory, 251

Complex-number line, 346

Complex-number plane, 13, 106, 265, 346

cyclotomic points, 110, 111, 331

Conchoid, 260, 345

Confucianism, 163

Confucius, 337

Conics, 241–248, 255, 260

Constant term, 36, 49, 107

Constantine, 43

Constructivism, 287

Continuous groups, 275–276

Conway, John, 32

Coolidge, Julian Lowell, 278, 347

Cosets, 216

Cossists and Cossick art, 92–93, 329

Counter-Reformation, 74

Courant, Richard, 23

Courant Institute, 287

Coxeter, H. S. M., 336, 339–340, 351

Crafoord Prize, 312, 352

Cramer, Gabriel, 170

Cramer’s rule, 170, 337

Crelle, August, 128–129, 341

Cremona, Luigi, 277

Crunode, 260

Crusades, 53, 56, 326

Cubic equations

algebraic solutions, 57–62, 71, 72, 75, 76–77, 78, 79–80, 81, 105, 109, 111, 118–121, 123, 130

classification, 55

complex numbers in, 58, 61, 75, 77, 79

del Ferro’s solution, 75, 80

depressed (reduced), 58, 118–119, 123

discriminants of, 62, 77

Fibonacci’s analysis and solution, 69–70

Fiore–Tartaglia duel, 76–77, 80, 329

Greek solutions, 33, 40, 54–55

irreducible case, 61–62, 79, 327

Khayyam’s, 54–55, 326

permutation of general solutions, 118–120, 121, 123

proof of general solution, 62–63

resolvent, 123–124, 126

roots of unity, 61, 109, 110, 111, 113, 119, 190, 215

symmetry of coefficients and solutions, 90, 120, 121, 202–203

trigonometric solution, 89

types (Pacioli’s classification), 75–78

Cubic polynomials, 81, 84

Cuneiform

mathematical texts, 23, 24–26, 32

modern problem texts compared, 26–27

Plimpton 322 tablet, 24

reading, 13

writing, 21–23

Cusps, 260, 345

Cyclic group of order n, 219

Cyclotomic integers, 112, 228–229

Cyclotomic points, 110, 111, 331

Cyril of Alexandria, 45

Cyrus the Great, 22

D

da Coi, Zuanne de Tonini, 76

da Vinci, Leonardo, 328–329, 343

Dardi of Pisa, 71

Dark Ages, 56

Darnley, Lord, 73

data (“things given”), 14, 88

De Moivre’s theorem, 104

De Morgan, Augustus, 175, 177–178, 180, 181, 182–183, 185, 294, 338, 339

Decimal point, 97

Dedekind, Richard, 230, 231, 233, 235, 268, 340, 341

del Ferro, Scipione, 75–76, 82

Deligne, Pierre, 310

Descartes, René, 81, 91–94, 97, 106, 168, 178, 242, 255, 287, 338

Determinants, 188

adding, 171

in algebraic geometry, 245, 250

defined, 167, 173

discovery, 168

interactions, 171

of matrices, 167, 172–173, 174, 245

multiplying, 171–172

in solving simultaneous linear equations, 168–169

theory of, 142, 172

Vandermonde, 117–118

Dewdney, A. K., 146, 334

Differential equations, 105, 185, 276, 285, 339

Differential geometry, 159

Dihedral groups, 219–220, 221, 271–272

Dilatations, 347

Dimensionality

fiction, 145–147, 148, 334

fields, 199

multidimensional geometry, 185, 247, 248

n-dimensional algebras, 156

as topological invariant, 283, 284, 285

of vector space, 138–140, 144, 152–156, 158–159, 199, 335

Diocletian, 43

Diophantine analysis, 31, 39, 54

Diophantus, 31, 34–46, 39–41, 43, 49, 51, 56, 70, 83, 88, 93, 162, 224, 289, 314, 324, 328

Dirichlet, Lejeune, 224–225, 230

Disney, Catherine, 149, 335

Disney, Walt, 335

Disquisitiones Arithmeticae, 112, 131, 172

Distributive law, 150

Divine Right of Kings, 333

Division

complex numbers, 13

rule of signs, 9

vectors, 144

Division algebra, 196

Duality concept, 351

Dumotel, Stéphanie, 210, 211

E

École Polytechnique, 208, 255

École Preparatoire (Normale), 209–210

Edict of Milan, 43

Edict of Nantes, 85

Edward VI, King of England, 73

Edwards, Harold, 287

Effi Briest, 235

Egyptians, ancient, 19–20, 23

mathematics, 29–30, 32

Eilenberg, Samuel, 302, 304, 307

Einstein, Albert, 236–237, 239–240, 268, 301, 343, 352

Eisenstein, Ferdinand Gotthold Max, 230

Electrical Age, 157

Electromagnetic field concept, 157

The Elements, 32–33, 76, 163

Ellipses, 241, 242–245, 289, 344

Enriques, Federigo, 277, 293

Epitrochoid, 260, 345

Equations.

See also Polynomial equations;

specific types of polynomial equations

defined, 37

theory of, 103–104

Erlangen program, 275–276, 293, 347

Erlangen University, 234, 236, 261, 275

Euclid, 32–33, 76, 163, 322, 332, 343

Euler, Leonhard, 104–105, 106, 110, 113, 117, 122, 129, 130, 224, 226, 332, 336

Euler’s number, 244

Everest, George, 185

Extension fields, 198–199, 201–202, 204–205, 340–341, 351

F

Factorials, 124, 165

Factorization

of cyclotomic integers, 229, 292

of integers, 234

non-unique, 299

polynomial equations, 11, 41, 111

unique, 227, 228, 229

zero vector, 159–160

Faraday, Michael, 157

Feit, Walter, 298

Fermat, Pierre de, 38, 224

Fermat’s Last Theorem, 37–38, 40, 224–225, 227, 228, 297

Ferrari, Lodovico, 79–80, 116

Fertile Crescent, 19–20

Feuerbach, Karl, 255

Fibonacci, Leonardo, 65–71

Fibonacci sequence, 65–67, 327–328

Fields and field theory, 133, 332

addition table, 197

axioms, 196, 223, 340

class, 299

closure rule, 196

commutativity, 196

dimensionality, 199

examples, 196

extension fields, 198–199, 201–202, 204–205, 340–341, 351

finite (Galois) fields, 196–198, 199, 203, 297

fraction field of rational numbers, 291

function fields, 2, 204–205, 231

limits, 291, 349

multiplication tables, 202, 203

p-adic numbers, 291

permutations of solution fields, 202–203

properties, 195–196, 291

rational-function field, 204–205, 231

rules, 196

solving equations with, 198–202, 340–341

as vector space, 199

Fields Medal, 309–310, 312, 352

Finite (Galois) fields, 196–198, 199, 203, 297

Finite groups, 219, 221–222, 271, 272, 275

Fiore, Antonio Maria, 75–77

Fitzgerald, Edward, 52

Flat plane, 13

Flatland, 145–147, 153, 340

Flatterland, 146

Flos, 69, 328

Fontane, Theodor, 235–236

Forgetful functor, 306

Foundations of Geometry, 294, 296

Foundations of mathematics, 181, 187

Fourier, Jean Baptiste Joseph, 105, 208, 209

Fractions, 9

Babylonian, 24–25

France, mathematical culture in, 81–82, 85–94, 117–118, 122–125, 178, 180, 208–209, 210, 231

Francis I, King of France, 85

Frank Nelson Cole Prize, 2, 298

Frederick II, Holy Roman Emperor, 65, 68–69

Frederick the Great, 104–105, 117, 122

Frege, Gottlob, 187

French Institute, 125, 172, 174

French Revolution, 119, 122, 125, 256

Frend, William, 179–180

Frost, Percival, 259

Function fields, 2, 204–205, 231

Functions, 1, 129

characteristic, 149

in differential equations, 276

theory of, 267–268, 281, 284, 305, 344

Functors, 301, 306, 307–308, 351

Fundamental theorem of algebra (FTA), 105–106, 264

proof of, 106–108

G

Galileo, 91

Galois, Évariste, 131, 154, 191, 195, 198, 202, 203, 206–207, 208–211, 218, 276, 288, 340

Galois theory, 89, 209, 212, 214–217, 276, 305

Gandy, Robin, 307

Gauss, Carl Friedrich, 36, 106, 112–113, 117, 126, 128, 131, 148, 159, 164, 172, 211, 224, 230, 231, 256, 258–259, 280, 292, 331, 332, 335, 337

Gaussian elimination, 163–164, 337

Gaussian integers, 226–227

Gaussian ring, 226–227

Gell-Mann, Murray, 316–317, 352

General theory of algebraic structures, 305–306

General theory of relativity, 159, 236–237, 268, 316

Geometry, Euclidean.

See also Algebraic geometry;

Analytical geometry;

Non-Euclidean geometry

ancient Greek approach to algebra, 32, 33, 88–89

Kant’s philosophy, 256–257, 344–345

nth roots of unity, 109, 113

of polytopes, 158–159

German, Sophie, 224

Germany

Cossists and Cossick art, 92–93, 329

mathematical culture in, 92–93, 129–130, 153–155, 179, 230–231, 235–239, 260–261, 301, 329

Nazi period, 235, 239, 292–293, 301

women’s status in, 235–236, 237

Ghaznavid dynasty, 53

Gibbon, Edward, 43, 44, 45, 324, 325

Gibbs, Josiah Willard, 157, 315

Girard, Albert, 90–91

Gödel, Kurt, 352

Goethe, Johann Wolfgang von, 179

Gordan, Paul, 261

Gordan’s problem, 261

Grassmann, Hermann Günther, 154–156, 157, 159, 316, 317

Grassmann algebras, 142, 154–155

Graves, John, 152–153

Great Quaternionic War, 158

Greek (ancient) mathematics, 10, 32–41, 56, 70, 324

Gregory, Duncan, 177, 185, 294, 338–339

Grotefend, Georg, 323

Grothendieck, Alexander, 299, 304, 308–314, 352

Groups and group theory, 3, 268

alternating group of index 2, 219

applications, 289, 293, 315

associativity, 212

axioms, 212–213, 223

Cayley tables, 189–191, 212, 213

closure, 212

commutative (Abelian), 214, 216, 341

continuous groups, 275–276

cyclic group of order n, 219

dihedral groups, 219–220, 221, 271–272

Erlangen program, 275–276

extensions, 302

finite groups, 219, 221–222, 271, 272, 275

founders, 132, 181, 187, 191, 270, 271

fundamental, 282–283, 284, 302, 348

Galois theory, 89, 209, 212, 214–217, 276

generators, 220–221

homology groups, 302–304

homotopy groups, 302, 304, 351

index of the subgroup, 215

infinite groups, 271–273, 348

invariants, 273–274

inverse elements, 212

isometries of Euclidean plane, 271–273

Jordan’s Treatise, 270, 271

Klein 4-group, 214

Lagrange’s theorem, 124, 125, 218

left and right cosets, 216

Lie group, 284, 297, 316–317, 347

normal subgroup, 215–216, 219, 221, 223, 305

order, 213–214, 215

permutations and, 132, 190–191, 202–203, 212, 213, 215, 216, 217–218, 221, 270

pth roots of unity, 213, 221, 228–229

quaternion group, 158, 221

simple groups, 221–222

simplexes, 302–303, 304

structure, 214–216, 218, 299

subgroups, 214–215, 218–219

Sylow p-subgroup, 218–219

symmetric group of order n!, 219

taxonomy, 219–222

in topology, 282–283, 284, 348

transformations, 219–220, 270, 275–276, 347

trivial, 283

unity in, 212

Grunwald, Eric, 319–320

Gupta dynasty, 47

Gustavus Adolphus, King of Sweden, 91, 94

H

Hadrons, 316–317, 352

Hamburg University, 301

Hamilton, John, 73

Hamilton, Sir William, 184

Hamilton, Sir William Rowan, 144, 148–152, 154, 155, 156, 157, 158, 174, 175–176, 184, 287, 315, 316, 317, 319, 334

Hamiltonian operator, 149

Hammurabi, 21, 22, 23, 323

Harriot, John, 93–94, 97, 330

Harvard University, 297, 301, 350

Hasse, Helmut, 292

Heath, Thomas, 33

Heaviside, Oliver, 157, 158, 315

Heemskerck, Martin, 42

Heisenberg, Werner, 316

Heisenberg’s uncertainty principle, 334

Helmholtz, Herman Ludwig von, 230

Henry III, King of France, 86

Henry IV, King of France, 86, 87, 330

Hensel, Kurt, 289, 292, 297

Heptadecagon, ruler-and-compass construction, 112

Heraclius, 46, 325

Hermite, Charles, 231

Herschel, John, 180

Hessian, 260, 345

Hilbert, David, 234, 236–238, 261, 262, 264, 276, 277, 286, 289, 294, 295, 296, 297, 305, 315, 316, 317, 343

Hilbert’s Basis Theorem, 295, 345

Hindus, 47

Hinton, Charles Howard, 340

Hinton, James, 340

Hirst, Thomas, 175

Hitler, Adolph, 235

Holmboë, Bernt Michael, 128

Homeomorphism, 280, 347–348

Homogeneity, law of, 88–89, 91–92

Homogeneous coordinates, 153–154, 248, 251, 255, 259, 260, 345

Homology theory, 302–303, 305, 309

Hopf, Heinz, 348

L’Hôpital, Marquis de, 168

Housman, A. E., 52

Hugo, Victor, 206

Huguenots, 85–86, 330

l’Huilier, Simon, 284

Hurewicz, Witold, 351

Huxley, Aldous, 346

Hyksos dynasty, 29

Hypatia, 44–45, 224, 324–325

Hyperbolas, 241, 256, 273, 344

Hypercomplex numbers, 160

Hyper-loops, 302

Hyper-mappings, 306

Hyperspheres, 283, 348

I

ibn Abd-al-Muttalib, uncle of Mohammed, 46

ibn al-’As, Amr, 46

ibn Qurra, Thabit, 51

Ideae mathematicae, 87

Ideal, 228, 231, 232–233, 238, 262–263, 264, 268, 305

Ideal factor, 228, 229, 268

Identity permutation, 100, 119

Imaginary numbers, 12, 40, 79

Indeterminate equations, 36–38, 39

Indians (Asian)

number system, 8, 47, 48, 68

Industrial Revolution, 104, 181

Infinite groups, 271–273, 348

Institut des Hautes Études Scientifiques (IHÉS), 308, 309, 310, 313

Institute for Advanced Study, 308

Integers ()

addition, 282, 348

cyclotomic, 112, 228–229

division, 226

factorization, 234 5-adic, 289–290

in fundamental group of a torus, 282–283, 348

Gaussian, 226–227

ideals in, 232, 343

Online Encyclopedia of Integer Sequences, 66

polynomials and, 225–226, 342

properties, 8–9, 10, 225

Integral domain, 306

International Congress of Mathematicians, 312

Intuitionism, 286–287, 348–349

Invariants, 181, 237, 243–244, 245, 250, 260–261, 273–274, 284, 285, 295, 305

Inverse functions, 265, 267

Iran, 22.

See also Persians

Irrational numbers, 32, 198, 292

Irreducible equations, 201, 327

Isagoge, 88

Islam, 53.

See also Muslims

Isometries of Euclidean plane, 271–273

Italy, mathematical culture in, 65–80, 82–85, 277–278

J

Jacobean variety, 2

James II, King of England, 331

Japan, mathematical culture, 168, 169–170

al-Jayyani, Mohammed, 51

Jefferson, Thomas, 342, 347

Jerrard, G. B., 130

Jews, 23–24, 69, 238, 292, 323, 324

Johannes of Palermo, 69

Johns Hopkins University, 296

Johnson, Art, 93

Jordan, Marie Ennemond Camille, 231, 270, 271, 273, 281

Jordan loops, 281–282

Journal of Pure and Applied Mathematics, 129, 341

K

Kähler, Erich, 318

Kant, Immanuel, 153, 256–257, 286, 287, 307, 344–345, 348–349

Kelvin, Lord (William Thomson), 158, 185

Kepler, Johannes, 344

Kepler’s laws, 171

Keratoid cusp, 260

Khayyam, Omar, 52, 53–54, 54–55, 56, 67, 69

al-Khwarizmi, 31, 46, 48–51, 52, 54, 67, 325

Kings College, Cambridge, 259

Kingsley, Charles, 324–325

Kizlik, S. B., 321

Klein, Felix, 147, 236–237, 257, 261–262, 271, 273, 275, 276, 277, 293, 294–295, 347

Klein 4-group, 214

Kneebone, G. T., 348

Kobori, Akira, 169

Kronecker, Leopold, 230, 287, 289, 331, 343

Kummer, Ernst Eduard, 228–230, 233, 258–259, 292

L

La géométrie, 91, 93, 94, 106

Lagrange, Joseph-Louis, 121–125, 127, 129, 130, 208, 332

Lagrange’s theorem, 124, 125, 215

Lamé, Gabriel, 225, 227, 228, 299

Landau, Edmund, 23, 238

Lang, Serge, 298

Laplace, Pierre Simon, 208

Laplace’s equation, 105

Lasker, Emanuel, 234–235

Lasker–Noether theorem, 235

Laws of Thought, 186

Lawvere, F. William, 307

Layard, Austen Henry, 22

Leaning Tower of Pisa, 328

Lectures on Quaternions, 155

Lefschetz, Solomon, 283, 295, 296

Lefschetz’s fixed-point theorem, 295

Legendre, Adrien-Marie, 224–225

Leibniz, Gottfried Wilhelm von, 3, 94, 104, 106, 168, 179, 352

Lemniscate, 260, 345

Leonardo of Pisa. See Fibonacci

Leray, Jean, 304, 309

Levi-Civita, Tullio, 277

Liber abbaci, 66, 67, 328

Lie, Sophus, 269–271, 273, 275, 276, 284, 297, 346, 350

Lie group, 284, 297, 316–317, 347

Limaçon, 260, 345

Lindemann, Ferdinand von, 288

Line at infinity, 247, 248

Line geometry, 251–252, 257, 270, 297

Linear dependence, 136–137, 336

Linear equations, finite fields applied to, 199–200

Linear extensions, theory of, 153–154

Linear functional, 142

Linear independence, 82, 136–139, 140, 141

Linear transformations, 142, 172

Lineland, 146

“Lines of force” concept, 157, 336

Liouville, Joseph, 211, 217, 218, 227, 228, 231

Lisker, Roy, 313–314

Listing, Johann Benedict, 280

Literal symbolism, 116.

See also Notation systems

and abstraction, 3

adoption, 3, 56, 88, 169

ancient Greek, 35–36, 38, 41–42, 56, 88, 93, 324, 328

Chinese, 169

Descartes’ contribution, 93, 97, 104, 132

invention, 81, 314

in logic, 133

Viète’s contribution, 88, 93, 104, 132

x, 5, 36, 93

zero, 38

Liu Bang, 162

Lobachevsky, Nikolai, 153, 255–256, 273

Logarithms, 97

Logic, algebraization of, 183–187

Longfellow, William Wadsworth, 335

Loop families, 282–283

Loop quantum gravity, 317

Lorentz group, 352

Lorentz transformation, 236, 316

Louis Philippe, King of France, 207–208, 209, 210

Luoxia Hong, 337

M

Mac Lane, Saunders, 299–301, 302, 304, 307, 350

Malik Shah, 53, 326

al-Mamun, 48

Manifolds, 3, 268, 269, 276, 293, 297, 302, 303–304, 317–318, 348, 352

Manzikert, Battle of, 53, 56

Maple, 254

Marinus, 324

Mary Queen of Scots, 73

Mathematica, 254

Mathematical Institute at Göttingen, 24, 292

Mathematical objects, 3, 167, 173, 321.

See also individual objects

Mathenauts, 334

Matrices, 3, 133, 299, 316

addition and subtraction, 173

algebraic geometry and, 245, 250

algebras, 173

ancient Chinese origins, 162

applications, 315

complex numbers represented by, 173

for conic equation, 245

defined, 174

determinants, 167, 172–173, 174, 245

discovery, 167, 181, 187

factorials, 165

Gaussian elimination, 163–164, 337

multiplication, 160, 173, 174, 337–338

permutations, 166–167, 174

product array, 171–172, 337–338

quaternions represented by, 158, 174

signs of terms, 165–166

Maxwell, James Clerk, 157

Mazur, Barry, 310, 350

McColl, Hugh, 187

Mesopotamia, 20–21.

See also Babylonians

cuneiform mathematical texts, 23, 24–26, 32

history, 19, 20–21, 32, 52, 322–323

Metoposcopy, 73

Metric system of weights and measures, 122

Michael VII, Byzantine emperor, 328

Michel, Louis, 310

Michigan Papyrus, 36, 620

Middle Ages, 52, 325

Mill, James, 339

Millennium Prize Problems, 283, 349

Mitchell, Charles William, 325

Möbius, August, 153–154, 155, 255, 280, 345

Möbius strip, 250, 280

Module, 233, 304

Moduli, law of, 150

Modulus

of bracketed triplet, 150

complex numbers, 13, 107

Mohammed, 46

Monasticism, 43

Monophysitism, 46, 325

Mori, Sigeyosi, 169

Morphisms, 306, 307

Motchane, Léon, 308, 309

Motivitic cohomology, 2, 350

M-theory, 317

al-Mulk, Nizam, 53–54

Multidimensional geometry, 185, 247, 248

Multiplication

associative rule, 153

complex numbers, 13, 84, 149–151, 173

determinants, 171–172

fields, 202, 203

matrices, 160, 173, 174, 337–338

quadruplets, 151

rule of signs, 8–9

triplets, 150–151

vectors, 135, 141, 143–144

Murray, Charles, 332

Muslims. See also Islam

Assassins, 326

Imams, 326

Ismailites, 52, 53, 326

medieval mathematics, 46, 48, 49–51, 54, 56, 67, 70, 314

Shias, 46, 52, 53, 67, 325–326

Sunnis, 46, 52, 53

Muwahid (Almohad) dynasty, 67

N

See Natural numbers.

Nahin, Paul, 158

Nakajima, Hiraku, 298

Napier, John, 97

Napoleon, 105, 180, 206

Napoleonic Wars, 182, 230, 255–256, 339

Nash, John, 351

Natural equivalences, 304

Natural numbers ()

properties, 7–8, 9–10, 322

Neal, John, 335

Nebuchadnezzar, 323

Negative numbers, 39, 54, 72, 79, 87

discovery and acceptance, 8, 27, 40, 83, 89, 94, 179–180

properties, 11, 58, 226

whole, 37

Neoplatonism, 44–45, 324

Nestorian heresy, 325

Neugebauer, Otto, 23–24, 27, 28, 30, 292

New Discoveries in Algebra, 90–91

Newman, James R., 30

Newton, Sir Isaac, 3, 91, 94, 97–99, 104, 105, 120, 168, 178, 179, 180, 181, 330–331, 332, 333, 338, 352

Newton’s theorem, 99, 102, 103

Niebuhr, Carsten, 323

Nietzsche, Friedrich Wilhelm, 286

Nightingale, Florence, 175

Nine Chapters on the Art of Calculation, 162, 163, 169

Nodes, 260, 345

Noether, Emmy, 23, 234, 235–240, 261, 296, 297, 301, 316, 317, 351

Noether, Fritz, 237, 239

Noether, Max, 235, 261

Noncommutativity, 153, 157, 174, 191, 343, 338

algebras, 152, 159, 160, 335

quaternions, 152–153, 157, 174

Non-Euclidean geometry, 153, 255–256, 293

Normal subgroup, 215–216, 219, 221, 223, 305

Notation systems. See also Literal symbolism

algebraic geometry, 242, 245–246, 344

brackets, 84–85

calculus dots and d’s, 179, 180, 338

cycle notation for permutations, 188–191

decimal point, 97

Déscartes’, 92–93, 97

Diophantine, 35–36, 38, 41, 70, 93, 324

equals sign, 93, 329

Euclidean approach, 32–33

exponentiation, 93

hollow letters, 7, 148

in logic, 183–184, 185–186

multiplication sign, 97

Muslim algebraists, 70, 71

Pacioli’s, 71

for permutations, 188–191

plus and minus signs, 72, 92, 329

in polynomials, 14–15

for powers, 71–72

square root sign, 93

superscripts, 71–72, 93

Nullstellensatz (Zero Points Theorem), 262–264, 295, 345, 346

Number theory, 31, 51, 109, 114, 226, 289.

See also Algebraic number theory

Numbers and number systems. See also individual families

abstraction, 2–3

Arabic (Hindu) numerals, 47, 68, 70

Babylonian, 24–25, 28

closed under division, 9

closed under subtraction, 8

countable, 7, 10–11, 322

decimal system, 47, 48, 342

dense, 9, 10, 108

geometric representations, 92

million, 329

mnemonic, 7

nested Russian dolls model, 7–13

rule of signs, 8–9

sexagesimal system, 24, 70

O

Octonions, 152–153, 335

Odoacer the Barbarian, 325

Omar (second Caliph), 46

Omayyad dynasty, 46

On Conoids and Spheroids, 33

Online Encyclopedia of Integer Sequences, 66

Operator, 149, 179

Orestes, Prefect of Egypt, 45

Origin

in complex plane, 13

in vector space, 134

Othman (third Caliph), 46

Oughtred, William, 97

P

Pacioli, Luca, 71, 75, 328–329

p-Adic numbers, 289–292, 306, 349

Palestine, 20, 29

Pallas (asteroid), 164, 337

Parabola, 241, 244, 344

Pasch, Moritz, 293–294

Pazzi, Antonio Maria, 83

Peacock, George, 177, 180–181, 182, 294, 302

Peano, Giuseppe, 187, 294

Peirce, Benjamin, 160, 187

Peirce, Charles Sanders, 187

Pentatope, 351

Perelman, Grigory, 283

Permutations

Cayley tables, 189–190, 212, 213

compounding, 131–132, 152, 188–190, 212

cubic equations, 118–120, 121, 123

cycle notation, 188–191

even and odd, 166–167

and group theory, 132, 190–191, 202–203, 212, 213, 215, 216, 217–218, 221, 270

identity, 100, 119, 188, 203, 341

matrix, 166–167, 174

quadratic equations, 118

of solution fields, 202–203

solving equations using, 118–121, 122–124, 130, 203, 217–218

structure of groups, 218

Persians, 45, 46, 47, 48, 52, 53, 325

Pesic, Peter, 333, 334

Petrie, John Flinders, 336

Petsinis, Tom, 206

Phillip II of Spain, 85, 86–87

Philosophical Society of Cambridge, 181

Physics, applications of algebra to, 299

Plane curves, 257

The Planiverse, 146

Plato, 33, 324

Platonic solids, 351

Plotinus, 324

Plücker, Julius, 251, 255, 257, 259, 270, 277, 297

Poincaré, Henri, 281–282, 283, 284, 285, 289, 302

Poincaré conjecture, 283, 349

Pointland, 146

Points at infinity, 247, 248, 344

Poisson, Siméon-Denis, 210

Poitiers University, 91

Polyhedra, 158, 253

Polynomial equations

classification, 50, 71

completion and reduction, 50–51

expansion and factorization, 11, 41, 111

FTA and, 105–106

nth-degree, 117

rational-number solutions, 41, 289

symmetries of solutions, 89–90, 103–104, 118, 119–120

Polynomials

asymmetric, 100, 101, 120

coefficients (givens), 14-15

composition of quadratic forms, 131

cubic, 81, 84

defined, 14

Diophantine analysis, 39

elementary symmetric, 89–90, 101–103

etymology, 13

graphs of, 59–60, 81

importance in algebra, 15

integers and, 225, 342

invariants of, 260–261, 305

“irreducible,” 81–82, 84

literal symbolism, 14–15

partially symmetric, 120

powers of unknowns, 14

properties, 225

quadratic, 59–60, 131

recipe for, 15

symmetric, 99–105, 124

unknowns (data), 14

vector space, 140–141, 142, 144

Polytopes, 158–159

Poncelet, Jean-Victor, 254–255, 256, 277, 309

Posets, 301

Positive numbers, 11, 37, 39

Powers of unknowns, 14, 36

Primes, 112–113, 213

Fermat’s Last Theorem, 225, 226, 228, 230

finite fields for, 197

Grothendieck’s, 310–311

irregular, 342

p-adic numbers, 289–292, 306, 349

powers of, 234

primitive root of, 114, 331–332

regular, 228, 230, 342

Princeton University, 295, 296, 297

Projective geometry, 247–248, 255, 256, 270, 273, 293

Projective plane, 249–250

Psellus, Michael, 328

Ptolemy, 33

Pythagoras, 32, 199

Pythagoras’s theorem, 13, 24, 322, 327

Pythagorean triples, 24

Q

See Rational numbers.

Quadratic equations

algebraic solution, 57, 58, 60–61, 64, 79, 118, 120, 323

Babylonian, 27, 51, 55

and conics, 242, 244

Diophantus’s solution, 38–39

Euclidean Propositions 28 and 29, 32–33

field theory applied to, 199–201, 340–341

irreducible, 201, 327, 340–341

al-Khwarizmi’s solutions, 49

symmetry of solutions, 89–90, 118

Quadratic polynomials, zero set of, 242

Quadruplets, 151

quaesita (“things sought”), 14, 88

Quantum theory, 149

Quartic equations, 40

algebraic solution, 62–63, 75, 80, 81, 104, 105, 115, 116

resolvent, 123, 126

symmetry of coefficients and solutions, 90

Quaternion group, 221

Quaternions, 144, 154, 156, 315, 336

as an algebra, 151, 152

matrix representation of, 174

noncommutativity, 152–153, 157, 174

Queen’s College, Cork, 185

Quintic equations

algebraic solution, 64, 103–104, 125, 130–131

elementary symmetric polynomials in, 90, 103

numerical solution, 115

proving unsolvability of general equation, 116, 117, 125, 126–130, 222

resolvent, 123, 125

severely depressed, 333

R

See Real numbers.

Rabbit number problem, 66–67, 68

Raleigh, Walter, 94

Ramphoid cusp, 260

Rational expressions, 15

Rational function, 204–205, 231, 332

Rational numbers (), 37, 39, 41

completion of, 292

fraction field of, 291

properties, 9, 10, 196, 288, 322

Rawlinson, Henry, 48, 323

Real line, 11

Real numbers ()

as an algebra, 160

discovery, 10

as “honorary” complex numbers, 12, 106

as points, 148

properties, 10–11, 12, 196, 322

Recorde, Robert, 329

Regular Polytopes, 336, 339–340, 351

Representation, 299

Resolvent equations, 123–124, 126

Rhind, A. Henry, 29

Rhind Papyrus, 23, 29–30

Riccati equation, 336

Ridler, Ann, 334–335

Riemann, Bernhard, 159, 230, 231, 265, 267–268, 273, 277, 281, 293, 297, 305, 316, 318, 348

Riemann hypothesis, 333–334

Riemann sphere, 344

Riemann surfaces, 265–267, 281, 346, 348

Riemann–Roch theorem, 268

Rig Veda, 156

Rings and ring theory, 133, 276

applications, 223–224, 234

commutative, 238, 262, 306

of complex numbers, 226, 233, 260, 264

defined, 226

Fermat’s Last Theorem and, 224–225, 226, 229

of 5-adic integers, 290–291

Gaussian ring, 226–227

ideal, 228, 231, 232–233, 238, 262–263, 264, 268, 305

internal structure, 233–234, 238, 260–261, 262–263

invariants, 305

Lasker ring, 234–235

Noetherian ring, 235, 297

primary ideal, 234–235

principal ideal, 233

properties, 226, 227, 232, 342

subring, 232, 262–263

unique factorization in, 227, 228, 229

units, 226, 342

Roch, Gustav, 346

Roman Empire, 43, 45–46

Roots of unity

complex numbers, 109–114

cube, 61, 109, 110, 111, 113, 119, 190, 215

fifth, 110, 112, 130

fourth, 110, 214

nth, 109–110, 111, 113–114, 331, 332

primitive nth, 113–114, 331–332

properties, 113

pth, 213, 221, 228–229

seventeenth, 112

sixth, 113, 190, 191

Rosenlicht, Maxwell A., 298

Rothman, Tony, 206, 208

Royal Society, 98, 125, 185, 259, 331, 332

Royal Swedish Academy of Sciences, 312

Rubaiyat, 52

Rucker, Rudy, 146–147

Ruffini, Paolo, 116, 126–127, 129, 154, 333

Rule of signs, 8–9, 11, 40–41, 59–60, 83, 84

Russell, Bertrand, 186–187, 286, 324, 351

S

Sabbah, Hassan, 54

Sachs, Abraham, 24, 27, 28

Saint-Venant, Jean Claude, 155

Salmon, George, 259–260

Sargon the Great, 21, 22

Scalar product, 142

Scalars, 134

Schläfli, Ludwig, 158–159

Segre, Corrado, 277, 293

Seki, Takakazu, 169, 170

Seljuk, 53

Serge, Victor, 311

Serre, Jean-Pierre, 304, 309

Sets and set theory, 187, 213, 231, 233, 264, 287, 301

Severi, Francesco, 277, 293

“Sevener” Shiites, 326

Shakespeare, William, 72, 330

Shang Yang, 162–163

Shing-tung Yau, 318

Simultaneous linear equations, 40, 161, 163–164, 166, 168–169, 170–171

Solvability, 299

Space–time, 335–336

Special theory of relativity, 236, 250–251, 316

Sphereland, 146

Spinode, 260

Spinors, 156, 352

Square root function, 265, 267

Squaring function, 265–266, 267

St. Bartholomew’s Eve massacre, 86

St. Petersburg Academy of Sciences, 255

Stalin, Joseph, 235

Steinerian, 260, 345

Steinitz, William, 234

Stevin, Simon, 342

Stewart, Ian, 146

Stott, Alicia Boole, 185, 339–340

String theory, 317, 318

Subgroups, 214–215, 218

Subtraction, 8

complex numbers, 13

Sumerian language and mathematics, 21, 28

Summa (Pacioli), 71, 328

Supersymmetric string theory, 317

Suslin, Andrei, 298

Swan, Richard G., 5

Sylow, Ludwig, 217–218, 269, 270

Sylow p-subgroup, 218–219

Sylvester, J. J., 174, 175, 181, 260, 317

Symmetric functions, 90, 120, 121, 333, 336

Symmetry

in algebraic geometry, 248, 250, 251

of coefficients and solutions in polynomial equations, 90, 120, 121, 202–203

group of order n!, 219

principles, 168–169

Synthetic geometry, 255, 344–345

T

Tartaglia, Nicolo, 76–80

Tensors, 82, 142, 159, 277, 316

Thompson, John G., 298

The Time Machine, 147

Topology. See also Algebraic topology

defined, 2, 279–280

founders, 280, 284

video on, 344

Transcendental numbers, 288

Transformations

affine, 250

in algebraic geometry, 250–251, 271–273

groups, 219–220, 270, 275–276, 347

isometries, 250, 271–273, 275

Lie group, 347

Lorentz, 237, 250–251, 316

Möbius, 251

projective, 250, 275

topological, 250

Treaty of Cateau-Cambrésis, 85

Trigonometry, 87, 104, 330

Trilinear coordinates, 260

Trinity College, Cambridge, 174, 180, 182

Trinity College, Dublin, 149

Triplets

an algebra for, 150–151

permutation, 166

Turkish empire, 52, 53

“Twelver” Shiites, 326

U

Universal algebra, 351

Universal arithmetic, 99, 131

Universal constructions, 300

University College London, 181–182

Universities

Berlin, 129, 234

Bologna, 75

Bonn, 257

Chicago, 5

Christiana (Kristiana, Oslo), 128, 217, 269

Göttingen, 234, 236–239, 261–262, 265, 270, 271, 280, 294, 316, 323

Heidelberg, 234

Königsberg, 294

London, 175, 182, 339

Michigan, 36, 302

Minnesota, 344

Montpelier, 313

São Paolo, 296

St. Andrews (and math website), 332

Tübingen, 156

Virginia, 175

Unknown quantity

in polynomials, 14

Sumerian, 28

Unramified class field theory, 2

V

Van der Waerden, B. L., 288, 301

van Roomen, Adriaan, 87

Vandermonde, Alexandre-Théophile, 117–118, 121, 122, 123, 130, 332

Variety concept, 263–264

Vector analysis, 157

Vector space, 133

abstract, 159

algebras, 142, 143–144, 154, 157, 347

attaching to a manifold, 304

basis, 140, 154

complex numbers as, 143–144

dimension of, 138–140, 144, 154, 199

embedding, 142, 335

extended field as, 199

four-dimensional, 152–156, 335

inner (scalar) product, 142

linear dependence and independence, 82, 136–139, 141, 154

linear functional, 142

linear transformations, 142

matrices, 173

module, 233, 304

n-dimensional, 158–159

pair mapping, 142

polynomial representations, 140–141, 142, 144

projections, 142, 154

subspace, 154

Vectors

adding, 136, 140, 141

characteristics, 134, 135

dividing, 144

factorization of zero vector, 159–160

inverse, 134, 135, 140

multiplying by scalars, 135, 141

multiplying by vectors, 143–144, 154, 347

Venn, John, 186

Vesalius, Andreas, 73

Viète, François, 13, 81, 85–91, 92, 94, 105, 120

Voevodsky, Vladimir, 310

Vogel, Kurt, 31, 41, 328

W

Wallis, John, 97

Walsh, James J., 65, 69

Waring, Edward, 333

Weber, Heinrich, 231, 268

Wedderburn, Joseph Henry Maclagen, 160

Weierstrass, Karl, 160, 231, 293, 343

Weil, André, 296–297, 309, 310, 350

Wells, H. G., 147

Wesson, Robert G., 20

Weyl, Hermann, 2, 238, 292–293, 296

Whiston, William, 99

Widman, Johannes, 72

Wiener, Hermann, 294

Wiener, Norbert, 239

Wigner, Eugene, 315

Wiles, Andrew, 38, 228, 297

Wilhelm II, German emperor (Kaiser), 235

Wittgenstein, Ludwig, 23

World War I, 237

X

Xuan-zang, 47

Y

Yale University, 24

Z

See Integers.

Zariski, Oscar, 295–296, 297, 310, 318

Zero

constant term in polynomials, 107

countability, 322

discovery, 8, 47, 68

division by, 11

position marker, 32

vector, 134, 159–160