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a k , 518 algebraic numbers, 388 a,...,b , 85 alternating harmonic series, 457, 460, 467, { } π, 373, 389 473, 475, 514 e, 363 Alternating Series Test, 456 antiderivative, 204 Abel’s Antiphon the Sophist, 313 Formula, 283 Apollonius, 226 Theorem, 524 Arbogast, Louis, 177 Abel, Niels Henrik, 537 arc length, 306 absolute value, 71 Archimedean absolutely convergent, 460 Property, 99, 372 Abu Kamil Shuja ibn Aslam, 53 spiral, 226 Abu’l-Wafa, Mohammad, 396 Archimedes, 53, 172, 226, 313, 392, 439, 483 accumulation point, 145 arcsine, 373, 389 addition area, 297 integers, 12 rectangle, 295 natural numbers, 5 signed, 302 rational numbers, 28 special polygon, 295 real numbers, 42 Aristotle, 52, 172, 439 Addition Law for Order, 13, 20, 29, 43, 63 Aryabhata, 393 Al-Baghdadi, Abu Mansur ibn Tahir, 53 Associative Law al-Battani, Abu Abdallah Mohammad ibn for Addition, 6, 13, 20, 29, 43, 63 Jabir, 396 for Multiplication, 6, 13, 20, 29, 43, 63 al-Haytham, Abu Ali al-Hasan ibn, 314 asymptote Al-Kashi, Ghiyath al-Din Jamshid Mas’ud, horizontal, 322 54 vertical, 322 al-Khashi, Ghiyath al-Din Jamshid Mas’ud, average value, 271, 427, 428 393 axioms Al-Khwarizmi, Abu Ja’far Muhammad ibn integers, 21 Musa, 53 ordered ﬁeld, 62 Al-Samawal, Ibn Yahya al-Maghribi, 54 ordered integral domain, 20 al-Tusi, Nasir al-Din, 396 real numbers, 64 Al-Uqlidisi, Abu’l Hasan Ahmad ibn Ibrahim, 54 Barrow, Isaac, 227, 228, 316 546 Index base p representation, 121, 431 Cauchy eventually repeating, 122 Completeness Theorem, 419 Berkeley, George, 229 Condensation Test, 458 Bernoulli product, 463 Daniel, 485, 524 sequence, 417 Jakob, 177, 394, 485 Cauchy’s Mean Value Theorem, 201 Johann, 177, 318, 355, 394 Cauchy, Augustin Louis, 57, 178, 230, 318, beta function, 283 355, 440, 465, 486, 518, 536 Bhaskara II, 53, 226, 354, 395 Cauchy–Schwarz Inequality, 311 Bhaskaracharya, 53, 395 for Integrals, 266 binary operation, 2 Cavalieri , Bonaventura, 175, 315 closed, 2 closed Binet’s formula, 524 bounded interval, 70 binomial non-degenerate, 70 coefﬁcient, 518 interval, 2, 70 series, 518 unbounded interval, 70 Bisection Method, 436 commutative diagram, 5 Bolzano, Bernard, 57, 178, 440, 486, 527, Commutative Law 536 for Addition, 6, 13, 20, 29, 43, 63 Bolzano–Weierstrass Theorem, 417 for Multiplication, 6, 13, 20, 29, 43, 63 Bonnet, Pierre, 230 compactness, 103, 163, 417 bound Comparison Test function, 137 series, 451 greatest lower, 47, 63 Type 2 Improper Integrals, 349 least upper, 47, 63 computer lower, 47, 63 science, 86 upper, 47, 63 concave up, 222 bounded, 47, 63, 294 conditionally convergent, 460 above, 47, 63 connectedness, 163 sequence, 405 constant, 90 away from zero, 260 Euler’s, 436 below, 47, 63 Euler–Mascheroni, 436 sequence, 405 sequence, 403 function, 137 content sequence, 405 innter, 296 Bradwardine, Thomas, 397 outer, 296 Brahe, Tycho, 397 continuous, 147 Brahmagupta, 53, 395 at a point, 147 Briggs, Henry, 397 uniformly, 158 Brouncker, William, 393 continuously Bryson of Heraclea, 313 differentiable, 189 Burgi,¨ Jost, 397 differentiable of order n, 189 convergence Calculator, see Swineshead, Richard interval of, 477 Cancellation Law convergent for Addition, 6, 15, 21, 66 absolutely, 460 for Multiplication, 6, 15, 21, 66 conditionally, 460 Cantor set, 286, 430, 435 improper integral, 342–344, 346 Cantor, Georg, 58, 429 pointwise, 490, 503 Index 547

sequence, 402 series, 445 series, 445 diverges uniformly, 493, 503 to infinity, 325, 331 converges, 132, 141, 323, 402, 445 sequence, 408 pointwise, 490 series, 445 uniformly, 493 to infinity from the left, 326 convex function, 225 to infinity from the right, 326 Copernicus, Nicolaus, 396 to negative infinity, 325 cosine, 375 sequence, 408 Cunha, Anastacio´ da, 486 series, 445 cut to negative infinity from the left, 326 Dedekind, 35 to negative infinity from the right, 326 irrational, 36 division lower, 40, 49 rational numbers, 31 rational, 36 real numbers, 66 Division Algorithm, 121 d’Alembert, Jean, 229 Darboux, Gaston, 319 element de Moivre, Abraham, 486 greatest, 92 decreasing, 207, 412 endpoint, 70 strictly, 207, 412 Dedekind cut, 35 left, 70 Dedekind, Richard, 58, 102, 441 right, 70 Definition by Recursion, 5, 86 equation Democritus of Abdera, 172 differential, 379, 507 density, 103 Euclid, 52, 226, 392, 439, 483 derivative, 183 Eudoxus of Cnidus, 53, 313, 439 nth, 188 Euler’s constant, 436 one-sided, 189 Euler, Leonhard, 177, 355, 394, 440, 486, second, 188 524, 536 symmetric, 191 Euler–Mascheroni constant, 436 Descartes, Rene,´ 56, 227, 316, 393 even diagram function, 265, 523 commutative, 5 integers, 128 dictionary order, 25 eventually repeating base p representation, differentiable, 183, 189 122 continuously, 189 existence infinitely, 189 theorem, 121 symmetrically, 191 explicit description, 86 differential equation, 379, 507 exponential function, 361 Dirichlet’s Test, 459 with base a, 366 Dirichlet, Lejeune, 177 extended real numbers, 328 discontinuity, 147 extension discontinuous, 147 function, 109 at a point, 147 periodic, 372 Distributive Law, 6, 13, 20, 29, 43, 63 Extreme Value Theorem, 163, 212 divergent extremum improper integral, 343, 344, 346 global, 209 sequence, 402 local, 209 548 Index factorial, 91, 475, 514 Gregory, James, 227, 316, 393 Fermat, Pierre de, 56, 227, 315 Fibonacci, 53, 393 half-open interval, 70 Fibonacci numbers, 431 Halley, Edmond, 534 field, 30, 62 Hamilton, William Rowan, 57 ordered, 30, 47, 62 harmonic series, 447, 455, 475, 514 fixed point, 170 Heine, Eduard, 58, 178 form Heine–Borel Theorem, 103 indeterminate, 332 Henstock–Kurzweil integral, 231 formal sum, 444 Hermite, Charles, 57 Fourier, Joseph, 177, 318, 486 Heron of Alexandria, 315 Frege, Gottlob, 59 Hilbert, David, 59 function Hipparchus bound, 137 of Nicaea, 394 bounded, 137 of Rhodes, see Hipparchus of Nicaea convex, 225 horizontal asymptote, 322 even, 265, 523 Hudde, Johann, 227 extension, 109 Huygens, Christiaan, 227 odd, 265 periodic, 371, 528 Identity Law polynomial, 90 for Addition, 13, 20, 29, 43, 63 sawtooth, 527, 533 for Multiplication, 6, 13, 20, 29, 43, 63 step, 247 improper integral, 342, 344, 346 Fundamental Theorem Type 1, 342 of Algebra, 33 Type 2, 345 of Arithmetic, 150 improperly integrable, 342, 344, 346 of Calculus increasing, 207, 412 Version I, 269 strictly, 207, 412 Version II, 272 indefinite integral, 277 indeterminate form, 332 Galilei, Galileo, 174, 315 induction, 83 gamma function, 354 inductive gauge integral, 231 hypothesis, 84 Gauss, Carl Friedrich, 57, 178, 355, 394, 440, reasoning, 83 486 set, 76 generalized Riemann integral, 231 step, 84 geometric series, 285, 447 infimum, 47, 63 Gerbert of Aurillac, 54 infinitely differentiable, 189 global inner content, 296 extremum, 209 integers, 12, 79 maximum, 209 addition, 12 minimum, 209 axioms, 21 golden ratio, 432, 524 even, 128 greatest less than, 12 element, 92 less than or equal to, 12 lower bound, 47, 63 multiplication, 12 Greatest Lower Bound Property, 47, 97 negative, 12, 14, 23 Gregoire´ de Saint-Vincent, 174, 397, 484 odd, 128 Gregory of Rimini, 440 positive, 14, 23 Index 549 integrable, 235 Kronecker, Leopold, 58 improperly, 342, 344, 346 locally, 342 l’Hopital’sˆ Rule, 334 integral l’Hopital,ˆ Guillaume de, 355 domain, 19 Lagrange Form of the Remainder Theorem, gauge, 231 519 generalized Riemann, 231 Lagrange, Joseph-Louis, 202, 229, 536 Henstock–Kurzweil, 231 Lambert, Johann, 394 improper, 342, 344, 346 Laplace transform, 341 indefinite, 277 Laplace, Pierre-Simon, 485 Lebesgue, 231, 297 least lower, 254 element, 20 Riemann, 231, 235 upper bound, 47, 63 Riemann–Stieltjes, 242 Least Upper Bound Property, 48, 64 upper, 254 Lebesgue Integral Test, 454 integral, 231, 297 interior, 70, 294 measure, 284, 297, 310 Intermediate Value Theorem, 163 Lebesgue’s Theorem, 287 interval, 70 Lebesgue, Henri, 319 closed, 2, 70 left closed bounded, 70 endpoint, 70 closed unbounded, 70 unbounded interval, 70, 322 endpoint, 70 left-hand limit, 141 half-open, 70 Leibniz, Gottfried von, 56, 175, 228, 236, interior, 70 317, 393, 484, 535 left unbounded, 70, 322 Leonardo of Pisa, 53, 393 non-degenerate, 70 less than non-degenerate closed bounded, 70 integers, 12 non-degenerate open bounded, 70 natural numbers, 8 of convergence, 477 rational numbers, 28 open, 70 real numbers, 42 open bounded, 70 less than or equal to open unbounded, 70 integers, 12 right unbounded, 70, 322 natural numbers, 8 Inverses Law rational numbers, 28 for Addition, 13, 20, 29, 43, 63 real numbers, 20, 42, 66 for Multiplication, 29, 43, 63 Levi ben Gerson, 54 irrational limit, 132, 322, 323, 402 cut, 36 left-hand, 141 numbers, 80 one-sided, 141 right-hand, 141 Johann Muller¨ of Konigsberg,¨ see superior, 479 Regiomontanus to infinity, 322 Jones, William, 394 Type 1, 322 Jordan measure, 297, 310 Type 2, 322 Jordan, Camille, 319 Limit Comparison Test, 453 Jyesthadeva, 534 Lindemann, Ferdinand von, 57, 394 Liouville, Joseph, 57 Kepler, Johannes, 174, 227, 314 Lipschitz 550 Index

condition, 162, 508 Multiplication Law for Order, 13, 20, 29, 43, constant, 162 63 Liu Hui, 393 multiplicative inverse local rational numbers, 28 extremum, 209 real numbers, 43 maximum, 209 minimum, 209 nth root, 171 locally integrable, 342 Napier, John, 397 logarithm function, 359 natural logarithm function, 359 with base a, 367 natural numbers, 3, 23, 76 lower addition, 5 bound, 47, 63 less than, 8 greatest, 47, 63 less than or equal to, 8 cut, 40, 49 multiplication, 6 integral, 254 negative, 69 integers, 12, 14, 23 part, 469 Machin, John, 393 rational numbers, 28 Maclaurin real numbers, 42 polynomial, 515 Neile, William, 316 series, 515 Nested Interval Theorem, 428 Maclaurin, Colin, 486, 536 Newton’s Method, 437 Madhava of Sangamagramma, 534 Newton, Isaac, 175, 228, 317, 393, 484, 534 Mathematical Induction Nicholas of Cusa, 172 Principle of, 83 No Zero Divisors Law, 13, 20, 66 Variant, 85 non-degenerate Maurolycus, Franciscus, 55 interval, 70 maximum, 92 rectangle, 294 global, 209 non-negative, 69 local, 209 Non-Triviality, 13, 20, 29, 44, 63 Mean Value Theorem, 200 numbers measure algebraic, 388 Jordan, 297, 310 Fibonacci, 431 Lebesgue, 284, 297, 310 irrational, 80 zero, 284, 285 natural, 3, 23, 76 Mengoli, Pietro, 450, 484 rational, 27, 28, 80 Meray,´ Charles, 58 real, 42 Mercator, Nicolaus, 398, 534 transcendental, 33, 388, 522 minimum numeral global, 209 Roman, 113 local, 209 monotone, 208, 412 odd strictly, 208, 412 function, 265 Monotone Convergence Theorem, 412, 414 integers, 128 multiplication one-sided integers, 12 derivative, 189 natural numbers, 6 limit, 141 rational numbers, 28 open real numbers, 43 bounded interval, 70 Index 551

non-degenerate, 70 Product Rule, 193 interval, 70 proof by induction, 83 unbounded interval, 70 Ptolemy, Claudius, 393 operation Pythagoras of Samos, 52 binary, 2 Pythagorean Theorem, 304, 306, 382 closed, 2 unary, 2 Quotient Rule, 193, 260 closed, 2 order relation, 41 radius of convergence, 477 ordered Raphson, Joseph, 437 field, 30, 47, 62 Ratio Test, 461, 474 axioms, 62 rational cut, 36 integral domain rational numbers, 27, 28, 80 axioms, 20 addition, 28 set, 41 division, 31 Oresme, Nicole, 53, 172, 226, 314, 397, 447, less than, 28 483 less than or equal to, 28 outer content, 296 multiplication, 28 multiplicative inverse, 28 p-series, 455 negative, 28 Parmenides of Elea, 354 subtraction, 31 partial sum, 445, 503 real numbers, 42 sequence of, 445, 503 addition, 42 Pascal, Blaise, 55, 175, 227, 316 axioms, 64 Peano Postulates, 3, 23, 77 division, 66 Peano, Giuseppe, 59, 319 extended, 328 period, 371 less than, 42 periodic less than or equal to, 20, 42, 66 extension, 372 multiplication, 43 function, 371, 528 multiplicative inverse, 43 Picard iteration, 508 negative, 42 Picard, Charles Emile, 508 subtraction, 66 Pigeonhole Principle, 125 rearrangement, 467 place value system, 114 Recorde, Robert, 55 Plato, 52, 392 rectangle, 294 pointwise convergent, 490, 503 area, 295 polygonal sum, 304 interior, 294 polynomial function, 90 non-degenerate, 294 Pope Sylvester II, 54 rectifiable, 306 positive, 69 recursive integers, 14, 23 definition, 86 part, 469 description, 86 power series, 474 Regiomontanus, 396 represented by, 510 region Principle between the graphs, 299 of Mathematical Induction, 83 under the graph, 299 Variant, 85 remainder Well-Ordering, 9, 21, 39, 78, 100, 102, 114 Taylor polynomial, 519 probability, 341 represented by a power series, 510 552 Index

Rheticus, Georg Joachim, 396 sum, 445 Riemann Taylor, 515 integrable, 235 telescoping, 446 integral, 231, 235 set sum, 231, 234 Cantor, 286, 430, 435 Riemann, Georg Friedrich Bernhard, 319, inductive, 76 470 ordered, 41 Riemann–Stieltjes signed area, 302 integrable, 242 Simpson, Thomas, 437 integral, 242 sine, 375 sum, 241 slope of the secant line, 221 Ries, Adam, 55 Sluse, Rene´ de, 227 right smooth, 189 endpoint, 70 Somayaji, Nilakantha, 534 unbounded interval, 70, 322 special polygon, 294 right-hand limit, 141 area, 295 Robert of Chester, 395 spiral Roberval, Gilles de, 227, 315 Archimedean, 226 Robinson, Abraham, 179 squarable, 297 Rolle’s Theorem, 199 square, 66 Roman numeral, 113 root, 101, 171 root step function, 247 nth, 171 Stevin, Simon, 55, 173, 314 square, 171 Stirling, James, 486, 536 Root Test, 479 strictly Russell, Bertrand, 58 decreasing, 207, 412 increasing, 207, 412 Sarasa, Alfonso Antonio de, 397 monotone, 208, 412 sawtooth function, 527, 533 subsequence, 415 secant line, 220 subtraction slope of, 221 rational numbers, 31 second derivative, 188 real numbers, 66 sequence, 115, 400 Suiseth, see Swineshead, Richard bounded, 405 sum above, 405 formal, 444 below, 405 of series, 445 Cauchy, 417 partial, 445, 503 constant, 403 supremum, 47, 63 of functions, 490 Swineshead, Richard, 483 partial sums, 445, 503 symmetric derivative, 191 series, 116, 444 symmetrically differentiable, 191 alternating harmonic, 457, 460, 467, 473, 475, 514 Taylor geometric, 285, 447 polynomial, 515 harmonic, 447, 455, 475, 514 remainder, 519 Maclaurin, 515 series, 515 of functions, 502 Taylor’s Theorem, 202, 283, 519 power, 474 Taylor, Brook, 536 rearrangement, 467 telescoping series, 446 Index 553

term limit to infinity, 322 sequence, 400 Type 2 sequence of functions, 490 improper integral, 345 series, 444 limit to infinity, 322 series of functions, 502 ternary expansion, 431 unary operation, 2 Test closed, 2 Ratio, 461, 474 uniformly Root, 479 continuous, 158 Theaetetus of Athens, 52 convergent, 493, 503 Theodorus of Cyrene, 52 upper Theorem bound, 47, 63 Abel’s, 524 least, 47, 63 Bolzano–Weierstrass, 417 cut, 40, 49 Cauchy Completeness, 419 integral, 254 Cauchy’s Mean Value, 201 Extreme Value, 163, 212 Valerio, Luca, 173 Heine–Borel, 103 van Heuraet, Hendrik, 316 Intermediate Value, 163 van Schooten, Frans, 484 Lagrange Form of the Remainder, 519 Varahamihira, 395 Lebesgue’s, 287 variable, 90 Mean Value, 200 vertical asymptote, 322 Monotone Convergence, 412, 414 Viete,` François, 55, 393, 484 Nested Interval, 428 Volterra, Vito, 434 Pythagorean, 304, 306, 382 Rolle’s, 199 Wallis , John, 175, 316, 355, 393, 484, 535 Taylor’s, 202, 283, 519 Weierstrass, Karl, 58, 178, 440, 528, 537 Torricelli, Evangelista, 227, 316, 355, 484 Well-Ordering Principle, 9, 21, 39, 78, 100, transcendental numbers, 33, 388, 522 102, 114 Transitive Law, 13, 20, 29, 43, 63 Wren, Christopher, 316 Triangle Inequality, 71, 311 Trichotomy Law, 8, 13, 20, 29, 43, 63 Yi Xing, 395 twice differentiable, 188 Type 1 Zeno of Elea, 172, 354, 439 improper integral, 342 Zu Chongzhi, 393 4 4 4 Ethan Bloch was born in 1956, and spent part of his childhood in Con- necticut and part in Is- rael. He received a B.A. in mathematics in 1978 from Reed College, where he developed a firm belief in the value of a liberal arts edu- cation, and a Ph.D. in mathematics in 1983 from Cornell University, under the supervision of Prof. David Henderson. He was an Instructor at the University of Utah for three years, and arrived at Bard College in 1986, where he has, very fortunately, been ever since. He is married and has two children; his family, his work and travel to Israel more than fill his time.

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