Index

A imaginary part of —, 74 absolute value, 166 modulus of —, 75 acute angle, 148 polar form of —, 78 aleph naught, ℵ0, 105 real part of —, 74 algebraic number, 104 complex plane, 76 complex —, 112 composite number, 3 angle bisector, 134 congruence, see modular arithmetic angle-angle-side, 122 Congruence Axiom, 116 angle-side-angle, 117 congruent geometric figures, 116 modulo m,23 B triangles, 116 bounded linear transformation, 213 conjugate, 75, 151 Brun, Viggo, 185 constructible angles, 148 numbers, 139 C points, 145 C, complex numbers, 75 polygon, 154, 155 Cn, n-dimensional complex space, 211 continuous, 213 calculus, 72 continuum canonical factorization, 33 cardinality of —, c, 105 Cantor’s Paradox, 109 Continuum Hypothesis, 110 Cantor–Bernstein Theorem, 98 convergence cardinality, 90–92, 97 conditional —, 189 Cartesian product of sets, 111 of a sequence, 191 Cauchy–Schwarz Inequality of an infinite series, 169 in 2, 206 absolute —, 188 in Rn, 202 corresponding angles, 120 central angle, 154 cosine, see trigonometric functions characteristic function, 108 countable set, 94 Chinese Remainder Theorem, 61 closed ball, 208 Comparison Test, 178 D compass, 133 De Moivre’s Theorem, 79, 88 complex numbers, C,74 decryptor, 45, 53, 54 argument of —, 77 Dedekind cuts, 71, 190

© Springer Nature Switzerland AG 2018 215 D. Rosenthal et al., A Readable Introduction to Real Mathematics, Undergraduate Texts in Mathematics, https://doi.org/10.1007/978-3-030-00632-7 216 Index degrees, 119 H Diophantine equation, 54–56 harmonic series, 181 disjoint sets, 90 alternating —, 189 distance Hilbert Space, 204 in 2, 207 in R2, 194 in R3, 197 I divergent series, 169 induction, see Mathematical Induction divisible, 2 infinite decimal, 179 divisor, 2 infinite series, 168 dot product, see inner product injective function, see function, one-to-one duplication of the cube, 153 inner product, 204 in 2, 205 in Cn, 211 E in R2, 200 e, 104, 183 in R3, 201 empty set, ∅,89 in Rn, 201 encryptor, 45, 52, 54 inscribed angle, 127 Enumeration Principle, 103 integers, Z,1 equilateral triangle, 117 interior angles, 120 erect a perpendicular, 136 alternate —, 120 Euclidean Algorithm, 49 intersection of sets, 90 Euclidean plane, R2, 111 interval Euler φ function, 47, 58, 61 closed —, 94 Euler’s Theorem, 59 half-open —, 95 exponential function, 188 open —, 95 irrational numbers, 66 isosceles triangle, 116 F base angles of —, 116 factor,2,86 Factor Theorem, 86 Fermat numbers, 22 L Fermat prime, 158 1, 211 Fermat’s Little Theorem, 38, 59 2, 205 field, 141 Law of Cosines, 131 extension of —, 143 Law of Sines, 131 finite sequence, 102 least upper bound, 175 four-dimensional Euclidean space, R4, 198 construction of —, 190 function, 90 Least Upper Bound Property, 176 domain of —, 91 limit, 191 inverse of —, 91 Limit Comparison Test, 191 one-to-one —, 91 line, 115 onto —, 91 segment, 115 range of —, 91 linear combination, 51 Fundamental Theorem of Algebra, 83 linear Diophantine equation, 54–56 Fundamental Theorem of Arithmetic, 31, 52 linear transformation, 212 lower bound, 190 greatest —, 190 G Gauss-Wantzel Theorem, 158 geometric constructions, 138 M geometric series, 171 Mathematical Induction Goldbach Conjecture, 6 Generalized Principle of —, 12 greatest common divisor, 49 Generalized Principle of Complete —, 17 Index 217

Principle of —, 11 degree of —, 73 Principle of Complete —, 16 long division, 84 modular arithmetic, 23 Poonen, Bjorn, 103 modulus, 23 power set, 105 multiplicative inverse, 64, 75 prime number, 3 modulo p, 28, 34, 39 private exponent, 54 private key, 54 product of a number and a vector in Rn, 201 N public exponent, 54 N, natural numbers, 1 public key, 54 n-dimensional complex space, Cn, 211 public key cryptography, 44 n-dimensional Euclidean space, Rn, 199 Pythagorean Theorem, 125, 131 2 natural numbers, N,1 in , 207 negative infinite decimals, 179 Nim, 22 norm, 204 Q Q in 1, 211 , rational numbers, 64 in 2, 205 Quadratic Formula, 87 in Cn, 211 quadratic residue, 28 in R2, 200 quadrilateral, 129 in R3, 201 quotient, 2, 24, 50 in Rn, 201 R R, real numbers, 66 O R2, two-dimensional Euclidean space, 196 origin R3, three-dimensional Euclidean space, 198 R2 in , 193 R4, four-dimensional Euclidean space, 198 R3 in , 196 Rn, n-dimensional Euclidean space, 199 R4 in , 199 radians, 76 orthogonal lines, see perpendicular lines ratio of a geometric series, 171 Ratio Test, 188, 191 rational numbers, Q,63 P Rational Roots Theorem, 68 parallel lines, 120 real numbers, R,65 Parallel Postulate, 120 rearrangement of a series, 189 parallelogram, 130 rectangle, 123 Parallelogram Law, 207 relatively prime, 51 partial sum, 169 remainder, 24, 50 perfect square, 7, 67 repeating decimal, 189 perpendicular bisector, 134 right angle, 118 perpendicular lines, 123 right triangle, 123 perpendicular vectors hypotenuse of —, 123 in 2, 207 legs of —, 123 in R2, 195 root, 66, 74 in R3, 198 of multiplicity m,86 in R4, 199 of a polynomial, 68 in Rn, 199 Root Test, 189, 191 plane, see Euclidean plane RSA, 44, 52 polygon, 153 decryptor, 45, 53, 54 regular —, 153 encryptor, 45, 52, 54 polynomial, 68, 73 Procedure for Encrypting Messages, 54 coefficients of —, 73 ruler, 133, 160 constant —, 73 Russell’s Paradox, 109 218 Index

S trigonometric functions, 77, 131 sequence, 190, 204 general definition, 77 set, 89 trisect, 133 element of —, 89 tromino, 14 labeled by —, 102 twin primes, 6 ordinary —, 109 Twin Primes Problem, 6, 185 side-angle-side, see Congruence Axiom two-dimensional Euclidean space, R2, 196 side-side-side, 118 Typewriter Principle, see Enumeration Sigma notation, 187 Principle similar triangles, 126 sine, see trigonometric functions Spivak, Michael, 72 U square, 130 uncountable set, 94 diagonals of —, 130 union of sets, 90 straight angle, 118 unit circle, 77 straightedge, 133 unit square, 108 subfield of R, 141 unity, 81 subset, 89 roots of —, 81, 82 sum of an infinite series, 169 upper bound, 175 surd, 145, 148 plane, 146 surjective function, see function, onto V vector in 2, 205 T in R2, 195 tangent, see trigonometric functions in R3, 197 terms of an infinite series, 168 in Rn, 199 three-dimensional Euclidean space, R3, vertical angles, 119 198 tower of fields, 144 transcendental number, 104 W transversal, 120 Well-Ordering Principle, 11 trapezoid, 130 Wilson’s Theorem, 39 height of —, 130 triangle, 115 area of —, 124 Z base of —, 124 Z, integers, 1 height of —, 124 Zermelo–Fraenkel Set Theory, 110 sides of —, 115 zero, see root vertices of —, 115 zero vector Triangle Inequality, 170 in 2, 205 n in 2, 206 in R , 199 in Rn, 203 Zhang, Yitang, 185