Numbers and GeometryNUMBERS AND GEOMETRY is a beautiful and relatively elementary account of a part of mathematics where three main fields--algebra, analysis and geometry--meet. The aim of this book is to give a broad view of these subjects at the level of calculus, without being a calculus (or a pre-calculus) book. Its roots are in arithmetic and geometry, the two opposite poles of mathematics, and the source of historic conceptual conflict. The resolution of this conflict, and its role in the development of mathematics, is one of the main stories in the book. The key is algebra, which brings arithmetic and geometry together, and allows them to flourish and branch out in new directions. Stillwell has chosen an array of exciting and worthwhile topics and elegantly combines mathematical history with mathematics. He believes that most of mathematics is about numbers, curves and functions, and the links between these concepts can be suggested by a thorough study of simple examples, such as the circle and the square. This book covers the main ideas of Euclid--geometry, arithmetic and the theory of real numbers, but with 2000 years of extra insights attached. NUMBERS AND GEOMETRY presupposes only high school algebra and therefore can be read by any well prepared student entering university. Moreover, this book will be popular with graduate students and researchers in mathematics because it is such an attractive and unusual treatment of fundamental topics. Also, it will serve admirably in courses aimed at giving students from other areas a view of some of the basic ideas in mathematics. There is a set of well-written exercises at the end of each section, so new ideas can be instantly tested and reinforced. |
Contents
II | 1 |
V | 4 |
VI | 7 |
VII | 9 |
VIII | 13 |
IX | 17 |
X | 20 |
XI | 23 |
XLIX | 159 |
L | 161 |
LI | 165 |
LII | 168 |
LIII | 171 |
LIV | 177 |
LV | 179 |
LVI | 183 |
XII | 26 |
XIII | 30 |
XIV | 37 |
XV | 40 |
XVI | 44 |
XVII | 47 |
XVIII | 50 |
XIX | 53 |
XX | 56 |
XXI | 59 |
XXII | 64 |
XXIII | 69 |
XXVI | 72 |
XXVII | 77 |
XXVIII | 82 |
XXIX | 85 |
XXX | 89 |
XXXI | 93 |
XXXII | 95 |
XXXIII | 100 |
XXXIV | 105 |
XXXV | 111 |
XXXVI | 113 |
XXXVII | 116 |
XXXVIII | 120 |
XXXIX | 124 |
XL | 127 |
XLI | 131 |
XLII | 136 |
XLIII | 143 |
XLVI | 147 |
XLVII | 152 |
XLVIII | 156 |
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Common terms and phrases
a₁ addition formula algebra arithmetic axiom b₁ calculus called coefficients complex numbers congruence classes conic sections construction coordinates cose cosh cube cutting and pasting Dedekind Deduce from Exercise defined dihedral angle Diophantus divides ellipse equation Euclid's Euclidean algorithm Euclidean plane Euler's criterion example fact Fermat Fermat's little theorem Figure finite follows fractions function Gaussian integers Gaussian prime gcd(a geometry gives hence hyperbola induction infinite integer points intersection inverse irrational number isometries lengths mathematicians mathematics mod q multiples natural numbers non-Euclidean plane norm number theory P₁ pairs parabola polygon polynomial prime divisor problem proof prove Pythagorean theorem Pythagorean triples quadratic curves rational numbers rational operations rational points rational triangle real numbers rectangle reflections result right-angled triangle rotation Show sine sinh square mod square roots tetrahedron unique prime unique prime factorization values