What is mathematics / Richard, Courant - Courant, Richard [et. al] (1941)
Preface to first editon, preface to second edition, how to use the book, weat is mathematics, the natural numbers introduccion; calculation with integers, laws of arithmetic, the representation of integers computation in systems otber than the decimal, the infinnitude of the number, system, mathematical induction,the principle of mathematical induction, the arithmetical progrecion, the geometrical progression, the sum of the first n squares, an important inequality, the binomial theorem, further remarks on mathematical induction, suplement to chepter the theory of numbers; the prime numbers, fundamental facts, the distribution of the primes, formulas producing primes, b, primes in arithmetical progressions, e, the prime number theorem , d, two unsolved problems concerning prime number, congruences, general concepts, fermat's quadratic residues, pythagorean number and fermat' s last theorem, te euclidean algorithm, general theory, application to the fundamental theorem of arithmetic, euler's function, fermat's theorem again, continued fractions, diophantine equations; the number, system of mathematics, the rational numbers, rational numbers a device for measuring, intrinsic need for the rational numbers principle of generalization, geometrical interpretation of rational numbers, incommensurables segments, irrational numbers, and the concept of limit, introduction, decimal fractions, infinite decimals, limits, infinite geometrical series, rational numbers and periodic decimals, general definition of irrational numbers by nested intervals, alternative methods of defining irrational numders, dedekind cuts, remarks on analytic geometry, the basic principle, equations of lines and curves, geometrical constructions, the algebre of number, more about inversion and its applications, invariance of angles, families of cireles, application to the problem of apollonius, repeated reflrctions; projective geometry, axiomatics, non-euclidean geometries, classification of geometrical properties, invariance under, transformations, projective transformation; topology, the five color theorem, the jordar curve theorem for polygons, thefundametal theorem of algebra; functions and limits, functions, continuity, functions of several variables functions and transformations.