Number: from Ahmes to Cantor

"In his successor and companion volume to Gnomon: From Pharaohs to Fractals, Midhat Gazale takes us on a journey from the ancient worlds of the Egyptians, the Mesopotamians, the Mayas, the Greeks, the Hindus, up to the Arab invasion of Europe and the Renaissance. Our guide introduces us to some of the most fascinating and ingenious characters in mathematical history, from Ahmes the Egyptian scribe (whose efforts helped preserve some of the mathematical secrets of the architects of the pyramids) through the modern era of Georg Cantor (the great nineteenth-century inventor of transfinite numbers). As he deftly blends together history, mathematics, and even some computer science in his characteristically compelling style, we discover the fundamental notions underlying the acquisition and recording of "number," and what "number" truly means." "Number will be indispensable for all those who enjoy mathematical recreations and puzzles, and for those who delight in numeracy."--Jacket. 1. The genesis of number systems -- Foundations -- Matching -- Naming -- Counting -- Grouping -- Archaic number systems -- The Egyptians -- The Mesopotamians -- The Greeks -- The Mayas -- Two current number systems -- The Hindus -- The Arabs -- The decimal number system -- Fractional numbers -- Uttering versus writing -- Units -- The binary number system -- 2. Positional number systems -- The division algorithm -- Codes -- Mixed-base positional systems -- Finding the digits of an integer -- Addition -- Uniform-base multiplication -- Mixed-base multiplication -- Construction 1 : a parallel adder -- Construction 2 : a digital-to-analog converter -- Construction 3 : a reversible binary-to-analog converter -- Positional representation of fractional numbers -- Going to infinity -- How precise is a mantissa? -- Finding the digits of a fractional number -- Finding the digits of a real number -- Periodic bases -- A triadic (ternary) yardstick -- Marginalia -- Unit fractions revisited -- Appendix 2.1 -- Appendix 2.2. 3. Divisibility and number systems -- The fundamental theorem of arithmetic -- Congruences -- Pascal's divisibility test -- Euler's function and theorem -- Euler's theorem -- Exponents -- Primitive roots -- A generalization of Euler's theorem -- The residue sequence -- Indices -- Conjugates and conformable multiples -- Positional representation of rational numbers -- Mixed bases -- Bases 2 and 10 -- Cyclic numbers -- Strings of ones and zeros -- Marginalia -- Mersenne primes -- On Dirichlet's distribution principle -- Appendix -- Carmichael's variation on Euler's theorem -- 4. Real numbers -- Rational numbers -- The integral domain -- The rational numbers field -- Marginalia : on the axiomatic method -- Commensurability -- Irrational numbers -- Pythagoras's theorem -- Pythagorean triples -- The Plimpton 322 tablet -- The ladder of Theodorus of Cyrene and Diophantine equations -- A variation on the ladder of Theodorus -- Fermat's last theorem -- The irrationality of (square root of) 2 -- A (theoretically) physical impossibility -- Dedekind -- Eudoxus -- Marginalia -- Three ancient problems -- Appendix -- Proof of the irrationality of e. 5. Continued fractions -- Euclid's algorithm -- Continued fractions -- Regular continued fractions -- Convergents -- Terminating regular continued fractions -- Periodic regular continued fractions -- Spectra of surds -- Nonperiodic, nonterminating regular continued fractions -- Two celebrated irregular continued fractions -- Appendix -- 6. Cleavages -- The number lattice -- Prime nodes -- Cleavages -- Coherence -- A definition of real numbers -- Some properties of fractions -- Contiguous fractions -- The mediant -- Affine transformations -- The Stern-Brocot tree -- Pencils and ladders -- Cleavages and continued fractions -- Klein's construction -- The greatest common divisor revisited -- Marginalia -- Cleavages and positional number systems -- Cleavages and automata -- Cleaving crystals -- Cleavages and replicative functions -- Gaussian primes -- Appendix 6.1 -- Proof of test (6.7) -- Appendix 6.2 -- The increment sequence. 7. Infinity -- Convergence -- Paradoxes of infinite series -- Further paradoxes of infinity -- You are always welcome at the Hilbert Hotel -- Zeno's paradoxes -- Horror infiniti? -- Potential versus actual infinity -- Cantor -- The power of the continuum -- Geometrical metaphors -- Transfinite cardinal numbers -- Cantor dust -- Beyond aleph 1 -- Postscript : the balance is improbable but the night sky is black