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.
This resource is supported by the Institute of Museum and Library Services under the provisions of the Library Services and Technology Act as administered by State Library of Iowa.