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
Print Book, English, ©2000
Princeton University Press, Princeton, N.J., ©2000
History
xv, 297 pages : illustrations ; 24 cm
9780691005157, 069100515X
41628217
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