## Number: From Ahmes to CantorWe might take numbers and counting for granted, but we shouldn't. Our number literacy rests upon centuries of human effort, punctuated here and there by strokes of genius. In his successor and companion volume to Gnomon: From Pharaohs to Fractals, Midhat Gazalé 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.Gazalé tackles questions that will stimulate math enthusiasts in a highly accessible and inviting manner. What is a natural number? Are the decimal and binary systems the only legitimate ones? Did the Pythagorean theorem and the discovery of the unspeakable irrationals cost the unfortunate mathematician Hippasus his life? What was the Ladder of Theodorus of Cyrene and how did the ancient Greeks calculate square roots with such extraordinary proficiency? An original generalization of Euler's theorem is offered that explains the pattern of rational number representations. Later on, the field of Continued Fractions paves the way for another original contribution by Gazalé, that of cleavages, which sheds light on the mysterious nature of irrational numbers as it beautifully illustrates Dedekind's famous Schnitt. In the end the author introduces us to the Hilbert Hotel with its infinite number of rooms, guests, and an infinite number of people waiting to check in, where he sets the debate between Aristotle and Cantor about the true nature of infinity.This abundantly illustrated book, remarkable for its coherency and simplicity, will fascinate all those who have an interest in the world of numbers. Number will be indispensable for all those who enjoy mathematical recreations and puzzles, and for those who delight in numeracy. |

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### Contents

Preface | xi |

INTRODUCTION | 3 |

The Genesis of Number Systems | 9 |

Positional Number Systems | 59 |

Divisibility and Number Systems | 107 |

Real Numbers | 152 |

Continued Fractions | 185 |

Cleavages | 199 |

Infinity | 257 |

### Common terms and phrases

abscissa affine transform al-Khawarizmy algebraic algorithm ancient Arabic Archimedes arithmetic belongs bers calculate Cantor centuries coherent conformable multiple congruence consists construction contains Continued Fractions continuum convergent corresponding cyclic number decimal denote digits division Egyptian equal equation Euclid's Euclid's Elements Eudoxus Euler's theorem example exists exponent expressed factor factorial base Fermat's finite follows Gaussian primes geometric given Greeks increment sequence infinite number infinite set infinity input integer integer pair irrational numbers irreducible fraction Ladder larger largest conformable length mantissa mathematical mathematicians Mayan Mersenne primes Mesopotamian modd modulo namely natural numbers number lattice number representations obviously Papyrus periodic positional number system positional system prime node primitive roots proof Pythagoras Pythagorean Pythagorean triple rational number reader real number relatively prime relatively prime residues represented residue sequence smallest square statement subset symbol Table Theodorus of Cyrene tion TT_R unit fractions universe verify words write zero