## Numbers: Computers, Philosophers, and the Search for MeaningNumbers deals with the development of numbers from fractions to algebraic numbers to transcendental numbers to complex numbers and their uses. The book also examines in detail the number pi, the evolution of the idea of infinity, and the representation of numbers in computers. The metric and American systems of measurement as well as the applications of some historical concepts of numbers in such modern forms as cryptography and hand calculators are also covered. Illustrations, thought-provoking text, and other supplemental material cover the key ideas, figures, and events in the historical development of numbers. |

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### Contents

The First Problems | 3 |

Early Counting Systems | 11 |

Our Place Value Number System | 34 |

Analytical Engines | 46 |

Extending the Idea of a Number | 65 |

An Evolving Concept of a Number | 67 |

Negative Numbers | 77 |

Algebraic Numbers | 85 |

Early Inisights | 119 |

Galileo and Bolzano | 130 |

Georg Cantor and the Logic of the Infinite | 143 |

Cantors Legacy | 165 |

Chronology | 181 |

Glossary | 199 |

203 | |

213 | |

### Common terms and phrases

algebraic equations algebraic numbers algorithm analytical engine ancient arithmetic axiom of choice base 60 bers Brahmagupta calculations Cantor’s Cardano century column complex numbers concept of number cultures d’Alembert decimal point Dedekind described developed digits discovered discovery Egyptians ENIAC Euclid Euler example exist fact finite fixed-point arithmetic French mathematician Galileo geometry German mathematician Gödel Greek Hilbert Hindu history of mathematics ideas important Indian mathematics infinite sets infinity insight irrational numbers logical mathe maticians matics Mayan Mesopotamians method multiply natural numbers negative numbers number system Pascaline pebble perfect squares philosopher pile positional notation positive number positive rational numbers prime numbers problem proof publishes rational numbers real number line represent Richard Dedekind rod numerals roots Russell paradox set of axioms set of numbers set of real sexagesimal solutions solve symbol system of numeration Tartaglia tion transcendental numbers transfinite numbers Turing machine Turing’s unit fractions whole numbers write written