Irrational numbers

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  • The aim of this book, announced in the first edition, is to give a bird’seye view of undergraduate mathematics and a glimpse of wider horizons. The second edition aimed to broaden this view by including new chapters on number theory and algebra, and to engage readers better by including many more exercises. This third (and possibly last) edition aims to increase breadth and depth, but also cohesion, by connecting topics that were previously strangers to each other, such as projective geometry and finite groups, and analysis and combinatorics....

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  • Di®erent types of numbers ¯nd di®erent application in the physical world. Whole numbers work well for counting discrete objects, such as the number of resistors in a circuit. Integers are needed when negative equivalents of whole numbers are required. Irrational numbers are numbers that cannot be exactly expressed as the ratio of two integers, and the ratio of a perfect circle's circumference to its diameter (¼) is a good physical example of this.

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  • The nine mathematicians whose works are represented in the following pages are among the most famous in the whole history of mathematics. Each of them made a significant con- tribution to the science-a contribution which changed the succeeding course of the development of mathematics. That is why we have called this book Breakthroughs in Mathe- matics. ...

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  • Let b ≥ 2 be an integer. We prove that the b-ary expansion of every irrational algebraic number cannot have low complexity. Furthermore, we establish that irrational morphic numbers are transcendental, for a wide class of morphisms. In particular, irrational automatic numbers are transcendental. Our main tool is a new, combinatorial transcendence criterion. 1. Introduction Let b ≥ 2 be an integer. The b-ary expansion of every rational number is eventually periodic, but what can be said about the b-ary expansion of an irrational algebraic number? ...

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  • If α is an irrational number, Yoccoz defined the Brjuno function Φ by Φ(α) =n≥0 α0α1 · · · αn−1 log 1 αn , where α0 is the fractional part of α and αn+1 is the fractional part of 1/αn. The numbers α such that Φ(α)

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