Elements of Mathematics: From Euclid to Gödel by John Stillwell

By John Stillwell

Elements of Mathematics takes readers on a desirable travel that starts in easy mathematics--but, as John Stillwell exhibits, this topic isn't really as ordinary or effortless as one may possibly imagine. no longer all subject matters which are a part of contemporary user-friendly arithmetic have been regularly regarded as such, and nice mathematical advances and discoveries needed to take place to ensure that definite topics to develop into "elementary." Stillwell examines straight forward arithmetic from a particular twenty-first-century standpoint and describes not just the sweetness and scope of the self-discipline, but in addition its limits.

From Gaussian integers to propositional good judgment, Stillwell delves into mathematics, computation, algebra, geometry, calculus, combinatorics, chance, and common sense. He discusses how every one quarter ties into extra complicated themes to construct arithmetic as a complete. via a wealthy number of simple rules, brilliant examples, and fascinating difficulties, Stillwell demonstrates that simple arithmetic turns into complex with the intervention of infinity. Infinity has been saw all through mathematical heritage, however the contemporary improvement of "reverse arithmetic" confirms that infinity is vital for proving recognized theorems, and is helping to figure out the character, contours, and borders of ordinary arithmetic.

Elements of Mathematics provides readers, from highschool scholars to expert mathematicians, the highlights of straight forward arithmetic and glimpses of the elements of math past its boundaries.

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Extra resources for Elements of Mathematics: From Euclid to Gödel

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This is not fast. However, repeatedly subtracting b from a until the difference r becomes less than b is the same as dividing a by b and obtaining the remainder r . This gives the following fact, on which we will base the Euclidean algorithm from now on. 1: Visualizing quotient and remainder. Division property. For any natural numbers a and b = 0 there are natural numbers q and r (“quotient” and “remainder”) such that a = qb + r where |r | < |b|. 1, because any natural number a must lie between successive multiples of b.

This is a fundamental unsolved problem in computation and logic, as we will explain further in chapters 3 and 9. 5 Quadratic Integers Let us begin with a curious fact about the numbers 25 and 27. 25 is a square and 27 is a cube, so these numbers give a solution of the equation y3 = x2 + 2; namely, x = 5 and y = 3. Nearly 2000 years ago, Diophantus specifically mentioned this equation and this solution in his Arithmetica, Book VI, Problem 17. After reading this passage in Diophantus, Fermat (1657) claimed that it is the only solution in the positive integers.

11: Fibonacci numbers in the Liber abaci. known that his numbers would have a long career in number theory and combinatorics, though perhaps he wondered whether there was an explicit formula for the nth number. This question was not answered for more than 500 years. Finally, in the 1720s, Daniel Bernoulli and Abraham de Moivre showed that 1 Fn = √ 5 √ 1+ 5 2 n − √ 1− 5 2 n , where Fn is the nth Fibonacci number (starting, for convenience, with F0 = 0 and F1 = 1). For more about this, see chapter 7.

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