By Mark C. Chu-Carroll
Mathematics is beautiful--and it may be enjoyable and fascinating in addition to sensible. Good Math is your consultant to a few of the main fascinating issues from thousand years of arithmetic: from Egyptian fractions to Turing machines; from the genuine which means of numbers to evidence bushes, workforce symmetry, and mechanical computation. If you have ever questioned what lay past the proofs you struggled to accomplish in highschool geometry, or what limits the services of laptop in your table, this is often the e-book for you.
Why do Roman numerals persist? How will we recognize that a few infinities are greater than others? and the way do we comprehend for definite a application will ever end? during this fast paced journey of recent and not-so-modern math, machine scientist Mark Chu-Carroll explores many of the maximum breakthroughs and disappointments of greater than thousand years of mathematical suggestion. there's pleasure and sweetness in arithmetic, and in additional than dozen essays drawn from his renowned "Good Math" web publication, you can find ideas, proofs, and examples which are frequently incredible, counterintuitive, or simply undeniable weird.
Mark starts his trip with the fundamentals of numbers, with an wonderful journey in the course of the integers and the typical, rational, irrational, and transcendental numbers. The voyage maintains with a glance at many of the oddest numbers in arithmetic, together with 0, the golden ratio, imaginary numbers, Roman numerals, and Egyptian and carrying on with fractions. After a deep dive into glossy common sense, together with an advent to linear good judgment and the logic-savvy Prolog language, the journey concludes with a journey of recent set concept and the advances and paradoxes of recent mechanical computing.
in the event that your highschool or collage math classes left you greedy for the interior that means in the back of the numbers, Mark's e-book will either entertain and enlighten you.
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Additional info for Good Math. A geek's guide to the beauty of numbers, logic, and computation
Sample text
He did not accept that it had any meaning at all: thus he termed it an “imaginary” number as part of an attempt to discredit the concept. Complex numbers built using i finally came into wide acceptance as a result of the work of Leonhard Euler (1707–1783) in the eighteenth century. Euler was probably the first to truly fully comprehend the complex number system created by the existence of i. And working with that, he discovered one of the most fascinating and bizarre mathematical discoveries ever, known as Euler’s equation.
Let’s start with the irrational numbers. These are numbers that aren’t integers and also aren’t a ratio of any two integers. You can’t write them as a normal fraction. If you write them as a continued fraction (which we’ll describe in 11, Continued Fractions, on page 69), then they go on forever. If you write them in decimal form, they go on forever without repeating. They’re called irrational because they can’t be written as ratios. Many people have claimed that they’re irrational because they don’t make sense, but that’s just a rationalization after the fact.
The fact that there is such a close relationship between i, π, and e is just shocking on the face of it. report erratum • discuss 8. i: The Imaginary Number • 49 What i Does Once the reality of i as a number was accepted, mathematics was changed irrevocably. Instead of the numbers described by algebraic equations being points on a line, suddenly they become points on a plane. Algebraic numbers are really twodimensional; and just like the integer 1 is the unit distance on the axis of the real numbers, i is the unit distance on the axis of the imaginary numbers.