By Mark C. Chu-Carroll
Mathematics is beautiful--and it may be enjoyable and fascinating in addition to sensible. Good Math is your advisor to a few of the main fascinating issues from thousand years of arithmetic: from Egyptian fractions to Turing machines; from the true which means of numbers to evidence bushes, workforce symmetry, and mechanical computation. If you've ever puzzled what lay past the proofs you struggled to accomplish in highschool geometry, or what limits the features of desktop in your table, this is often the booklet for you.
Why do Roman numerals persist? How will we be aware of that a few infinities are better than others? and the way do we be aware of for definite a software will ever end? during this fast moving travel of contemporary and not-so-modern math, computing device scientist Mark Chu-Carroll explores the various maximum breakthroughs and disappointments of greater than thousand years of mathematical inspiration. there's pleasure and sweetness in arithmetic, and in additional than dozen essays drawn from his renowned "Good Math" weblog, you'll locate ideas, proofs, and examples which are usually staggering, counterintuitive, or simply simple weird.
Mark starts his trip with the fundamentals of numbers, with an unique journey during the integers and the average, rational, irrational, and transcendental numbers. The voyage keeps with a glance at a number 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 smooth common sense, together with an creation to linear good judgment and the logic-savvy Prolog language, the journey concludes with a travel of contemporary set concept and the advances and paradoxes of contemporary mechanical computing.
in the event that your highschool or university math classes left you greedy for the internal 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 (Pragmatic Programmers)
Not too long ago, an interesting fellow (and former coworker of mine) named Gregory Chaitin (1947–), showed that the irrational numbers are even worse than we thought. Not only are most numbers not rational, not only are most numbers not algebraic, most numbers cannot even be described in any way. It’s not a big surprise that they can’t be written down, because we already know that we can’t really write down any irrational number—the best we can do is write a good approximation. In fact, for most numbers, we can’t write a description, an equation, or a computer program to generate them.
Many histories will tell you that the pyramids of Egypt are built on the golden ratio or that the proportions of various features of Greek temples were built to fit the golden ratio. But like so much else about the lore of the golden ratio, it’s just apophenia—finding patterns where there are none. Let’s look at an example of the supposed connection between the pyramids and the golden ratio. 57. The Great Pyramid is famous for the precision of its construction, but on the scale of the pyramid, that error corresponds to about 6 feet.
The reals are defined by a tuple: (R, +, 0, ×, 1, ≤), where R is an infinite set, “+” and “×” are binary operators on members of R, “0” and “1” are special distinguished elements of R, and “≤” is a binary relation over members of R. The elements of the tuple must satisfy a set of axioms, called the field axioms. The real numbers are the canonical example of a mathematical structure called a field. A field is a fundamental structure used all over the place in mathematics; it’s basically the structure that you need to make algebra work.
Good Math: A Geek's Guide to the Beauty of Numbers, Logic, and Computation (Pragmatic Programmers) by Mark C. Chu-Carroll