By John L. Casti
Compliment for 5 Golden Rules
"Casti is among the nice technology writers of the Nineteen Nineties. . . . if you would like to have a good time whereas giving your mind a firstclass exercise session, then money this e-book out."-Keay Davidson within the San Francisco Examiner.
"Five Golden ideas is caviar for the inquiring reader. . . . there's pleasure right here in staring at the unfolding of those problematic and gorgeous options. Casti's reward is as a way to permit the nonmathematical reader percentage in his knowing of the great thing about a great theory." -Christian technology Monitor.
"Merely understanding in regards to the lifestyles of a few of those golden principles could spark new, interesting-maybe revolutionary-ideas on your mind." -Robert Matthews in New Scientist (United Kingdom).
"This e-book has meat! it's strong fare, nutrition for idea. 5 Golden ideas makes math much less forbidding and masses extra interesting." -Ben Bova within the Hartford Courant
"With this groundbreaking paintings, John Casti indicates himself to be a good arithmetic author. 5 Golden ideas is a dinner party of infrequent new delights all made completely comprehensible." -Rudy Rucker, writer of The Fourth Dimension.
"With the lucid informality for which he has develop into recognized, John Casti has written an attractive and articulate exam of 5 nice mathematical theorems and their myriad applications." -John Allen Paulos, writer of A Mathematician Reads the Newspaper.
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Additional resources for Five Golden Rules: Great Theories of 20th-Century Mathematics--and Why They Matter
Yes. 5 that is not a rational number but that, when squared, gives 2. As we have already said, 2 is deﬁned by the equation 2 ¥ 2 = 2, which is the mathematical way of saying that 2 is the positive number that squares to give 2. So 2 is a new type of number. Yes, new or different, but we have not proved this yet. Because it is not a rational number, it is called an irrational number. Not that there is anything unreasonable about it. It is so named because it cannot be expressed as the ratio of two integers in the way that the fractions are.
But the fractions we skipped 3 17 99 , , 2 12 70 on the other hand, are three overestimates of 2, which become progressively closer to 2. Right again. When these fractions are squared, they give 2 plus something positive. Note that this time the ﬁrst fraction is the largest and the last one the smallest. 23 24 CHAPTER 1 This is the opposite of the previous case. I think I see what you’re driving at. The underestimates are creeping up on 2 from the left while the overestimates are creeping back toward 2 from the right.
Yes, and what is wrong with this? Didn’t we say at the start that the numbers m and n have no factor in common? Other than 1, we did. So this is the contradiction then? It is. We began by assuming that 2= m n with the natural numbers m and n having no common factor other than 1. But the argument we have just given shows convincingly that this assumption forces m and n to be both even. Which means that m and n have 2 as a common factor. But this contradicts the original assumption that they have no factors in common except 1, which is a pain to have to keep mentioning.
Five Golden Rules: Great Theories of 20th-Century Mathematics--and Why They Matter by John L. Casti