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The Annotated Turing: A Guided Tour Through Alan Turing's Historic Paper on Computability and the Turing Machine by Charles Petzold

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2  The Irrational and the Transcendental

We begin counting 1, 2, 3, and we can go on as long as we want. These are known as the counting numbers, the whole numbers, the cardinal numbers, the natural numbers, and they certainly seem natural and intuitive enough because the universe contains so many objects that we can count. Natural numbers were likely the first mathematical objects conceived by early humans. Some animals, too, it seems, have a concept of numbers, as long as the numbers don't get too large.

For many centuries, zero was not included among the natural numbers, and even now there is no firm consensus. (Text books on number theory usually tell you on the first page whether the author includes zero among the natural numbers.) On the other side of zero are the negative whole numbers. To refer to all the positive and negative whole numbers as well as zero, the word integer does just fine. The integers go off into infinity in two different directions:

images

To refer to only the positive whole numbers starting at 1, the term positive integers works well. For positive numbers starting with zero (that is, 0, 1, 2, 3, . . .) the term non-negative integers is unambiguous and not too wordy.

Rational numbers are numbers that can be expressed as ratios of integers, except that a denominator of zero is not allowed. For example,

is a rational number, also commonly written in the decimal ...

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