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Irrational numbers

by adjacent
Tags: irrational, numbers
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stevendaryl
#19
Jun14-14, 12:35 PM
Sci Advisor
P: 2,097
Quote Quote by adjacent View Post
So funny. But I asked you to count it :p and ##\pi !## returns a math error


You will have to do that forever then.
In mathematics, there are different "sizes" of infinity. There are infinitely many integers: 0, 1, 2, ... It never stops, so it's infinite. There are also infinitely many real numbers. But there is a sense in which there are more real numbers than there are integers. The sense is this:

Two sets [itex]A[/itex] and [itex]B[/itex] are said to be "the same size" (technically, the same cardinality) if you can set up a correspondence between the two sets, so that every element of [itex]A[/itex] is matched with exactly one element of [itex]B[/itex], and vice-verse (technically, a one-to-one mapping). For example, the sets

[itex]A = \{ cat, dog, pig \}[/itex]
[itex]B = \{ red, yellow, blue\}[/itex]

are the same size because they can be put into correspondence many different ways, but here's one: [itex]cat \leftrightarrow red,\ dog \leftrightarrow yellow,\ pig \leftrightarrow blue[/itex]

Infinite sets can be put into a one-to-one correspondence, also. For example, the set [itex]A = [/itex] the positive integers and the set [itex]B = [/itex] all integers:

[itex]1 \leftrightarrow 0[/itex]
[itex]2 \leftrightarrow -1[/itex]
[itex]3 \leftrightarrow +1[/itex]
[itex]4 \leftrightarrow -2[/itex]
[itex]5 \leftrightarrow +2[/itex]
etc.

You can also set up a one-to-one correspondence between the integers and the rationals. That's a little harder to describe, but it can be done.

Any set that can be put into a one-to-one correspondence with the positive integers is called a "countable" set.

Some sets are not countable. The easiest example is the set of reals. There is no way to set up a one-to-one correspondence between the positive integers and the reals.
adjacent
#20
Jun15-14, 03:34 AM
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I see, thank you so much


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