# Irrational numbers

Tags: irrational, numbers
P: 1,945
 Quote by adjacent 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 $A$ and $B$ 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 $A$ is matched with exactly one element of $B$, and vice-verse (technically, a one-to-one mapping). For example, the sets

$A = \{ cat, dog, pig \}$
$B = \{ red, yellow, blue\}$

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

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

$1 \leftrightarrow 0$
$2 \leftrightarrow -1$
$3 \leftrightarrow +1$
$4 \leftrightarrow -2$
$5 \leftrightarrow +2$
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.
 PF Gold P: 1,463 I see, thank you so much

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