## rational

A rational number is a number which can be represented as the quotient (a fraction) of two integers.

[AMG: I'm guessing that the name derives from the fact that a rational number is a ratio between two integers. Even if this isn't true, it's a great mnemonic. I wish somebody had bothered to say this in high school when I first learned the term, because as it was I was left with the impression that rational numbers were somehow better than irrational numbers. Negative pedagogy!]

There are an infinite number of such rationals: take the arrangement:

```   ...     ...     ...     ...     ...
5/1     5/2     5/3     5/4    (5/5)   ...
4/1    (4/2)    4/3    (4/4)    4/5    ...
3/1     3/2    (3/3)    3/4     3/5    ...
2/1    (2/2)    2/3    (2/4)    2/5    ...
1/1     1/2     1/3     1/4     1/5    ...
0/1    (0/2)   (0/3)   (0/4)   (0/5)   ...
-1/1    -1/2    -1/3    -1/4    -1/5    ...
-2/1   (-2/2)   -2/3   (-2/4)   -2/5    ...
-3/1    -3/2   (-3/3)   -3/4    -3/5    ...
-4/1   (-4/2)   -4/3   (-4/4)   -4/5    ...
-5/1    -5/2    -5/3    -5/4   (-5/5)   ...
...     ...     ...     ...     ...
```

etc., and you have clearly written down all the rational numbers that are possible. Quotients in parenthesis are equal to a quotient with smaller denominator, i.e., in a column further left.

You can also place these fractions in correspondence with the natural numbers, following for example a curve such as the one below.

```   3/1  --  3/2  --------  3/4
|                       |
2/1  ----------  2/3     |
|      |
1/1  --  1/2     1/3    1/4
|        |       |      |
0/1       |       |      |
|       |      |
-1/1  -- -1/2    -1/3   -1/4
|                |      |
-2/1  ---------  -2/3     |
|
-3/1  -- -3/2  -------  -3/4
|```

Rational number No. 0 is 0/1=0, rational number No. 1 is 1/1=1, rational number No. 2 is 1/2, rational number No. 3 is -1/2, rational number No. 4 is -1/1=-1, and so on. From this follows that the number of fractions is the same as the number of natural numbers (for every natural number there is a fraction).

In the same way (actually a bit simpler, since one doesn't have to skip the repeated numbers) one can construct a 1-1 correspondence between natural numbers and pairs of integers. This is the basis of the proof that infinity * infinity is still only infinity. (pow(2,infinity) is however strictly larger than infinity.)

A rational function is a function that can be represented as the quotient of two polynomials.