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.
See also Playing with rationals