Rational Numbers

If one takes two integers $a$ and $b$ and divides $a$ by $b$ to form $\frac{a}{b}$, the result will often not be an integer. For example, $1/2$ is not an integer, nor is $1/3, 1/4, 1/5...$, nor $2/3,4/7,129/37$ and so on, although it may be; $4/2 = 2$ is an integer, for example. These numbers are all the ratios of two integers and are hence called rational numbers^2.3 .

The rational numbers do close under division, with the single exception of division by zero, which is undefined. They can therefore be factored and form a division algebra.

Rational numbers in physics are generally expressed in base 10, which simply extends the representation for integers above with negative powers of ten: $3012.694 = 3\times 10^3 + 0\times 10^2 + 1\times 10^1 + 2\times 10^0 + 6\times 10^{-1} + 9\times 10^{-2} + 4\times 10^{-3}$.

Rational numbers expressed in a base have an interesting property. Dividing one out produces a finite number of non-repeating digits, followed by a finite sequence of digits that repeats cyclically forever, for example $1/6 = 0.1666...$. Note that finite precision decimal numbers are precisely those that are terminated with an infinite string of $0$ digits - this is the special class of numbers over which we do much of our arithmetic especially on digital computers.

If all rational numbers have digit strings that eventually cyclically repeat, what about all numbers whose digit strings do not cyclically repeat? These numbers are not rational, and cannot be expressed as the ratio of two finite integers, since the integers would require an infinite number of digits and hence would be infinite themselves.