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 $2/3,4/(-7) = -4/7,129/37$ and so on. These numbers are all the ratios of two integers and are hence called rational numbers⁶ .

Rational numbers when expressed in a base⁷ e.g. base 10 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:

$$\frac{1}{3} = 0.3333...$$ (8)

or

$$\frac{11}{7} = 1.571428 571428 571428...$$ (9)

Note that finite precision decimal numbers are precisely those that are terminated with an infinite string of the digit 0 . If we keep numbers only to the hundredths place, e.g. 4.17, -17.01, 3.14, the assumption is that all the rest of the digits in the rational number are 0 - 3.14000...

It may not be the case that those digits really are zero. We will often be multiplying by $1/3 \approx 0.33$ to get an approximate answer to all of the precision we need in a problem. In any event, we generally cannot handle an infinite number of digits, repeating or not, in our arithmetical operations, so truncated, base two or base ten, rational numbers are the special class of numbers over which we do much of our arithmetic, whether it be done with paper and pencil, slide rule, calculator, or computer.

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.