1.1 Real numbers: algebra essentials  (Page 2/35)

The set of rational numbers    is written as $\left\{\frac{m}{n}|m\text{ and }n\text{ are integers and }n\ne 0\right\}.$ Notice from the definition that rational numbers are fractions (or quotients) containing integers in both the numerator and the denominator, and the denominator is never 0. We can also see that every natural number, whole number, and integer is a rational number with a denominator of 1.

Because they are fractions, any rational number can also be expressed in decimal form. Any rational number can be represented as either:

1. a terminating decimal: $\frac{15}{8}=1.875,$ or
2. a repeating decimal: $\frac{4}{11}=0.36363636\dots =0.\overline{36}$

We use a line drawn over the repeating block of numbers instead of writing the group multiple times.

Irrational numbers

At some point in the ancient past, someone discovered that not all numbers are rational numbers. A builder, for instance, may have found that the diagonal of a square with unit sides was not 2 or even $\frac{3}{2},$ but was something else. Or a garment maker might have observed that the ratio of the circumference to the diameter of a roll of cloth was a little bit more than 3, but still not a rational number. Such numbers are said to be irrational because they cannot be written as fractions. These numbers make up the set of irrational numbers    . Irrational numbers cannot be expressed as a fraction of two integers. It is impossible to describe this set of numbers by a single rule except to say that a number is irrational if it is not rational. So we write this as shown.