As described in Review of past work, a number is a way of representing quantity. The numbers that will be used in high school are all real numbers, but there are many different ways of writing any single real number.

This chapter describes rational numbers.

Figure 2

The term whole number does not have a consistent definition. Various authors use it in many different ways. We use the following definitions:

The following numbers are all rational numbers.

You can see that all denominators and all numerators are integers.

Definition 1: Rational Number

A rational number is any number which can be written as:

a b a b

(2)

where aa and bb are integers and b≠0b≠0.

Note that because we can write a - b a - b as -a b -a b (in other words, one can always find an equivalent rational expression where b>0b>0) mathematicians typically define rational numbers not as both aa and bb being integers, but rather that aa is an integer and bb is a natural number. This avoids having to worry about zero in the denominator.

This means that all integers are rational numbers, because they can be written with a denominator of 1.

Therefore

are not examples of rational numbers, because in each case, either the numerator or the denominator is not an integer.

A number may not be written as an integer divided by another integer, but may still be a rational number. This is because the results may be expressed as an integer divided by an integer. The rule is, if a number can be written as a fraction of integers, it is rational even if it can also be written in another way as well. Here are two examples that might not look like rational numbers at first glance but are because there are equivalent forms that are expressed as an integer divided by another integer:

All integers and fractions with integer numerators and denominators are rational numbers. There are two more forms of rational numbers.

You can write a rational number as a decimal number. Two types of decimal numbers can be written as rational numbers:

For example, the rational number 5656 can be written in decimal notation as 0,83˙0,83˙ and similarly, the decimal number 0,25 can be written as a rational number as 1414.

A decimal number has an integer part and a fractional part. For example 10,58910,589 has an integer part of 10 and a fractional part of 0,5890,589 because 10+0,589=10,58910+0,589=10,589. The fractional part can be written as a rational number, i.e. with a numerator and a denominator that are integers.

Each digit after the decimal point is a fraction with a denominator in increasing powers of ten. For example:

This means that:

When the decimal is a repeating decimal, a bit more work is needed to write the fractional part of the decimal number as a fraction. We will explain by means of an example.

If we wish to write 0,3˙0,3˙ in the form abab (where aa and bb are integers) then we would proceed as follows

And another example would be to write 5, 4˙ 3˙ 2˙ 5, 4˙ 3˙ 2˙ as a rational fraction.

For the first example, the decimal was multiplied by 10 and for the second example, the decimal was multiplied by 1000. This is because for the first example there was only one digit (i.e. 3) recurring, while for the second example there were three digits (i.e. 432) recurring.

In general, if you have one digit recurring, then multiply by 10. If you have two digits recurring, then multiply by 100. If you have three digits recurring, then multiply by 1000. Can you spot the pattern yet?

The number of zeros is the same as the number of recurring digits.

Not all decimal numbers can be written as rational numbers. Why? Irrational decimal numbers like 2=1,4142135...2=1,4142135... cannot be written with an integer numerator and denominator, because they do not have a pattern of recurring digits. However, when possible, you should try to use rational numbers or fractions instead of decimals.