Common Number Sets

Integers

"Integer" is just a posh word for a whole number. An integer can be positive, negative or zero. That's all it is.

Integers are whole numbers. Integers can be positive, negative or zero.

A rational number is a number that can be written in the form \(\frac{p}{q}\) where \({p}\) and \({q}\) are both integers (whole numbers) and \({q} \ne {0}\).

A decimal which terminates (i.e. one which does not go on forever) can easily be written in the form \(\frac{p}{q}\), for example 1.23 can be written as \(\frac{{1.23}\times{100}}{100} = \frac{123}{100}\),  1.234 can be written as \(\frac{{1.234}\times{1000}}{1000}=\frac{1234}{1000}\). Surprisingly, recurring decimals can ALWAYS be written in the form \(\frac{p}{q}\) - click here to see how

All integers are rational numbers since any integer, \(N\), they can easily be written in the form \(\frac{p}{q}\) if \({p}={N}\times{q}\), for example 12 can be written \(\frac{{12}\times{10}}{10}=\frac{120}{10}=12\) (not a very efficient way of writing 12 but it is mathematically true).

An irrational number is a number that can NOT be written in the form \(\frac{p}{q}\) where \({p}\) and \({q}\) are both integers (whole numbers) and \({q} \ne {0}\).

π is an irrational number. Although it is often written as 3.142 or \(\frac{22}{7}\) these are only approximations; the exact value cannot be written as a fraction or decimal. Similarly \(\sqrt{2}\) and \(\sqrt{3}\) are irrational numbers since it is not possible to write the exact value as a fraction or decimal. Your calculator will give you a decimal value for \(\sqrt{2}\) but this only an approximation to the real value.

Not all square roots are irrational, for example \(\sqrt{4}\) = 2 which is both rational and an integer.

There are other number sets but you don't need to know about those for GCSE Maths.