Rational and Irrational numbers (including surds)

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  • Rational numbers

    A rational number is any number that can be expressed in the form \(\dfrac{p}{q}\) where \(p\) and \(q\) are integers. Rational numbers include all integers, since they can easily be expressed as fractions by suing a denominator of 1 e.g. \(3= \dfrac{3}{1}\). Rational numbers also include all fractions which have integer numerators and denominators, by definition. All terminating and recurring decimals can be expressed as fractions with integer numerators and denominators, and so are rational.

    Irrational numbers

    An irrational number is simply a number that is not rational (see above). That is, it cannot be expressed in the form \(\dfrac{p}{q}\) where \(p\) and \(q\) are both integers. Irrational numbers written as decimals would go on for ever without a recurring pattern. Surds (see below) are irrational, but there are also irrational numbers that are not surds. For example, \(\pi\) is irrational but not a surd. It is in fact an example of a transcendental number.


    A surd is an irrational root. A root (whether a square root, cube root or higher root) of any integer will either be an integer or a surd. The root will only be an integer if the original integer was a perfect power. For example, consider square roots: the square root of an integer n will only be an integer if n itself is a perfect square number such as 1, 4, 9, 16, etc. The square root of any other integers will always be surds. So for example, \(\sqrt{7}\) is a surd, and as it is irrational, its decimal expansion would go on forever without a recurring pattern. Note that square roots of decimals or fractions are not always surds. For example, \(\sqrt{6.25}=2.5\) which is rational and therefore not a surd. \(\sqrt{\dfrac{1}{4}}=\dfrac{1}{2}\) which is also rational and therefore not a surd.

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