Square Roots and Surds

Start off by referring to Key Skills

Square Roots and Surd Form

A statement like 'the square root of 5' would be represented in mathematical symbols as

Some square roots are integers, for example

Whereas some are more 'complicated'

If a square root is not actually an integer, then if an attempt is made to represent the square root as a decimal number, then in practice it would need to be approximated to a certain number of decimal places.

In reality, every number actually has two square roots. For example, the square roots of 16 are

\[ 4 \ \ \mbox{and} \ \ -4 \] .

It is worth noting in passing that just as every number has two square roots, so every number has three square roots, and four fourth roots, and so on. To understand this fully requires hanging on to A-Level where the subject of Complex Numbers is introduced - the number 1 obviously has one cube root (1 itself) but it also has two additional complex roots (i.e. numbers that don't exist!).

However, for the moment, you just need to know that every real, positive number has two square roots - both of the same magnitude but one positive and the other negative. I should stress the expression 'postive number' in the previous sentence because the square root of a minus number has no meaning in the world of real numbers.

The name surd is given to expressions like


Irrational Numbers

Rational numbers are numbers which can be represented as a fraction. These includes the integers because a number like 8 can always be represented as a fraction, i.e. \[ \frac{8}{1} \]

Irrational numbers cannot be expressed as a fraction, and square roots often fall into this category. For example

cannot be expressed as a fraction.

It is only recurring decimals that can be represented as fractions, and therefore represent rational numbers. For example

\[ 0.222222222222222222.......... = \frac{2}{9} \]

Likewise, for a more 'complicated' recurring fraction

\[ 0.57142857142857142857142857142857...... = \frac{4}{7} \]

Decimals like 0.5 and 0.125 represent rational numbers because they are really

\[ 0.500000000......, \ \ \ 0.12500000000......... \]

i.e. recurring decimals, with recurring zeros.

It is worth noting that a famous irrational number is


which has currently been calculated to an enormous number of places of decimals without any recurrence patterns occuring.

Manipulating Surds

The basic rule for Multiplication is

and for Division

These rules can be used to present a surd form in its 'simplest form', as in this example

I have included the multiplication signs here to aid initial understanding but oin practice they would tend to be left out.



a) 72 b) 216

Rationalising Denominators


The problem with this expression is that we normally like the denominator to be an integer.

In order to transform the above expression into the desired form, we can multiply top and bottom by the same number, in this case the number in the denominator. We are using two rules here

  • The fact that a fraction can be manipulated at will (and it will still represent the same number) as long as you either multiply top and bottom by the same number, or divide top and bottom by the same number
  • A square root multiplied by a square root will, by definition, give you the number which was being squared rooted originally, e.g.

So, for our original example, multiply top and bottom by the denominator, i.e.

to produce.

This is the same number as before, but in a 'nicer' form.