Beware of Surds

Surds are right at the start of C1.


They are easy to forget.

As I found out tonight, trying a C1 practice paper.

Surds are basically this. They are an exact representation of irrational numbers, where a fraction or a decimal will not do. Normally we have rational numbers, like 1, 2, 3.5, 4.54, and so on. These numbers can be expressed as a fraction or as a decimal. But sometimes we get numbers like √2 or π which would go on forever if we tried to express them as a decimal (3.14159265 is only the start of π) so a decimal or a fraction would only ever give us an approximation.

1.41.... will never really describe √2 exactly, because it has an infinite number of digits after the decimal point. Apparently there is a proof of this but you don't need to prove it for C1.

So we leave these numbers expressed as a square root, or combination of square root and integer or fraction, and we call these surds (because 3 or 5 times the irrational square root of 2 will be irrational itself, usually). They enable us to do exact calculations of numbers which cannot be represented exactly as fractions or decimals.

There are a few rules of surds.

1) You can only add or subtract like surds. ie 3√7 - 2√7 = √7

2) You can multiply surds but there are a couple of rules to remember:

√5 x √6 = √30

3 x √5 = 3√5

5√3 x √4 = 5√12

3) Sometimes a surd will reveal itself to be a rational number: watch out for this:

√3 x √3 = 3

3√4 x 2√4 = 6√16 = 24

4) surds multiply out of brackets just like anything else does.

5)The trickiest thing about surds is fractions with a surd as a denominator. You need to get rid of the surd.

If you think about it, there is a way to make rational numbers out of surds that uses a rule of brackets. (a-b)(a+b) always gives you a²-b².

So if you have 3+√5 as the denominator, multiply it by 3-√5 and you rationalise the denominator into 9-5 = 4.

Of course whatever you times the denominator by you need to times the numerator by.

This means you will often end up with your answer being a surd, but you will have eliminated the irrational denominator and so you will have been able to simplify your fraction.


I know I'm meant to be doing a series on geometry of the line but I keep getting distracted...