Surds - Higher

Rational and irrational numbers

A number is described as rational if it can be written as a fraction (one integer divided by another integer). The decimal form of a rational number is either a terminating or a recurring decimal. Examples of rational numbers are 17, -3 and 12.4. Other examples of rational numbers are \(\frac{5}{4} = 1.25\) (terminating decimal) and \(\frac{2}{3} = 0.\dot{6}\) (recurring decimal).

A number is irrational if it cannot be written as a fraction. The decimal form of an irrational number does not terminate or recur. Examples of irrational numbers are \(\pi\) = 3.14159… and √2 = 1.414213...


A surd is an expression that includes a square root, cube root or other root symbol. Surds are used to write irrational numbers precisely – because the decimals of irrational numbers do not terminate or recur, they cannot be written exactly in decimal form.


This square has an area of 3 m2. Write down the exact length of the side of the square.

The area of the square = 3m^2

The length of the side is √3 m.

This answer is in surd form. It is irrational and it is said to be "in exact form". A decimal answer, such as 1.73 (2 decimal places), is not exact. Even 1.732050807568877 is not exact. When an answer is required in exact form, you must write it as a surd, ideally simplifying it if possible.