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 (2dp), 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.