Surds Definition

A surd is a special kind of number that can’t be written exactly as a fraction or a whole number. To show this unique number, we use a symbol that looks like a checkmark with a line (√).

Surds are used to express numbers with precision. Irrational numbers, when represented in decimal form, lack a finite or repeating pattern. Hence, we retain these numbers as surds, capturing their exact values without approximation. A surd expression can include a square root, cube root, or other root symbol.

Basic Rules of Surds

1.1     Manipulation of Surds

A rational number is a number that can be expressed in the form ^a ⁄ _b , where a and b are integers.

An irrational number cannot be expressed as a ratio of two integers. When written as a decimal, irrational numbers are non-terminating and non-recurring. For instance, π, e (also known as Euler’s number), √2 and −√11  are irrational numbers.

A surd is an irrational number, written with a radical (√ ) sign (also known as root sign). When the square root of an integer when evaluated is an irrational decimal, for example,√2 = 1.414213, a surd will be used to represent the decimal in exact form. Other examples include √3, √5 and √10 .

Note: The square roots of perfect squares such as √9 and √25  are not considered as surds since the result is not an irrational number.

For a, b > 0, the following properties will be used to simplify surds.

We also manipulate surds in ways that are similar to algebraic terms.

1.2     Rationalisation

When a surd appears in the denominator of a fraction, for example, ¹ ⁄ _√2 , we can rewrite the fraction without the surd appearing in the denominator. This process is called rationalisation of the denominator.

1.3    How to Solve Surds in an Equation

1.3.1  When the equation involves the root of an unknown, we can remove the root by
squaring both sides of the equation. Note that this technique may give solutions
that do not satisfy the original equation or context, so reject solutions.

1.3.2   When an equation with unknown constants also has rational and irrational terms, we can use the following property of surds to find the unknown constants.

             If a + b√k = c + d√k , where a, b, c and d are rational and √k is irrational,
             then a = c and b =  d.

For more surds example problems and guidance, check out our free math eBooks and test papers for secondary school students!

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