# Home > Numbers and Pre-Algebra > Surds

# Surds

Surds, also called irrational roots, are examples of irrational numbers. We saw earlier that the real number system is made of rational and irrational numbers. Any rational number can be expressed as , where q 0, and p and q are integers.

## Rational numbers

A rational number can be an integer, fraction, mixed number, terminating or recurring decimals, such as , -2, 0, 1, 1 , 0.33333, 0.25. These numbers can all be written as      , and respectively.

## Irrational numbers

An irrational number cannot be written as a fraction. For example,   , e, 2 + , etc. An irrational number will be a decimal number that does not repeat (recur), nor terminate nor continues.

Surds are numerical expressions that involve irrational roots, and hence are irrational numbers. Examples of surds are –  , 24 , 7 + , etc.

## Rules for Simplification of Surds

Rule 1 : =  Let’s consider = 10  =  5 x 2

But can be expressed as  =  5 x 2  =  10

Here are some more examples of using the rule 1 for surds:

1. =  =  4 x 2  =  8

2. =  =  5 x =  5 Rule 2 : = Here 0, and x > 0 when x = 0

Let’s consider = =  2

Using Rule 2 of surds, the same expression can be written as = =  2

Rule 3 : =  x

Here and square are inverse of each other, hence they will cancel each other out, and the answer is simply the number under the square root symbol.

As an example, =  5