**Summary**

**Rational Number:**A number that can be written as a fraction, where both the numerator and the denominator are whole numbers. E.g 3/4**Irrational Number:**They cannot be written as a fraction. They can only be written as decimals numbers. E.g which is equal to 1.7320……- A
**surd**is a number that’s written with the (under root) sign. Remember the answer has to be irrational. - Rules for using surds:

i)

ii)

iii)

Before we move on to surds, let us recall what rational and irrational numbers are.

#### Rational Number

A rational number is a number that can be written as a fraction, where both the numerator and the denominator are whole numbers.

Every whole number is a rational number, because any whole number can be written as a fraction i.e number 8 is a rational number because it can be written as the fraction 8/1.

Some other examples of rational numbers include:

- 3/4
- 25, 144, 808/56, 103, 492
- 1/2 etc

#### Irrational Numbers

Numbers that are not rational are considered irrational. Irrational numbers cannot be written as a fraction. They can only be written as decimals numbers.

Moreover, they have endless digits to the right of the decimal point.

Some examples of irrational numbers include:

- The value of which is equal to 3.1415…….
- which is equal to 1.7320……

is not a irrational number as it is equal to 2 ( a whole number).

Now that we have revised what rational and irrational numbers are, let’s move on to Surds.

#### What are surds?

A surd is a number that’s written with the (under root) sign. It can be a square root , cube root or other *n* roots.

These are used when an exact answer is required i.e if the answer is , we can leave it like this instead of solving this on a calculator to give an answer in decimals after rounding off like 1.732.

No matter how many decimal places we use, we can never get exactly 3 after squaring 1.7320….. We will end up getting something like 2.99999 which is not very accurate.

Hence, we say that Irrational root of a rational positive number is called a surd. I.e or . Remember the answer has to be irrational, for instance and are not surds as they result in a rational number: and .

#### Rules for using Surds

#### Example #1

Q. Simplify

*Solution:*

Q. Find

*Solution:*

Q. Rationalise the denominator of

*Solution:*

Firstly multiply the top and bottom by the denominator, but change the sign in front of the surd.

**Ans: **

##### Reference

- https://www.factmonster.com/math/numbers/rational-and-irrational-numbers
- CGP Edexcel AS-Level Mathematics , Complete Revision and Practice.