**What is a Surd?**

A surd is an irrational number.

We know √4 = 2, √9 = 3, √16 = 4,

But what is √2 =? and √5 =?

√2 and √5 are not rational numbers such as ** √4, √9 and √16 **and others which are all rational numbers.

Rational numbers include terminating and non-terminating decimals.

½ is a terminating decimal, 0.5 and

1/3 is a non-terminating decimal, 0.3333….

But √2 = 1.414215…….. can be expressed as neither a terminating nor a non-terminating decimal.

So, √2 is a surd.

Definition of a Surd:

The *n ^{th}* root of a number

Let ** a** be a rational number and

Then *n**√a* is called a Surd if it, i.e. *n**√a*is an irrational number.

*n√a*is spoken as *“n ^{th} root of a” *

*n ^{th}*

Examples of Surds:

*2**√2** or just **√2* is a surd.

3√9, 3√16, 4√25, 5√100 are all numbers whose given roots are not rational numbers. They are therefore Surds.

Note:

*2*or just √8 is irrational and therefore a Surd, but 3√8 = 2, a rational number, so, 3√8 is not a surd.*√8*

What is Not a Surd?

In the surd *n*** √a, a** is a rational number.

In** ***n**√a, **if a is irrational, then **n**√a **is not called a Surd. *

From this definition:

- √8 is a surd as 8 is a rational number and
is irrational, but*√8* - √ (2 + √3) is not a surd as (2 + √3) is not a rational number, and also
*2*=*√ (2√81)**2*= 3, a rational number. Therefore, 2√ (2√81) is not a surd.*√9*

Note:

- Every surd is irrational, but every irrational number is not a surd.

Order of a Surd:

In the surd *n*** √a, n **is called order of the surd.

**Examples: **

*2*is a surd of order 2*√3**4*is a surd of order 4,*√12**100*is a surd of order 100.*√3*

Note:

If the order of surd is 2, it is optionally dropped.

*2**√3 **is same as **√3. *

Types of Surds:

The following are various types of Surds:

- Pure Surd:

Surds such as ** √3, 3√9 **which are entirely irrational numbers are called pure surds

**Mixed Surd:**

Surds such as 2** √3, 3√9 **are called mixed surds as they containrational numbers such as 2, 3 and surds such as

**Compound Surd:**

Surds such as √2 + √3, √3 - √2 are called Compound Surds.

Compound surds are sum or difference of two other surds.

**Like Surds or Similar Surds:**

Surds that are different multiples of same surds are called similar surds.

Example:

√80 and √45, because

√80 = √ (5 × 16) = 4√5 and

√45 = √ (5 × 9) = 3 √5

**Conjugate Surds:**

Two surds of the form ** x + √y** and

Example:

2 + √3 and 2 - √3 are called conjugate surds.

Note:

The sum or difference of two conjugate surds is a rational number.

(2 + √3) + (2 - √3) = 4, a rational number.