## Surds

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.

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 nth root of a number a is denoted as n√a (also as a1/n)

Let a be a rational number and n a positive integer.

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

n√ais spoken as “nth root of a”

nthroot of a number a is a surd, if it is an irrational number.

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

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

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

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 √3 and √9

• 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 x - √y are called conjugate surds.

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.