In a very simple way, A surd is a square root which cannot be reduced to a whole number. For example, is not a surd, as the answer is a whole number. But is not a whole number. You could use a calculator to find that but instead of this we often leave our answers in the square root form, as a surd.

Another definition: When we can’t simplify a number to remove a square root (or cube root etc) then it is a *surd*.

Positive Root: The positive root of a is denoted by or .

Formal Definition: If a is a positive rational number and n is a positive integer, such that the n^{th} root of a i.e. or is an irrational number, then or is called a surd.

Examples:

- or is a surd of order
- or is not a surd because it is equal to which is not an irrational number.
- . Clearly is a positive rational number such that is an irrational number. Therefore is a surd of the order
- Similarly, . Therefore is not a surd.

Type of Surds

*Quadratic Surds:* Surds of order .

Examples: etc.

*Cubic Surds:* Surds of order .

Examples: etc.

*Bioquadratic Surds:* Surds of order .

Examples: etc.

*Pure Surds:* A surd having no rational factor other than unity is called a pure surd.

Examples:

*Mixed Surds:* Surds having a rational factor other than unity is called mixed surd. Example

Laws of Surds:

*First Law:* If is a positive rational number and is a positive number then

*Second Law:* If is a positive rational number and is a positive number then

*Third Law:* If is a positive rational number and is a positive number then

*Fourth Law:* If is a positive rational number and are positive numbers then

*Fifth Law:* If is a positive rational number and are positive numbers then

Another way of classifying surds:

*Monomial Surds:* A surd containing only one term is called a monomial surd.

Example:

*Binomial Surds:* An expression consisting of the sum or difference of two monomial surds or sum of a monomial surd and a rational number is called a binomial surd.

Examples:

*Trinomial Surds: *An expression consisting of three terms of which at least two are monomial surds is called a trinomial surd.

Examples:

Rationalising Factor: If the product of two surds is a rational number, then each one of them is called the rationalizing factor of the other.

Example: is a rationalising factor of because which is a rational number.

Similarly, is a rationalising factor of because which is a rational number.

Conjugate Surds: Two binomial surds which differ only in sign between the terms connecting them are called conjugate surds. and are conjugate surds. Also and are conjugate surds.