Surds

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  • August 2, 2023
9th Maths Ch#03 Algebraic Expression & Formulas Surds

Surds

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1. Definition:

A surd is an expression involving roots (√) that cannot be expressed as a whole number or a fraction. It usually appears in the form √(n), where “n” is a non-perfect square number.

2. Irrational Numbers:

Surds are related to irrational numbers, which are numbers that cannot be written as fractions. Irrational numbers go on infinitely without repeating decimals. The square root of any non-perfect square number is an irrational number and can be expressed as a surd.

3. Examples of Surds:

  • √2
  • √3
  • √5
  • √7

4. Simplifying Surds:

While we can’t express surds as simple whole numbers or fractions, we can simplify them to make them more manageable. To simplify a surd, look for perfect square factors in the number under the radical (√). Then, take out the square root of those perfect square factors.

Example:

Simplify √18:

√18 = √(9 × 2) = √9 × √2 = 3√2

5. Operations with Surds:

You can perform basic arithmetic operations (addition, subtraction, multiplication, division) with surds just like with regular numbers.

Example:

Simplify √8 + √32:

√8 + √32 = √(4 × 2) + √(16 × 2) = 2√2 + 4√2 = 6√2

6. Rationalizing the Denominator:

When a surd appears in the denominator of a fraction, it’s often desirable to eliminate the surd from the denominator. This process is called rationalizing the denominator.

Example:

Rationalize the denominator of 1 / √3:

Multiply both the numerator and denominator by √3:

(1 / √3) × (√3 / √3) = √3 / 3

7. Real-Life Applications:

Surds are essential in various fields, such as engineering, physics, and computer science. They help describe quantities and measurements that cannot be expressed as rational numbers.



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