What are Surds

What are Surds



In this particular article, you will be able to know about the surds. You will also be able to understand what are surds and how the problems on surds are solved.

Before studying about surds, let us know about the rational numbers, irrational numbers and real numbers in brief.


Rational Numbers


A number which can be expressed in the form a/b where a, b Z and b  0 is called a Rational Number, where a and b are in the lowest form. That means a and b do not have any more common factor other than unity.


Example 1: Express 0.075 as a rational number in the lowest form.

Solution: 0.075 = 0.075  1000/1000 = 75/1000 = 3/40


Important Points


(i) When b = 1, the rational number = a/1  a, i.e., all integers belong to the set of rational numbers.

(ii) Since 0  0/b, where 0, b  Z and b  0 thus 0 is a rational number.

(iii) a/0 is not a rational number since its denominator = 0 and division by 0 is not defined.

Irrational Numbers


A non-terminating and non-recurring decimal is an irrational number.

e.g. (i) 0.23953356...  (ii) 2.5623045... and so on.


The square roots, cube roots etc. of natural numbers are irrational numbers if their exact values cannot be obtained.

e.g. √2, √3, √5, √7are all irrational numbers.


Numbers which are not rational that means if any number that cannot be expressed in the form a/b, where a, b are integers, b > 0, a and b have no common factor (except 1) is called an irrational number but can be plotted on the Number Line.


Real Numbers (R)


The Real Numbers constitute the union of the set of rational numbers and set of irrational numbers. Thus, the totality of all rational and all irrational numbers form the set of all real numbers.


Important Points


1. Every real number is either rational or irrational.

2. To every real number there corresponds a point on the number line and to every point on the number line there corresponds a real number.


What are Surds?


When a root of a positive real number cannot be exactly determined, that root is called a Surd.

Thus, √2, √3, √5, √6 are all surds.

Positive nth root of a real number


Let a be a real number and n be a positive integer. Then a number which when raised to the power n gives a is called the nth root of a and it is written as na or a1/n.


Thus, nth root of a real number a is a real number b. Such that bn a. The real number b is denoted by a1/n or na. The cube root of 3 is the real number whose cube is 3. The cube root of 3 is denoted by the symbol 31/3.

Thus, if a is a rational number and n is a positive integer such that the nth root of a, i.e. a1/n or na is an irrational number, then a1/n or na is called a Surd or radical of order n and a is called the radicand.


Important Points


(i) a is a rational number and

(ii) na is an irrational number.

(iii) If n is a positive integer and a is a real number, then na is not a Surd if a is irrational or na is rational.


Types of Surds


Pure Surds


Surds having no rational co-efficient except unity are called pure Surds. Thus, √5, 3√3, 4√8 are all pure Surds.


Mixed Surds


A Surd having a rational co-efficient other than unity is called a mixed Surd. Thus, 2√3, 53√3, 54√8 are all mixed Surds.


 Similar Surds


Surds having the same irrational factor are called similar or like Surds.

For example, √3, 2√3, 5√3, 7√3 are all similar surds.


Solved Examples


Example 1: Rationalise the denominator of 2/√3.

Solution: The rationalising factor of denominator is √3.

2/√3 = 2√3/√3.√3 = 2√3/3


Example 2: Rationalise the denominator of 8/√2.

Solution: The rationalising factor of denominator is √2.

8/√2 = 8√2/√2.√2 = 8√2/2 = 4√2


Example 3: Simplify: √50 – √98 + √162.

Solution: √50 – √98 + √162 = √(25 × 2) – √(49 × 2) + √(81 × 2)

                                                 = 5√2 – 7√2 + 9√2

                                                 = 7√2

Related Topics:

 What is an Addend in Maths

Minuend and subtrahend

Multiplicand and multiplier

Dividend, divisor, quotient and remainder

Natural numbers

Whole numbers

Properties of rational numbers

Are all integers rational numbers?

Find five rational numbers between 3/5 and 4/5

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