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, √7are 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 n√a
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 n√a.
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 n√a is an irrational number, then a1/n
or n√a 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) n√a is an
irrational number.
(iii) If n is a positive integer and a
is a real number, then n√a is not a Surd if a
is irrational or n√a 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