IRRATIONAL NUMBERS - DEFINITION, SURDS, RATIONAILSING FACTOR, EXAMPLES, EXERCISE

Rational Numbers before Irrational Numbers,
if you have not already done so.

Need for Extension of Set of Rational Numbers

Can you find a rational number which when
multiplied by itself (square) gives 4 ?

You know the answer is 2.
i.e.4 = 2

What is the rational number whose square is 9⁄25 ?

As you know (3⁄5) x (3⁄5) = 9⁄25,
i.e.(9⁄25) = 3⁄5.

Example 1

Now I ask you another question,
Can you find a rational number whose square is 2 ?

In other words, Is √2 a rational number ?

If √2 is rational,
let it be equal to ab where a and b(≠0) are integers.
Let us say ab reduced to
the simplest form be pq,
which means p and q are integers which
do not have common factors other than 1.

2 = pq⇒ (√2)2 = (pq)2⇒ 2 = p2q2
p2 = 2q2p2 is divisible by 2.
p is divisible by 2.-----(i)

Since p is divisible by 2,
we can take p = 2r where r is an integer.

Substituting this in p2 = 2q2, we get
(2r)2 = 2q2⇒ 4r2 = 2q2q2 = 2r2q2 is divisible by 2.
q is divisible by 2.-----(ii)

From (i) and (ii), p and q have common factor 2.
But p and q do not have any common factor other than 1.

This contradiction has arisen because of our assumption
that √2 is a rational number.

So, we conclude that √2 is not a rational number.

Thus there is a need for extending the rational numbers.

Definition of Irrational Numbers

A number which can not be expressed in
the form ab, where a and b are integers,
b≠0 is called an Irrational Number.

We have seen every terminating or
repeating decimal is a rational number.

This gives us another way of defining Irrational Number.

Every number which when expressed in decimal form
is expressible as a non-terminating and
non-repeating decimal, is called an Irrational Number.

When a positive rational number is not a power of n, then
nth root of the number is irrational and is called a surd.or radical.

Examples :

3,91⁄3, 101⁄4 etc are surds.

All surds (or radicals) are irrational numbers.

We dealt with radicals of numbers in

Example 2 of Irrational Numbers

Prove that √3 is irrational.

Solution:

We first take the statement that √3 is rational
and prove that the statement is wrong.

Let √3 be rational,
Let it be equal to ab where a and b(≠0) are integers.
Let us say ab reduced to
the simplest form be pq,
which means p and q are integers which
do not have common factors other than 1.

3 = pq⇒ (√3)2 = (pq)2⇒ 3 = p2q2
p2 = 3q2p2 is divisible by 3.
p is divisible by 3.-----(i)

Since p is divisible by 3,
we can take p = 3r where r is an integer.

Substituting this in p2 = 3q2, we get
(3r)2 = 3q2⇒ 9r2 = 3q2q2 = 3r2q2 is divisible by 3.
q is divisible by 3.-----(ii)

From (i) and (ii), p and q have common factor 3.
But p and q do not have any common factor other than 1.

This contradiction has arisen because of our assumption
that √3 is a rational number.

So, we conclude that √2 is irrational number.

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Examples of irrational numbers which are not surds

(i) Example of irrational number which is not a surd is
π, which is defined as the ratio
of circumference and diameter of a circle.

The value of π = 3.1416.... which is a
non-terminating and non-repeating decimal.

We some times take π = 22⁄7 which
is only an approximate value of π.

(ii) Another Example of irrational number which is
not a surd is e, the base of natural logarithm.

e is defined as the sum of the infinite series
e = 1 + 1⁄1! + 1⁄2! + 1⁄3! + 1⁄4! + .....
where, ! stands for factorial
(n! = product of natural numbers from 1 upto n)

The value of e = 2.71828.... which is a
non-terminating and non-repeating decimal.

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Rationalising Factors

If the product of two irrational numbers is
a rational number, then each of the irrationals
is called a rationalising factor of the other.

Examples :
(i) √5 x √5= 5, so √5is the rationalising factor of √5.

(ii) (√3+ √2)x (√3- √2)= [(√3)2- (√2)2]= 3 - 2 = 1 which is rational.
So, (√3+ √2)and (√3- √2)are rationalising factors of each other.

(iii) (6 + √5)x (6 - √5)= 62 - (√5)2= 36 - 5 = 31 which is rational.
So, (6 + √5)and (6 - √5)are rationalising factors of each other.

Rationalisation of Denominator of a Fraction

Multiplying the numerator and denominator of
the fraction by the rationalising factor of
the denominator and then simplifying is called
Rationalisation of the Denominator of a Fraction.

Example 3 of Irrational Numbers

Rationalise the denominators of each of the following.

(i) 2⁄√6

(ii) 3⁄(3 + √7)

(iii) 16⁄(√11- √3)

Solution:
(i) 2⁄√6= (2 x √6)⁄(√6 x √6)
= (2 x √6)⁄6= √6⁄3. Ans.

(ii) 3⁄(8 + √7)= [3 x (3 - √7)]⁄[(3 + √7)x (3 - √7)]
= [3 x (3 - √7)]⁄[(32 - √7)2]
= [3 x (3 - √7)]⁄[(9 - 7)]= 3(3 - √7)⁄2. Ans.

(iii) 16⁄(√11- √3)= [16(√11+ √3)]⁄[(√11- √3)(√11+ √3)]
= [16(√11+ √3)]⁄[(√11)2- √3)2]
= [16(√11+ √3)]⁄(11-3)= [16(√11+ √3)]⁄8= 2(√11+ √3). Ans.

Exercise on Irrational Numbers

1. Prove that √5is irrational
2. Rationalise the denominators of each of the following.
1. 7⁄√14
2. 8⁄(5 - √23)
3. 27⁄(√10+ √7)

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Answers to Exercise on Irrational Numbers

1. Proof
2.
1. 14⁄2
2. 4(5 + √23)
3. 9(√10- √7)