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

Please study
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,
the answer is 3⁄5.
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 a⁄b where a and b(≠0) are integers.
Let us say a⁄b reduced to
the simplest form be p⁄q,
which means p and q are integers which
do not have common factors other than 1.

√2 = p⁄q ⇒ (√2)² = (p⁄q)² ⇒ 2 = p²⁄q²
⇒p² = 2q² ⇒ p² 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 = 2q² ⇒q² = 2r² ⇒ q² 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.

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Definition of
Irrational Numbers

A number which can not be expressed in
the form a⁄b, 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, 9^1⁄3, 10^1⁄4 etc are surds.

All surds (or radicals) are irrational numbers.

We dealt with radicals of numbers in

Rational Exponents

and

Fractional Exponents

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

√3 = p⁄q ⇒ (√3)² = (p⁄q)² ⇒ 3 = p²⁄q²
⇒p² = 3q² ⇒ p² 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 = 3q² ⇒q² = 3r² ⇒ q² 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.

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

Answers to Exercise on Irrational Numbers

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