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