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.
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.
Rationalise the denominators of each of the following.
8⁄(5 - √23)
Progressive Learning of Math : Irrational Numbers
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These also provide teachers and parents with lessons on how to work with the child on the concepts.
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