# CBSE / NCERT Maths Solutions Class 10 – Real Numbers – Irrational Theorems

## CBSE / NCERT Maths Solutions Class 10 – Real Numbers – Irrational Theorems

In this article, we will discuss Irrational Numbers from Class 10 Maths Chapter 1 Real Numbers.

Irrational Numbers

Irrational numbers which when expressed in decimal forms are expressible as non-terminating and nonrepeating decimals are known as irrational numbers.

0.1010010001…is a non-terminating and non-repeating decimal.

0.2020020002

0.12112111211112 is irrational.

0.232322223 is irrational.

22/7 is irrational.

Some Results on Irrationals

Theorem 1

Let P be a prime number and a be a positive integer. If p divides thenÂ show that p divides a.

Proof: – Let p be a prime number and a be a positive integer such that p divides

We all are aware that every positive integer can be expressed as the product of primes.

Let a = p1Â , p2 …….Â pn where Â  , ….. , are primesÂ p1,Â p2 …….Â pn
a2 = (p1, p2 …….Â pn) (p1, p2 …….Â pn)

a2 = (p12, p22 …….Â pn2)

P dividesÂ a2

P is one ofÂ Â p1, p2 …….Â pn

P divides a

Learn how to use Theorem 1 in solving questions, click CBSE Maths Class 10th for demos.

Theorem 2

Prove that âˆš2 is irrational.

Proof: – Let âˆš2 be rational and let its simplest form be a/b

Then a an b are integers having no common factors other than 1 and b is not equal 0

âˆš2=a/b

2=aÂ²/bÂ²

2bÂ²=aÂ²

2 divides a

2 is prime and divides aÂ²

Let a=2c for some integer c

Putting a=2c we get

2bÂ²=4cÂ²

bÂ²=2cÂ²

2 divide bÂ²

2 divide b

2 is prime and divides bÂ²

2 divide b

2 is a common factor of a and b

But this contradicts the fact that a and b have no common factor other than 1

The contradiction arises by assuming that âˆš2 is rational

Hence âˆš2 is irrational

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Theorem 3

Prove that âˆš3 is irrational

If possible âˆš3 be rational and let its simplest form be a/b

Then a and b are integers having no common factors other than 1 and b is not equal 0

âˆš3=a/b

3=aÂ²/bÂ² on squaring both sides

3bÂ²=aÂ²

3 divides a

3 is prime and 3 divides aÂ²

3 divides a

Let a=3c for some integer c

Putting a =3c we get

3bÂ²=9cÂ²

bÂ²=3cÂ²

3 divide bÂ²

3 divide b

3 is prime and 3 divide bÂ². Thus, 3 is a common factor of a and b

But this contradicts the fact that a and b have no common factors other than 1.

The contradiction arises by assuming âˆš3 is rational.

Hence âˆš3 is irrational.

In the same manner, we can prove that each of the numbers âˆš5, âˆš7 is irrational.

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May 1, 2018