## 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 then**Â 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 = p_{1}Â , p_{2 …….}Â p_{n} where Â , ….. , are primesÂ p_{1,}Â p_{2 …….}Â p_{n}

a^{2} = (p_{1}, p_{2 …….}Â p_{n}) (p_{1}, p_{2 …….}Â p_{n})

a^{2} = (p_{1}^{2}, p_{22 …….}Â p_{n}^{2})

**P** dividesÂ a^{2}

**P** is one ofÂ Â p_{1}, p_{2 …….}Â p_{n}

**P** divides** a**

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__Theorem 2__

*Prove that **âˆš2 is irrational.*

Proof: – Let **âˆš****2** 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 **a****Â²**

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 **âˆš**2 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 **âˆš**3 is irrational.*

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

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