Question

Prove that √2 is irrational.

Class 10 - cbse

Answer 1

Let us assume √2 is rational number.
a rational number can be written into he form of p/q
√2=p/q
p=√2q
Squaring on both sides
p²=2q²__________(1)
.·.2 divides p² then 2 also divides p
.·.p is an even number
Let p=2a (definition of even number,'a' is positive integer)
Put p=2a in eq (1)
p²=2q²
(2a)²=2q²
4a²=2q²
q²=2a²
.·.2 divides q² then 2 also divides q
Both p and q have 2 as common factor.
But this contradicts the fact that p and q are co primes or integers.
Our supposition is false
.·.√2 is an irrational number.

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Answer 2

solution:-

here , it is Given √2 is irrational number.
Let √2 = p / q wher p,q are integers q ≠ 0
we also suppose that p / q is written in the simplest form
Now √2 = p / q⇒ 2 = p2 / q2 ⇒ 2q2 = p2
∴ 2q2 is divisible by 2
⇒ p2 is divisible by 2
⇒ p is divisible by 2
∴ let p = 2r
p2 = 4r2 ⇒ 2q2 = 4r2 ⇒ q2 = 2r2
∴ 2r2 is divisible by 2
∴ q2 is divisible by 2
∴ q is divisible by 2
∴p are q are divisible by 2 .
this contradicts our supposition that p/q is written in the simplest form
Hence, our supposition is wrong
∴ √2 is irrational number.