Question

Prove that √2 is an irrational number
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Answer 1

√2 = 1.414....... (Non-terminating non-repeating decimal expansion)

Therefore, √2 is an irrational number.

Answer 2

Answer:

√2 is irrational

Step-by-step explanation:

Let us assume that,√2 is rational.

√2=a/b,where a and b are co-primes as their H.C.F=1

So,

√2=a/b

Squaring on both sides,

(√2)²=(a/b)²

      2=a²/b²

  2b²=a²→→→→→1

Here,2 divides a² also(if a prime number divides the square of a positive integer,then it divides the integer itself)

Let,

a=2c

Squaring on both sides,

a²=(2ac)²⇒a²=4c²→→→→→2

Substitute equation 1 in equation 2

2b²=4c²

 b²=2c²⇒2c²=b²

Here, a and b are divisible by 2 also.But our assumption that their H.C.F is 1 is being contradicted.Therefore,our assumption that √2 is rational is wrong.Thus,it is irrational.

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