Question

Prove that 1/√12
are irrational
​

Answer 1

assume √2 as rational number

the

√ 2 can be written as p/q ( where p&q are co- prime numbers and q is not equal to zero)

now square both sides

(√2)(√2)= (pxp)/(qxq)

2= p^2/q^2

2q^2=p^2 -----(I)

, p^2 is divisible by 2 there fire p is divisible by 2

hence , p=2m

now substitute p=2m in (I)

we get ,

2q^2=(2m)^2

q^2=4m^2/2

q^2=2m^2

q ^2 is divisible by 2 therefore q is divisible by 2

therefore q=2n

but p &q are co - prime numbers therefore are assumption is wrong

√2 = irrational number

now 1/√2 ,

any number divided by an irrational number is irrational number ,

therefore 1/√2 is an irrational number

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