Question

prove that 1/√2 is irrational
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Answer 1

Answer:

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To prove 1/√2 is irrational

Let us assume that √2 is irrational

1/√2 = p/q (where p and q are co prime)

q/p = √2

q = √2p

squaring both sides

q² = 2p² .....................(1)

By theorem

q is divisible by 2

∴ q = 2c ( where c is an integer)

putting the value of q in equitation 1

2p² = q² = 2c² =4c²

p² =4c² /2 = 2c²

p²/2 = c²

by theorem p is also divisible by 2

But p and q are coprime

This is a contradiction which has arisen due to our wrong assumption

∴1/√2 is irrational

Answer 2

Step-by-step explanation:

Assume that $\frac{1}{\sqrt{2} }$ is rational

∴ $\frac{1}{\sqrt{2} }$ =$\frac{p}{q}$, where p and q are integers and co-primes, q≠0

⇒  $\frac{\sqrt{2} }{1}= \frac{q}{p}$

⇒$\sqrt{2}=\frac{q}{p}$

We Know That $\frac{q}{p}$ i rational and $\sqrt{2}$ is irrational

This is a contradiction to the fact that $\sqrt{2}$ is irrational

hence our assumption is wrong

∴$\frac{1}{\sqrt{2} }$  is irrational