Question

prove that this is irrational

Answer 1

Let us assume to the contrary that √2 is rational.
hence
√2=r/s ( where S is not equal to 0)
Now,suppose r and s have a common factor other than 1
√2= a/b
b√2,= a
2b^2= a^2 (squaring both sides)
Hence 2 divides a^2 as well as a

Now let us consider
a=2C
Substituting the value of a in the earlier equation
2b^2= 4c^2
b^2 = 2c^2
Now, 2 divides b as well

But if √2 is rational a and b should be co primes.But in this case that have another factor other than 1.Hence a and b are not co primes.

This contradiction has arisen because of our wrong assumption that √2 is rational. Hence it is irrational

Now , let us assume that 3/√2 is rational
hence 3/√2 = a/b ( where a and b are rational numbers)
3b/a=√2...
we know that division of rational numbers always gives rational result.
But we know that √ 2 is irrational.
Hence our assumption was wrong .
Hence 3/√2 is irrational.

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