Question

Prove that 2√2-1 is an irrational number

Answer 1

Hey.. I think this can be ur answer!!

To prove: 2root2 - 1 irrational.

Let us assume that 2root2 - 1 is rational so,
2root2 - 1 = a/b (a and b are co prime)
2root2 = a/b+1 = a+b/b

Now,
Root2 = a+b/2b

BUT we know that root2 is irrational
So 2root2 - 1 is also irrational

Hence, our assumption was wrong.

Hence proved!!

Hope it helps you dear!!

Answer 2

let be 2√2-1 is an rational number.
then their exist a and b positive integer.
now

2√2-1 = a/b ( where a and b are co primes)

2√2=a/b +1
2√2= a+b/b
√2= a+b/2b ( by transposing LHS to rhs )

but we know √2 is an irrational no.
and a and b are positive integer so,
a+b/2b is an rational no.

this is a contradiction
because LHS is always equal to RHS.

therefore our assumption is wrong.

hence , 2√2-1 is an irrational number.