Question

Prove that `2-1` is an irrational number.​

Answer 1

Answer:

hey mate,  here is your answer

Step-by-step explanation:

i am writing two answers both of them are correct

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!!

or

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.

thank you