Question

prove that 2 rote 3 -1 is an irrational no.

Answer 1

x=27.142857142....
multipliying by 1000000 on both sides
1000000x=12.142857.1428
subtracting eq 1 by eq 2
999999x=27142857-27.142857
x=27142857  /  999999
hence the prime factorisation of the denominator contains factor more than 2 or 5

Answer 2

Actually we need to prove that 2√3-1 is a irrational no..
so for that let us take that 2√3 -1is a rational  num as a contradiction.
so let 2√3-1 be equal to a/b(where a,b are co primes)
we took it becoz a rational num ill be in the form a/b
so,2√3-1=a/b
2√3=a/b+1
√3=a+b/2b
here a+b/2b is a ratinal num but √3 is a irrational num.
so our assumption that 2√3-1 is rational num is a contradiction.

so,from this we can say that 2√3-1 is a irrational num.
.............................................hope it helped u:)))))