Question

prove that for any prime positive integer p,
$\sqrt{p}$
is an irrational number
















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Answer 1

Answer:

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Answer 2

TO PROVE :- For any prime positive integer p, √p is an irrational number.

Let us assume that √p is a rational number.

Then, there exist positive co-primes a and b such that :-

√p = a/b

p = a²/b²

b²p = a²

p divide a²

p divides a.

a = pc ( positive integer c. )

Now, b²p = a²

b²p = p²c²

b² = pc²

p divide b²

p divides b

Therefore, p/a and p/b

This contradicts the fact that a and b are co-primes.

HENCE , √p IS IRRATIONAL NUMBER !

\huge{\boxed{\sf{Hence\:Proved\:!}}}

HenceProved!