How to prove that 3 is an irrational number?

Let 3 = p q , p , q relatively prime. 3 = p 2 q 2 , so 3 divides p and so on.

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Armeninilu

2022-06-27

How to prove that

\sqrt{3}

is an irrational number?

Leland Ochoa

Beginner2022-06-28Added 25 answers

Let

\sqrt{3} = \frac{p}{q}, p, q

relatively prime.

3 = \frac{p^{2}}{q^{2}}

, so

3

divides

p

and so on.

aligass2004yi

Beginner2022-06-29Added 7 answers

Let

\sqrt{3} = \frac{a}{b}

, where a and b have no common factors besides

1

3 b^{2} = a^{2}

a^{2}

is a multiple of

3

, and hence a should be a multiple of

3

. Let

a = 3 k

, then

b^{2} = 3 k^{2}

, and b must also be a multiple of three. You will arrive at a contradiction to the earlier assumption that

a

and

b

have no common factors.

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