How do you prove that square root 15 is irrational?

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How do you prove that square root 15 is irrational?

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Answer 1 · 2015-09-20T14:21:34

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Explanation:

This proof uses the unique prime factorisation theorem that every positive integer has a unique factorisation as a product of positive prime numbers.

Suppose $sqrt(15) = p/q$ for some $p, q in NN$ . and that $p$ and $q$ are the smallest such positive integers.

Then $p^2 = 15 q^2$

The right hand side has factors of $3$ and $5$ , so $p^2$ must be divisible by $3$ and by $5$ . By the unique prime factorisation theorem, $p$ must also be divisible by $3$ and $5$ .

So $p = 3 * 5 * k = 15k$ for some $k in NN$ .

Then we have:

$15 q^2 = p^2 = (15k)^2 = 15*(15 k^2)$

Divide both ends by $15$ to find:

$q^2 = 15 k^2$

So $15 = q^2 / k^2$ and $sqrt(15) = q/k$

Now $k < q < p$ contradicting our assertion that $p, q$ is the smallest pair of values such that $sqrt(15) = p/q$ .

So our initial assertion was false and there is no such pair of integers.

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How do you prove that square root 15 is irrational?

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