Is the square root of -13 a rational number?

Answer 1

No, 13 is not a rational number. Yes, here's the proof.

Let's start out with the basic inequality 9 < 13 < 16.

Now, we'll take the square root of this inequality:

3 < √13 < 4.

If you subtract all numbers by 3, you get:

0 < √13 - 3 < 1.

If √13 is rational, then it can be expressed as a fraction of two integers, m/n. This next part is the only remotely tricky part of this proof, so pay attention. We're going to assume that m/n is in its most reduced form; i.e., that the value for n is the smallest it can be and still be able to represent √13. Therefore, √13n must be an integer, and n must be the smallest multiple of √13 to make this true. If you don't understand this part, read it again, because this is the heart of the proof.

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Answer 2

No, it is not.