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Is the area of a circle ever an integer?

I was trying to answer someone else's question on yahoo answers today and I got thumbs down from people on my answer and have come here to get a thorough proof on it because now I just must know! :)

My assertion: Suppose we have two integers a and b. We can form a rational numbers out of each of these: $\frac{a}{1}$ and $\frac{b}{1}$. Now forming another rational by dividing them: $\frac{\frac{a}{1}}{\frac{b}{1}}$ = $\frac{a}{b}$. Now if we divide this by itself: $\frac{\frac{a}{b}}{\frac{a}{b}}$ = $\frac{ab}{ab}$ = 1. Therefore, we can conclude that given any rational number, we can always describe it using only integers thus always resulting in an integer number.

However, even if our radius is $\sqrt{\frac{1}{\pi}}$ then we will have $\pi\sqrt{\frac{1}{\pi}}^2$ = $\frac{\pi}{\pi}$, by definition of area = $\pi r^2$.

Now I think this the trickiest part. Does $\frac{\pi}{\pi}$ = 1? Or can it only equal $\frac{\pi}{\pi}$? How do we prove that $\frac{\pi}{\pi} \neq$ 1? Or is my method sufficient at all? Thanks!

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I'm afraid you are wrong here. Cancellation occurs in the real numbers just as it does in the rationals. I think you should look into rational and real numbers a little more in-depth. – Alex Becker Mar 21 '11 at 2:13
up vote 18 down vote accepted

By definition, $\frac{1}{b}$ is the unique real number which, when multiplied by $b$, yields $1$.

By definition, $\frac{a}{b}$ is the product of $a$ and $\frac{1}{b}$.

Since $\frac{1}{\pi}$ is the unique real number that, when multiplied by $\pi$, yields $1$, then $\pi\left(\frac{1}{\pi}\right) = 1$. Hence, $\frac{\pi}{\pi}=1$.

If you allow any radius for a circle, then a circle has integer area if and only if its radius $r$ is the square root of an integer divided by $\pi$, that is, $r = \sqrt{a/\pi}$ for some nonnegative integer $a$.

Your first paragraph, though, is completely irrelevant.

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Thanks! I think I understand now! Sorry, I don't mean to be crazily irrelevant or anything...just an amateur mathematician trying to learn. :) Appreciate rigorous correction and proof. – Mr_CryptoPrime Mar 21 '11 at 2:22
@Mr_CryptoPrime: My point was merely that the argument in the first paragraph is correct, but is irrelevant to the question of whether a circle can have area $1$. Yes: every rational number can be described with two integers; that's the definition of rational number. And is as relevant to the question of whether a circle can have area one as the correct assertion that one meter equals 100 centimeters. True, but irrelevant to the question at hand. – Arturo Magidin Mar 21 '11 at 2:29

$\frac\pi\pi$ is $1$, because when you multiply $\pi$ by $1$ you get $\pi$.

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Ok, so the sqrt() and exponentiation should not affect $\pi$ in any way? Is this the only solution or are there others? How could we prove this is the only solution? – Mr_CryptoPrime Mar 21 '11 at 2:14
There are infinitely many others. Any circle of radius $a^{1/2}/\pi^{1/2}$, where $a$ is an integer, will have an area that is an integer (specifically, $a$). – Alex Becker Mar 21 '11 at 2:16

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