Is the circumference of a unit circle irrational?

Question

Let us assume that I have a unit circle, and there are no existence of errors in measurement of a quantity. Is the circumference irrational, like the diagonal of a unit square?

Edit : My apologies. For a moment the realisation that circumference is irrational was making me feel "How is that possible? Lengths are meant to be rational", until I happened to remember the formula of the diagonal of a square, and after reading a few comments I realised(for the lack of a better word) it is a usual fact when dealing with circles.

My apologies if it caused unnecessary waste of time on your behalf. Thank you for helping me.

Answer 1

The circumference of a unit circle would be $2 \pi$ (since $c = 2 \pi r$), so it would be irrational. I'm not sure what you mean by "change your outlook," but $\pi$ is a pretty universally understood quantity. It's the ratio between a circle's circumference and diameter.

Answer 2

For a moment the realisation that circumference is irrational was making me feel "How is that possible? Lengths are meant to be rational"

Ah. That's very important and yes, your outlook should change.

In geometric space, the world is a continuum and lengths can be any (non-negative) values. You can't have a distance or length that can't exist. There aren't any "atoms" where one distance is possible but another is not.

Or at least Euclid thought so when he claimed "Any straight line segment can be extended indefinitely in a straight line" in which he was claiming distances exist for every possible positive value (although Max Planck may have another opinion).

It's actual because of this irrational numbers became such a problem. It'd be fine to Pythagoras that all values are rational and although $r^2 = 2$ for rational $r$ is impossible, he'd accept that as $\sqrt 2$ simply need not exist and we could live in a universe without them.... except the diagonal of a square must be an $x$ where $x^2 =2$ and $x$ can't be rational. But it must exist.

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The diagonal of a unit square is $\sqrt 2$ which is irrational.

The circumference of a unit circle is $2\pi$ which is irrational.

I don't know what in your outlook made the concept difficult while the concept of a the diagonal of a square being irrational was okay. But given the options I'd say, yes, it is just as normally understood.

A geometric figure will have many measurable qualities (sides, diagonals, radii, angles, area etc) many interrelated. The ratios between them need not be rational. If one's outlook is that this is strange then, yes, it should be changed. If one's outlook is that this is normal (actually, inevetible) then it need not be changed.