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We know that the ratio between the circumference's length and diameter is equal to $\pi$, but can this be proved for every circle? Or is this an axiom?

bubba
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user108425
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    It can be proved. – copper.hat Dec 02 '13 at 03:34
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    This is a consequence of the fact that any two circles are similar – Ben Grossmann Dec 02 '13 at 03:35
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    This is how pi is defined. – user85798 Dec 02 '13 at 03:38
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    From the above two comments, since circles are similar, the ratio of the circumference and diameter has to be a fixed number. But then it turned out that, unlike with squares for example, that this number is not so "easy". And as Archimedes in very early ages (and others later on), started investigating this, it turned out that this ratio had a lot of decimals. And so it made sense to give this number some abbreviation. The pi symbol came much later though, somewhere in medieval times, give and take few centuries. It was proven transcendental in eighteenth century – imranfat Dec 02 '13 at 03:41
  • @Omnomnomnom $\pi$ is not defined, $\pi$ always existed, we did not calculate $\pi$ for all circles, so we cannot say that all circles will have the same ratio between lenght and diameter. However, if circles are similar, then we can say it. But can we prove that circles are similar? – user108425 Dec 02 '13 at 03:51
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    First of all, there's a difference that showing $\pi$ exists (i.e. that it's constant and consistent between circles) and calculating the value of $\pi$. Second of all, yes, we can prove that circles are similar. What we need for that is a definition of circles, and a definition of similarity that applies to circles. – Ben Grossmann Dec 02 '13 at 03:58
  • OK, Folks, just how do we prove that all circles are similar? – Robert Lewis Dec 02 '13 at 04:02
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    Oh, oh! This is going to take a while... – DonAntonio Dec 02 '13 at 04:24
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    Just keeps going round in circles... – copper.hat Dec 02 '13 at 04:26
  • @Omnomnomnom: A circle is the set of all points in a plane that are a given distance (called the radius) away from a given point (the center). – Dan Dec 02 '13 at 09:43
  • This is not true on a sphere. – jimjim Dec 02 '13 at 09:51
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    @Dan a circle is a metric notion: A circle is the set of all points in a metric space whose distance from a fixed point is given constant. There are circles that are not in the plane. – Ittay Weiss Dec 02 '13 at 10:08
  • @user108425 we can certainly compute the circumference of all circles in the plane. See my answer below. – Ittay Weiss Dec 02 '13 at 10:09
  • possible duplicate of How is the value of $\pi$ actually calculated? – Namit Sinha Dec 02 '13 at 14:00
  • I think Dan's definition of a circle is the one that's appropriate here. The generalizations might be fun, but they are probably not what the OP had in mind. – bubba Dec 03 '13 at 05:55
  • See https://math.stackexchange.com/questions/1766238/how-do-we-know-the-ratio-between-circumference-and-diameter-is-the-same-for-all?rq=1 – Watson Apr 03 '20 at 06:18

3 Answers3

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It is somewhat of a delicate matter. Firstly, the independence of the circumference-diameter ratio from the radius of the circle is not true in all geometries. It is in fact a characteristic of 'flat' geometries, or, to use the standard term, of Euclidean geometries. It is quite easy to see for instance that this ratio is not a constant for circles on a sphere. Also, using any of a number of models for the hyperbolic plane, the ratio in the hyperbolic plane is also not constant.

Next, the notion of length of a curve is very subtle, and requires a good definition. The issue with length is that it is very sensitive to small changes, and is not continuous (in the sense that for curves that uniformly converge to a given curve, the lengths of the curves need not converge to the length of the limiting curve). This makes approximating a length of a curve by geometric means subtle and error-prone.

One solid way to proceed is to accept the definition of the length of the graph of a differentiable function $f:[a,b]\to \mathbb R$ to be given by $\int _a^b\sqrt{1+f'(x)^2}dx$. This definition is obtained by using the Pythagorean theorem, which has hundreds of proofs. Now, the upper semi-circle of radius $r$ is the graph of the function $f:[-1,1]\to \mathbb R$ given by $f(x)=r\sqrt{1-x^2}$. Plug that into the definition of length and you'll get $2r\pi $. Thus the circumference is linear in the radius, which is why upon division by the diameter, you always get a constant. (To make this argument more rigorous, one also needs to prove that length is translation invariant, so that any circle can be translated to have its centre in the origin).

Remark: The ancient greeks argued differently about the constant $\pi$, since they did not have our modern definition of length. I'm not sure how aware they were of the subtleties of the length notion. Arguably, it is better to first consider area and not length. $\pi$ can then be defined as the ratio of the area of a circle of radius $r$ to the square of the radius. The proof of this ratio being independent of the radius uses integrals again, but to measure the area under the graph, rather than the length of the graph.

Ittay Weiss
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Here's what Euclid had to say on the matter. Ostensibly, an equivalent sort of argument can be made as follows:

We note that for any two circles, an inscribed square has perimeter proportional to the diameter of the circle. Similarly, we can show (inductively?) that an inscribed regular $2^{n}$-gon has perimeter proportional to the diameter of the circle.

We then may state that the sequence of lengths of successive inscribed regular $2^n$-gons of a given circle forms a monotonically increasing sequence that bounded above, since the length of an inscribed regular polygon is less than that of the circumscribed polygon (of the same number of sides).

Thus, there exists some limit of the lengths of these $2^n$-gons, and we may "reasonably" take this length to be the circumference of our circle. Since each $2^n$-gon has length proportional to the diameter of the circle, we may state that the circumference of the circle must also be proportional to its diameter. Thus, the ratio of a circle's circumference to its diameter must be constant.

Ben Grossmann
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  • since length of curve is not a continuous operation, the claim that 'there exists some limit of the lengths...' is problematic. – Ittay Weiss Dec 02 '13 at 14:05
  • @IttayWeiss not really. As I stated, the sequence of length is bounded above and monotonically increasing. – Ben Grossmann Dec 02 '13 at 14:07
  • sure, but that only shows some limit exists. But for a sequence of curves converging to a curve, the sequence of lengths need not converge to the length of the limiting curve. – Ittay Weiss Dec 02 '13 at 14:08
  • This approach doesn't presume a notion of length beyond the approximation I've provided. However, it is consistent with your approach since the integral formula for arc length works by taking the limiting length of successively improved piecewise approximations. The $2^n$-gon approximation can be thought of as a sort of "Riemann sum". – Ben Grossmann Dec 02 '13 at 14:13
  • yes, that is true, but it requires further proof than the mere existence of the limit of sequence of lengths. Length is a very delicate business, being lower semi-continuous only. – Ittay Weiss Dec 02 '13 at 14:17
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It can be proved that the ratio $\frac{\text{circumference}}{\text{diameter}}$ is the same for all circles, no matter how large or small; in other words, this ratio is a constant number. The symbol $\pi$ is used to denote this constant number. We need a special symbol because there's no other good way to write the number, really, since it is transcendental. You can easily write down approximations like $\tfrac{22}{7}$ or 3.14159265, but these are only roughly equal to $\pi$, not exactly $\pi$.

In short, then, $\pi$ is defined to be the ratio $\frac{\text{circumference}}{\text{diameter}}$. Using your term, this is an "axiom".

I'm guessing that your knowledge of mathematics is fairly elementary, so I gave you an elementary answer that ignores some subtle issues. For example, I assumed that we're working with standard planar Euclidean geometry, and that we have a workable intuitive notion of what "length" means.

bubba
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    I thought the OP's question isn't about how $\pi$ is defined, but why $\pi$ is well-defined, i.e. why it makes sense to say the ratio is the same for two circles of different sizes. This isn't an easy result: it depends on intrinsic characteristics of flat surfaces (e.g. a two circles of different sizes inscribed on a sphere will have different circumference:diameter ratio). – Clive Newstead Dec 02 '13 at 14:00
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    Well, as you point out, there are two somewhat independent questions, and I'm not really sure which one the OP was asking. I guessed that he was asking the easy one, so that's the one I answered. Other people tackled the harder one (which I glossed over). Presumably the OP will let us know which answer(s) he likes, at some point. – bubba Dec 02 '13 at 14:17
  • See the OP's comment on the question: "$\pi$ is not defined, $\pi$ always existed, we did not calculate $\pi$ for all circles, so we cannot say that all circles will have the same ratio between lenght and diameter. However, if circles are similar, then we can say it. But can we prove that circles are similar?" – Clive Newstead Dec 02 '13 at 14:19
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    (Don't get me wrong, I think your answer is good, but that it's answering a different question.) – Clive Newstead Dec 02 '13 at 14:23
  • I read that comment, but didn't notice that it was written by the OP. So, it looks like you're right, which means I answered the wrong question. – bubba Dec 02 '13 at 14:27