How do we know Pi is Constant?

[Jow]

As I am sure you know it was Pi day the other week and I found myself wondering how we know pi is constant. I decided it would be a fun problem for me to try and though I have spent a few nights awake in my bed thinking about it I cannot come up with a proof that pi is constant (although to be fair I haven't given it a lot of thought). So what is the proof? I am fluent in elementary calculus but that (and a little Linear Algebra) is the limit of my mathematical prowess (a problem which I will soon remedy). At any rate if you could keep your proof within my capabilities that would be much appreciated. If you cannot then I suppose I shall just have to be satisfied until I learn some more maths.

And for the sake of simplicity (complexity?) use pi and not tau. Thank you :)

 

[Vorde]

Well, it's a bit late for me to do it myself but I imagine it would be pretty easy to show with some basic arc length calculus that the ratio between the circumference of a function of a circle and its radius is independent of its radius.

The reason for this is pretty simple, the geometry of flat surfaces (our euclidean geometry) is scale-independent, aka it looks the same at all sizes, aka the ratio of anything can't depend on size.

In other, more complicated geometries, things aren't as simple, and I'm sure that the equivalents of pi in those geometries are not constants (though most of the simple geometries you can think of probably have the property that as radius decreases the ratio of c/d approaches pi).

 

[Vahsek]

Actually Pi is obtained by taking a limit so that it converges to 3.14... It's like e (base of natural log) which converges to a given value by taking a limit. And I'm not really sure what you mean by "pi is a constant". Anyway all I know is how to compute the value of pi (proof).

 

[arildno]

Congruence* arguments, alluded to by Vorde, can be used to show that there exists some constant PI, so that circumference=PI*diameter.

Similarly, such arguments can be used to show that there exists a constant pi, so that area=pi*r^2 (where r is the radius).

One of Archimedes' important results was to prove that PI=pi

Edit:
*Silly me, "similarity arguments"..:-(

 

[HallsofIvy]

Jow, you ask "how do we know pi is constant" but do not say what you take to be the definition of \pi. Vorde and arildno are assuming that you mean "How do we know that all circles have the same ratio of circumference to diameter" (classically the original definition of \pi). That was shown, using the "similar triangles" argument, in Euclid's Element's thousands of years ago.

You say you have taken Calculus. One of the things you should have done in Calculus is use the arclength formula and integration to find the circumference of the circle given by x^2+ y^2= R^2 and show that it is a constant times R.

 

[Jow]

When I say is Pi constant I do mean how do we know that all circles have the same ratio of circumference to diameter. I should clarify though, I haven't taken any course on Calculus, I simply worked through a textbook over the summer.

 

[Vahsek]

When I say is Pi constant I do mean how do we know that all circles have the same ratio of circumference to diameter. I should clarify though, I haven't taken any course on Calculus, I simply worked through a textbook over the summer.

This is just a very basic proof where you take a general circle of radius R. You'll find that the area will always converge to a value, Pi*(R^2). Similarly, you can do the same for the circumference and find a clever way to show how the 2 are related. All of this can be done without calculus. It's very simple to follow.( I'm in high school and I did it)

 

[tiny-tim]

Hello Jow!

Take any two circles, draw them on different flat pieces of paper, and place one directly above the other.

construct the cone that starts at the larger circle, passes through the smaller circle, and ends at a point

then use a bit of 2D geometry

 

[dodo]

This could be another way of visualizing it:

Draw a circular wheel, standing on a straight floor. Roll the wheel over the floor until it advances one full circumference; mark the distance traveled on the floor. Draw also a wheel's diameter as a line.

You can scale (enlarge or shrink) your drawing to fit any circle that you can think of. The two straight lines that you have drawn, a diameter and the distance traveled, will be each scaled by a constant when you scale your drawing. Your question is the same as asking if these two scaling constants are the same.

 

[Vahsek]

This could be another way of visualizing it:

Draw a circular wheel, standing on a straight floor. Roll the wheel over the floor until it advances one full circumference; mark the distance traveled on the floor. Draw also a wheel's diameter as a line.

You can scale (enlarge or shrink) your drawing to fit any circle that you can think of. The two straight lines that you have drawn, a diameter and the distance traveled, will be each scaled by a constant when you scale your drawing. Your question is the same as asking if these two scaling constants are the same.

That was brilliant! A very simple graphical proof. Good job dodo.

 

[bcrowell]

Vorde and arildno are assuming that you mean "How do we know that all circles have the same ratio of circumference to diameter" (classically the original definition of \pi). That was shown, using the "similar triangles" argument, in Euclid's Element's thousands of years ago.

I don't think that's quite right. http://mathoverflow.net/questions/7...of-c-d-is-independent-of-the-choice-of-circle

That was brilliant! A very simple graphical proof. Good job dodo.

What dodo gave wasn't a proof, and I assume dodo didn't intend it as one.

Mathematicians today don't define pi as the ratio C/d of the circumference to the diameter. A typical definition would be that \pi is the smallest positive real number such that e^{i\pi}=-1, where the exponential function is defined by its Taylor series.

If you want to use the C/d definition, then C/d doesn't have to be constant. In noneuclidean geometry, it isn't. You can take constancy of C/d as an axiom, and then it distinguishes euclidean geometry from noneuclidean geometry in the same way as the parallel postulate or Playfair's axiom does. If C/d isn't constant, then you have a noneuclidean geometry, and the way in which C/d varies with the size of the circle defines the curvature of space at a particular point.

 

[WannabeNewton]

The OP and others may find this interesting: http://en.wikipedia.org/wiki/Buffon's_needle

I sure loved it :) Math y u so beautiful

 

[Vahsek]

I don't think that's quite right. http://mathoverflow.net/questions/7...of-c-d-is-independent-of-the-choice-of-circle
What dodo gave wasn't a proof, and I assume dodo didn't intend it as one.

Mathematicians today don't define pi as the ratio C/d of the circumference to the diameter. A typical definition would be that \pi is the smallest positive real number such that e^{i\pi}=-1, where the exponential function is defined by its Taylor series.

If you want to use the C/d definition, then C/d doesn't have to be constant. In noneuclidean geometry, it isn't. You can take constancy of C/d as an axiom, and then it distinguishes euclidean geometry from noneuclidean geometry in the same way as the parallel postulate or Playfair's axiom does. If C/d isn't constant, then you have a noneuclidean geometry, and the way in which C/d varies with the size of the circle defines the curvature of space at a particular point.

I know what you mean but he offered a simple and effective graphical representation from which it can perfectly be grasped that C/d for any circle is just as good as scaling d/C by making the same circle smaller or bigger by any factor. So it gives a good idea, though it isn't rigorous at all but that could be extrapolated quite easily and intuitively in this sense.

 

[Jow]

I found a proof online that is nice. http://learni.st/learnings/27809-proof-pi-is-constant

 

[bcrowell]

I found a proof online that is nice. http://learni.st/learnings/27809-proof-pi-is-constant

There are several problems with it. (1) It claims that pi is "defined as the ratio of a circle's circumference to its diameter," which is not true in modern mathematics. (2) It's much, much more complicated than needed. (3) It doesn't make clear what its assumptions are.

Re 3, note that it's actually false in physical space that C/d is constant.

If you want a proof in that style, something like the following would eliminate the three problems above.

Axioms: All the axioms of Euclidean geometry.

Theorem: C/d is the same for all circles.

Proof: Since the geometry is Euclidean, the Pythagorean theorem holds, and arc lengths can therefore be calculated as \int_a^b \sqrt{1+f'(x)^2}dx. (Curves that aren't functions can be broken into parts that are functions.) Since the geometry is Euclidean, geometrical figures can be rescaled by any factor. Taking a circle and rescaling it by a factor k, we find by carrying out a change of variables that the integrals representing C and d both rescale by the factor k. QED.

 

[lavinia]

One can define pi geometrically without using ratios in Euclidean geometry.

The Gauss Bonnet theorem says that for any closed surface - no matter what its intrinsic geometry - the integral of its Gauss curvature is a constant time its Euler charateristic.
This constant is is defined as 2pi.

This theorem shows that pi is intrinsically related to the geometry of all surfaces and it does not specifically depend upon Euclidean geometry.

 

[jgens]

It claims that pi is "defined as the ratio of a circle's circumference to its diameter," which is not true in modern mathematics.

You are being needlessly picky here. Modern definitions of pi are entirely matters of convenience, not matters of substance. Pushing your argument on this point to the limit, it is like suggesting that there are problems with Euclid's Elements because it approaches geometry axiomatically, while modern mathematics (as you know) approaches geometry by introducing metrics and connections and then studying curvature and other assorted invariants of a space.

It doesn't make clear what its assumptions are.

Perhaps not, but in my opinion the intended audience of the proof will certainly assume that everything is taking place in the Euclidean plane geometry setting, and those who know better should certainly be able to identify the correct setting.

 

[SteamKing]

If pi wasn't constant, then potentially every day could be pi day.

 

[bcrowell]

This theorem shows that pi is intrinsically related to the geometry of all surfaces and it does not specifically depend upon Euclidean geometry.

Is this in response to my #15? I agree with what you said, and it doesn't contradict #15. #15 wasn't about pi, it was about C/d. What depends on Euclidean geometry is the statement that C/d is the same for all circles (in which case it then also equals pi).

 

[lavinia]

Is this in response to my #15? I agree with what you said, and it doesn't contradict #15. #15 wasn't about pi, it was about C/d. What depends on Euclidean geometry is the statement that C/d is the same for all circles (in which case it then also equals pi).

You ask a good question. I guess I was making a few points.

- I wanted to mention another way of seeing how pi is constant.

- I wanted to show that pi in an intrinsic property as well as extrinsic. For the circle in the Euclidean plane its length ( intrinsic) is compared to it radius(extrinsic) so the classic ratio is an extrinsic measure of pi. Gauss Bonnet gives it as intrinsic.

- it was said here that the definition of pi is a matter of convenience. I guess this means that pi is defined in the context of the subject. While true, I believe that pi is intrinsic to mathematics and appears in different guises. But it is hardly a matter of convenience. I wanted the example of the Gauss Bonnet theorem to suggest this. One arrives at pi by necessity. I wanted to show that the ratio example is only one way that this happens.

BTW:

- Your example of ratios on a curved surface seemed to suggest that pi is not constant.
It omitted the limit of these ratios as the polar radius goes to zero. This would have given another definition of pi since this limit is the same a every point on every surface, yet another proof that pi is constant.

- Another way to get at this would be though the idea of winding number.

 

[bcrowell]

Your example of ratios on a curved surface seemed to suggest that pi is not constant.

Not true. I clearly stated that I was not defining pi as C/d.

It omitted the limit of these ratios as the polar radius goes to zero. This would have given another definition of pi since this limit is the same a every point on every surface, yet another proof that pi is constant.

That wouldn't qualify as a proof unless you specified some set of axioms to be used in the proof. Riemannian geometry is locally Euclidean, so defining pi using this limit simply defines pi as C/d in the Euclidean case. It then remains to prove the constancy of C/d in the Euclidean case, presumably based on Euclid's postulates. That would then establish that the definition of pi as C/d was consistent.