Hi everybody, My question is: how do we prove that the ratio of a circle's circumference to its diameter is a certain real number, the same for any circle (which we call pi)? If the proof is difficult to post, could you suggest some books that may include it because i haven't found one yet. Thanks
The increase in the circumference of the circle is directly proportional to the increase in the radius. Therefore it is constant
According to: http://www.jimloy.com/geometry/pi.htm "Euclid proved that this ratio (C/d) is always the same, no matter the size of the circle. What he did was inscribe similar regular polygons in any two circles. Then, he increased the number of sides of the inscribed regular polygons. He reasoned that as the number of sides increased, the perimeter of the inscribed polygon gets closer and closer to the circumference of the circle. He also showed that the perimeters of the similar polygons were proportional to the radii of the circles in which they were inscribed. And so, C is proportional to r, in other words C/r is a constant. By convention, pi=C/2r. And we can use that as our definition of pi."
Such a proof must use, implicitly or explicitly, that fact that one is dealing with a Euclidean plane. What is your starting point? That is, what is your definition of a circle? and circumference? and radius? In polar coordinates, a circle at the origin is [tex]r=R[/tex] (a constant)... the circumference is [tex]C=\int_0^{2\pi} r(\theta) d\theta[/tex]. You can do a similar [but more obscure] calculation in rectangular coordinates.
I don't see what's wrong with Tide's answer in the first reply. Similar figures have corresponding linear dimensions in the same proportion. I think that's the best and simplist solution.
How do you DEFINE "similar"? If you are taking it to be "corresponding linear dimensions in the same proportion" then you are begging the question. You can't use "all circles are similar" to argue that circumference and diameter are in the same proportion if you need to show that prove that "all circles are similar".
My message was in the form of a question intended to prompt the OP to explore similarity and you (Halls) provided the details I was hoping the OP would pick up on.
What I liked about Tide's first reply is that it showed that the original problem was equivalent to proving that all circles are similar. Since a circle is fully defined by a single parameter "r" (apart from translation of the center which does not effect similarity) then it is obvious that the only non-translational changes that can be made is alteration of "r", which changes the size and not the shape. I hate trying to prove things that are so simple that they're self evident. Really how much proof does anyone really need that all circles are similar!
Just to amplify my earlier point... From a point, extend a rope of length r until it is taut... this is the radius. Now sweep around that point, travelling perpendicular to that "radius"... keeping the rope taut. The distance you travel until you return to your starting point is the circumference. On the earth's surface, if r="half of the earth's circumference" (that is, the distance from the north pole to the south pole), the circumference is zero. If r="one-fourth of the earth's circumference" (that is, from the north pole to the equator), the circumference is equal to the earth's circumference. Clearly, on the earth's surface, the ratio of circumference to radius is not a constant.
Thanks for all your answers and sorry for taking me so long to respond. I checked out the "similarity" to find out what you mean. It's seems that if circles are similar to each other then definitely the ratio of a circle's circumference to its diameter is a constant. (I use the classic euclidian definition of a circle) But I haven't seen any proof of this equivalent problem (that cirlces are all similar to each other). It does seem self evident, as uart mentioned, but that doen't mean that we shouldn't bother proving it. If you have any ideas for this proof, it would be appreciated Thanks again
Euclid did it by taking n points around the circumference of the two circles. Drawing the radii, you get n triangles in one circle similar to the triangles in the other circle. Since the ratio of the bases of the triangles in one circle to the bases of the triangles in the other are in the same proportion as the ratios of the sides (radii of the circle), so are the sums of the bases, which approximate the circumference of the circle. You need a limit argument to make that precise and Euclid didn't have that terminology.
One method of measuring the ratios involves empirical analysis; This is similiar to Euclids method with the use of geometric tools available Greeks (compass and straightedge) to show how ALL circles are similiar. Here is a seemingly long but simple proof with no math. If you take two circles of any arbitrary radius, and align the circles such that they are concentric (you can do this with any circle). Using Compass Rule and Straightedge Rule, you can draw 6 equilateral triangles, which show a basic relation exists between the path around any circle and the distance between collinear opposite points along the circle passing through its center. Here's how, Using a compass draw one circle with any radius, then draw another smaller/larger circle using the same center. The Compass rule says that for any circle there exists exactly one point as its center. So now there is a smaller circle: Circle A, and a larger circle: Circle B. Draw another circle using any point along Circle A as the center, then use the center of Circle A as the radius of Circle C, draw Circle C using a compass. By the Point Rule there will now be exactly two points where Circle A and Circle C intersect. These two points will serve as the apex of two triangles. Using a straightedge draw a line connecting center Circle C with center Circle A. Now, again using a straightedge, draw lines connecting the apex to the endpoints of the base for each triangle. The base shared by the two triangles, and one side of each triangle is equivalent to the radius of the Circle A (in fact all sides are equal to the radius). Extend all three radii to form three diameters. In extending these three lines (base, side, side) to infinity it shows that for all circles 6 equilateral triangles are formed. And the distance along any circumference is AT LEAST 6R and the distance between opposite points is exactly 2R. Therefore the ratio: Circumference / Diameter is AT LEAST 3. This can be repeated with infinitely smaller triangles to obtain greater accuracy to pi. Another crude method is to make a slightly larger circle and compare the hexagons formed in each circle. For a mathematically rigorous proof, you need integral calculus. - Arman Khos.