Pi is said to be discovered by ancients

[JamesU]

Pi is said to be discovered by ancients, but if even we don't know the EXACT number of pi, how could they, or was theirs just a close estimate of pi? :yuck:

 

[snoble]

Pi is just the constant defined by the circumference over the diameter. Recognizing that the ratio doesn't change between circles qualifies as the discovery of pi. It's an interesting exercise. Also try to show the same constant relates the radius to the area. And be careful that you're arguments aren't circular (poor pun intended). You can't use any theorems that depend on what you're trying to show.

Looking at the discoveries of the ancients often makes you feel like the lady who walks out of one of Shakespeare's plays and say "I don't get what the big deal is. All he did was string a bunch of famous cliches together." Some of what they did seems simple but try to be the first guy to do it without modern machinery.

 

[JamesU]

Good point, I never thought opf it that way

 

[inquire4more]

Pi is an irrational number. This means that it is non-repeating, non-terminating. In other words, it is infinite in length and, unlike 1/3 = 0.33333..., it does not have some infinitely repeating pattern of digits in its decimal expansion. So, in short, the ancients determination of the value of pi was, just as ours is, just an approximation. No one may know exactly the value of pi. But, for all practical purposes, an approximation is sufficient.

 

[Manchot]

How would one prove precisely (using geometry) that all circles are similar? The only way that I can think of is to imagine a circle as a regular polygon with an infinite number of sides. Since it's easy to prove that any two n-sided regular polygons are similar, it should therefore extend to circles. However, this isn't very rigorous. Anyone know of a way to prove it rigorously using classical geometry?

 

[Gokul43201]

How would one prove precisely (using geometry) that all circles are similar? The only way that I can think of is to imagine a circle as a regular polygon with an infinite number of sides. Since it's easy to prove that any two n-sided regular polygons are similar, it should therefore extend to circles. However, this isn't very rigorous. Anyone know of a way to prove it rigorously using classical geometry?

If you accept a definition of similarity in terms of scaling :

1. translate the center, c1 to new center c2
2. scale by a factor r2/r1 about center (since every point is an equal distance from the center, scaling the radius by a factor x scales the distance of every point from the center by this same factor)

As for the OP : http://www.geom.uiuc.edu/~huberty/math5337/groupe/overview.html