What is the derivation of pi and how is it defined?

[alexk307]

Who is pi derived, and what exactly is pi.

 

[Crazy Tosser]

Pi = circumference/(2xradius) or Area/(radius^2), or area of a unit circle.

My guess is that it's derived using the unit circle equation y=sqrt(1-x^2), integrating it from 0 to 1 and multiplying by 4. Since direct integration is (I think) impossible, we use different methods of finding areas under a curve, and the better these methods are, the more precise the answer is.

Hope I helped.

 

[Tac-Tics]

Pi is defined as the ratio of a circle's circumference to its diameter. That is the definition.

What's interesting is that it pops up all over the place outside of geometry.

It's value is something very fundamental to mathematics. It's not like the speed of light. We don't *measure* it. Using the axioms of geometry and calculus, we can produce its value exactly.

 

[Big-T]

Rudin defines pi/2 to be the smallest positive number such that cos(pi/2)=0.

 

[JG89]

Courant shows how to calculate pi to any degree of accuracy using the "method of exhaustion" (the same method Archimedes used, or slightly modified, I believe). Would you like me to post it?

 

[NoMoreExams]

I have a feeling if people used google and wikipedia, half the questions would disappear :) Check this out: http://en.wikipedia.org/wiki/Pi

 

[HallsofIvy]

That would ruin all the fun of feeling superior to the people who do ask those questions!

 

[robert Ihnot]

Crazy Tosser: My guess is that it's derived using the unit circle equation y=sqrt(1-x^2), integrating it from 0 to 1 and multiplying by 4. Since direct integration is (I think) impossible, we use different methods of finding areas under a curve.

It is a very easy integral with the substitution x=sin(u)

Goes on to be: \int_{0}^{\Pi/2}cos^2(u)du

(Of course that gives the answer in terms of pi, and I am not sure what you had in mind anyway.)

 

[Doodle Bob]

I have a feeling if people used google and wikipedia, half the questions would disappear :) Check this out: http://en.wikipedia.org/wiki/Pi

Actually, that wikipedia article leaves out some mathematically crucial aspects to the so-called "definition" of pi being the ratio of the circumference of the circle to its diameter.

1. It shouldn't be taken on faith that this is a well-defined concept. After all, such a constant doesn't necessarily exist in other geometries: why is it that this ratio is the same for all circles on the Euclidean plane?

2. Why does the same constant show up in both the circumference and area formulas for circles in the Euclidean plane? In fact, it is possible to use the formula of circumference/area to prove the formula of area/circumference.

I personally like the definition of pi to simply be the area of the unit circle.

 

[Tac-Tics]

1. It shouldn't be taken on faith that this is a well-defined concept. After all, such a constant doesn't necessarily exist in other geometries: why is it that this ratio is the same for all circles on the Euclidean plane?
...
I personally like the definition of pi to simply be the area of the unit circle.

An interesting point of view and a cute definition.

 

[gunch]

Actually, that wikipedia article leaves out some mathematically crucial aspects to the so-called "definition" of pi being the ratio of the circumference of the circle to its diameter.

1. It shouldn't be taken on faith that this is a well-defined concept. After all, such a constant doesn't necessarily exist in other geometries: why is it that this ratio is the same for all circles on the Euclidean plane?

2. Why does the same constant show up in both the circumference and area formulas for circles in the Euclidean plane? In fact, it is possible to use the formula of circumference/area to prove the formula of area/circumference.

I personally like the definition of pi to simply be the area of the unit circle.

I don't really see why those are issues. Wikipedia is not a math textbook and should therefore not adhere to the same standards. Under definition, it defines pi as,
"In Euclidean plane geometry, pi is defined as the ratio of a circle's circumference to its diameter"
So it specifically mentions that this is in Euclidian plane geometry (which includes the parallel postulate) so what exists in other geometries is irrelevant. They state that the ratio is constant and that this is a consequence of similarity. They don't provide a proof, but this is well-known so they shouldn't need to. Proofs are available in many geometry books, and with some knowledge of Euclidean geometry you could even reconstruct the proof yourself from the hint that it's a consequence of similarity. Your definition makes many assumptions as well, for instance why is the area of the unit circle well-defined? Also I don't see why they need to explain the appearance of pi in both formulas. The definition only speaks about one of them, and then you can derive the other formula as you said.
Personally I like to start with a formal definition of the trigonometric functions and then base my definition of pi upon those (like how Big-T defined it) since I hate working rigorously with Euclidean geometry, but use any definition that suits you and is formally valid.

 

[jim mcnamara]

In programming the value of pi is often generated with: atan 1 times 4.

// C example
double pi=atan(1.0) * 4.0;

 

[robert Ihnot]

I am not certain what it is that is being sought. There are many numerical series for pi. A very simple one, slow to converge is the Leibniz Series:

\Pi/4 =\int_{0}^{1}\frac{dx}{x^2+1} =\int_{0}^{1}1-x^2+x^4-+- = 1-1/3 +1/5-1/7+-+

 

[csprof2000]

Pi, eh... hmmm...

Well, I think it's true that Pi is always between the following two quantities:

2p(n)/r < Pi < 2p(n)/a(n),

where r is the distance from the "center" of a regular polygon to one of its vertices, a is the distance from the "center" of a regular polygon to the midpoint of one of its edges, and p is one half the total perimeter.

As n increases, p and a both change, but let's hold r fixed. Then this relationship remains true, but I suppose the terms on the left and right both get closer and closer together. I define Pi to be the number such that for all n, the above relationship holds.

This also gives a nice way of calculating it...

 

[HallsofIvy]

Well, it doesn't quite make sense to say "Pi is always between the following two quantities: 2p(n)/r < Pi < 2p(n)/a(n)" and then say "I define Pi to be the number such that for all n, the above relationship holds."

You could, of course, show that there is a single number always between those (by showing that 2p(n)/r < 2p(n)/a(n) for all n and showing that there is only one number between those. Of course, you would then also need to show that "your" pi has the properties other have mentioned, in particular that it is the ratio between circumference and diameter for any circle.

Or you could do it the other way around. That is define pi to be "ratio between circumference and diameter for any circle" (after showing that that ratio is always the same for all circle- I believe that is in Euclid) and then show that pi satifies the inequalities you give. That's essentially what Archmedes did.