by Anon on April 5th, 2007

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Can anyone REALLY prove that pi is really pi without using pi?

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by Quirkie on April 9th, 2007
But what does "prove that pi is pi" mean???

I can easily show that pi is approximately 3.14 by measuring a circle. (pi is defined as the ratio of the circumference to the diameter)

But if you say, but what about proving its exact value - that makes no sense!

pi is defined to be the ratio of circle to diameter. No proof needed. It's equal to that ratio by definition.

But what is that ratio? There are plenty of ways to get the number as accuratly as you want.
I gave one in another answer.

Here's another:
pi = 16 * arctan(1/5) - 4 * arctan(1/139)

And the formula for arctan(x) is
arctan(x) = x - x^3/3 + x^5/5 - x^7/7 + ...

The proof of the arctan formula can be done by differentiation.
The proof of the formula for pi is to find
(5+i)^4 / (1+i)
The answer is a multiple of (239+i).
The argand diagram for this gives the above formula for pi.
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- I think "prove that pi really is pi" must mean "calculate the exact value of pi", otherwise it is just a question of dictionary definitions as you say...
  
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  by TheKnife V2.1 - Grandiose and Obnoxious on April 9th, 2007
- I originally asked this question in "pondering" because I'm not a mathematician, or anywhere near one. I had a strong suspicion when they placed it in a geometry category that I would get answers with calculations that I probably would not understand. As I suspected you guys are WAY over my head. What I was actually trying unsuccessfully to ask was "How the heck did they figure that it was 3.14... and how can you prove it really is" It was meant to be a "Are things really what you think they are" kind of question. Thanks for all the answers though.
  
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  by Anon on April 10th, 2007
- Hey Quirkie if could explain what you wrote so a novice can understand I pomise I'll pay attention
  
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  by Anon on April 10th, 2007
- One other thing you gave the formula for arctan, but what is arctan suppose to represent? This has always been the problem I have with math, I just can't accept calculations, I need to know why those calculations work for that circumstance. Every math teacher I've ever had has rolled their eyes at me and said "Because it does" I just can't learn that way
  
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  by Anon on April 10th, 2007
- Here's another way to find pi: Draw a circle, cut it into strips. Each strip is near enough a rectangle. The length of each strip is approximately 2 sqrt(r^2-x^2) where x is the distance of the strip from the center. Add up the areas of all the strips. More accuracy? Use thinner strips.
  
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  by Quirkie on April 10th, 2007
- This picture: http://en.wikipedia.org/wiki/Image:Circle-trig6.svg might help with tan. a "tan" is the length of the line labelled "tan" for a given angle (labelled theta). The arctan is the angle theta in radians, that gives you a known tan.
  
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  by Quirkie on April 10th, 2007
- Hey if you feel the ground shaking it's just because I think I'm starting to get it! Thanks
  
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  by Anon on April 13th, 2007
- Quirkie, would you happen to know any similar series for calculating arc sine and/or arc cosine?
  .
  Good anser by the way :)
  
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  by POP Fan on November 18th, 2009
- for sufficiently small x, cos(x) = 1 - x^2/2 + x^4/4! - x^6/6! etc.
  sin(x) = x - x^3/3! + x^5/5! etc.
  
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  by Quirkie on November 23rd, 2009
- Oops misread your question. No I don't know the formula for arcsin/arccos.
  
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  by Quirkie on November 23rd, 2009
- Oh well, I appreciate your help anyway :)
  
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  by POP Fan on November 24th, 2009
- Arcsin is here: http://functions.wolfram.com/ElementaryFunctions/ArcSin/06/01/03/01/0003/ and arccos is obviously pi minus arcsin. The "one half subscript n" part is the "falling Pochhammer symbol" and means (1/2)((1/2)-1)((1/2)-2)... for n factors.
  
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  by Quirkie on November 26th, 2009
- Thanks quirkie.
  
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  by POP Fan on November 27th, 2009
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by empty_chilli on April 8th, 2007
Sure it can be proven...

A few years ago I spent some time trying to derive pi myself, and was very chuffed with myself when I did manage it. I only used basic trigonometry, no circles, and managed to derive that:

pi = Z x sin(180/Z) as Z tends to infinity
(I can email you my derivation if you really want)

but then I went off to see if anyone had seen anything like my derivation before, and was faced with both a little ignominy and pride to learn that it had already been done a few hundred years before me :)
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by LeMeThink on April 5th, 2007
3.1415926535 8979323846 2643383279 5028841971 6939937510..
This is what you mean?
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- Exactly (well not literally exactly) now prove that it's that and how you came up with it...please
  
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  by Anon on April 5th, 2007
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by LeMeThink on April 5th, 2007
Ratio of the circumference to the diameter of a circle; approximately equal to 3.14159265358979323846...
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- Yeah but are you saying that there aren't ANY circles that are EXACTLY exact? How did they come up with it, and how can you prove it's true? I'm not trying to be a jerk, I actually have had this conversation and no one can answer it. But then again it could be the circles I travel in.
  
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  by Anon on April 5th, 2007
- Unintentional pun! (circles and pi that is)
  
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  by Anon on April 5th, 2007
- circles are not circular at all, rather points drawn together through more points(lines) and they are bent so infitessimally small, it seems like a curve.
  
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  by Anonymous on April 9th, 2007
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by Gonzo on April 5th, 2007
No. Well, i cant. I swear these maths people mke this stuff up. How the hell would you decide that pi is 3. whatever!? It's beyond me.
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- That's kinda what I'm asking...So does this mean I'm not crazy or we're just equally crazy?
  
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  by Anon on April 5th, 2007
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by sunshineinabag on April 5th, 2007
what? lol you lost me.
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by cmb71129 on April 5th, 2007
I don't see how they could.
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by sweetiebabe on April 5th, 2007
i agree wtih preetypirate.
and what kinda question is that.
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- Hey the category IS pondering
  
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  by Anon on April 5th, 2007
- I think it's valid... only cause IDK the answer :P.
  
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  by Penny The Wise on April 5th, 2007
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by PrettyPirate on April 5th, 2007
No.

But I can offer you some pie.
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- Is it apple? Or Cherry? MMMMMM....
  
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  by Penny The Wise on April 5th, 2007
- Apple. Mmmmmmmmmmm.
  
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  by PrettyPirate on April 5th, 2007
- Mornin' Sunshine! Hows things in your neck of the woods?
  
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  by Penny The Wise on April 5th, 2007
- Bloody fantastic :) And you?
  
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  by PrettyPirate on April 5th, 2007
- Pretty alright. Little hungry, and this pie convo isn't helping, but I think it is going to be a nice day. It's gorgeus outside and I have an afternoon to myself. Should be fun!
  
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  by Penny The Wise on April 5th, 2007
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by archimedes on November 18th, 2009
First see Archimedes "Measurement of a Circle" Propositions 2 and 3 primarily( try booksforfree.com or onlinebooks fro free something like that)

or my method...BUY MY BOOK!

or integrate this
-(17,496,000〖sec(180/n)〗^2 〖tan(180/n)〗^2)/n^5 -(97,200tan(180/n) 〖sec(180/n)〗^2)/n^4 -(5,832,000〖sec(180/n)〗^2)/n^5 +(2,916,000cos(180/n))/n^5 +(48,600sin(180/n))/n^4
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- I should say...I can proove that pi is not 3.141592654........ as this is shy by quite the decrement
  
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  by archimedes on November 18th, 2009
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by purplecows on August 3rd, 2010
There's also a statistical method involving matches thrown down onto a flag.

Hey Quirkie, check this out: http://answerbag.com/q_view/2268856#ans9471384
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