i have derived a formula for pi which approximates the value of pi.
pi = lim n*cos((180/n) - 90)
n->infinity
higher the value of n, more accurate the value of pi.
i have derived it based on inscribing a polygon in circle. variable n represents number of sides of polygon.
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I don’t understand why pi is an irrational number. Can’t you just measure the circumference accurately and then divide by the diameter - there you have a rational fraction. I can’t see how a constant derived by real division can be an irrational number.
@rishi: Great question. The problem with drawing and measuring a circle is that there’s no such thing as a perfect circle. Anything you draw is just a collection of points (each drop of ink, or each molecule of ink!) and is therefore a very large polygon, maybe with billions of sides.
We can measure the circumference of this polygon, but it won’t be “pi”, just a very close guess. After all, we could have added more sides and got a better guess.
One way to see the irrational, neverending decimal is to consider pi the result of an infinite process (adding more and more sides to a polygon to approximate a circle), one we can approximate but never write out completely. Hope this helps!
I like the write-up very much, but find the title a little misleading.
You give a good description of applying Archimedes method of calculating the numerical value of pi. In fact, this type of successive approximation is useful for computing many other interesting values as well.
To many, however, the “discovery of pi” is the realization that the ratio of circumference to diameter is the same for ALL circles. Without that, we wouldn’t be talking about the circumference of a unit circle, nor would that value have a special name (pi).
Adding an intuitive description of that discovery to your write-up would really make it shine.
@Eric: Great question! I think another article would be warranted for that general idea of proving that all circles are similar (proportional to each other).
There is an ancient proof here:
http://school.maths.uwa.edu.au/~schultz/3M3/L6Euclid.html
but yes, it’d be a great topic. Thanks for the suggestion.
The idea that the newOutside is the harmonic mean and the newInside is the geometric mean is not very intuitive. Why is newOut being derived from the perimeter of the previous inside and previous outside, similarly with geometric mean?
Thanks for the great write up. I’m returning to calculus after 20 years and your article is helping me finally internalize something I’ve never grasped before.
@Simon: Awesome, glad it was helpful for you!
@Sapan: Yes, I struggle with that too – I don’t have an intuitive understanding of why it would be the geometric and harmonic mean to figure out those ratios. Right now my understanding is at the level of “the math works” :).
I actually came up with Archimede’s method on my own but I started with a triangle and kept going with more polygons (basically each side of the triangle got another triangle, and so on). Basic geometry got me from the perimeter of one poly to the next. Using my PC i was able to calculate pi to a million decimal places rather quickly (i did a text-compare with one i found online and it was right). I thought i may have stumbled on something new but later i found out it was not so.
The only interesting thing was that it was recursive and used only basic geometry (right triangles).
@Dedic: That’s a cool story – there’s always something to be said for the joy of discovery, even if you weren’t the first to do so :).
Hi Kalid, Wonderful post…
I’m actually a young guy and new to complex stuff but u make it look easy…
A question :
Is a straight line a part of a large circle ???
@Shankar: Glad you liked it! Hrm, I’m not sure what you mean – i.e., is a circle made up of straight line segments? A perfect circle seems never has two points on a perfect line (i.e. if you rotate the circle only one of the points will be “rightmost”, you can’t have both vertically above each other) but reality is quite different :).
Hi Kalid…What i meant was that a road seems perfectly straight to us…however its just a part of a large circle called earth…
So if we keep on extending a straight line on both sides infinitely, will we get a large circle ???
And one more thing…
How can we be sure that pi is an irrational number.??.
Maybe after the 100 billionth number after the decimal point, it may repeat itself, thus making it a rational number…
[…] Prehistoric Calculus: Discovering Pi describes the logic behind Archimedes’s method of calculating π . […]
Dear Khalid,
Grate article and will look out for other article by you.
Regarding: PI ~= 335/113
On PI day (3.14) a french lady emailed me a gift that further to my Quran and Prime Numbers reseach, the 355 days in a Hijri leap year divided by the chapters of The Message (113 chapters) is a very close approximation to PI.
PI ~= days in a year cycle (circumference) divide by the number of chapters of the message (stright path, diameter)
Here is a summary for all your readers about the prime numbers in the Quran.
Quran = Key + Message
114 chapters = 1 (Al-Fatiha) + 113 (Remaining chapters)
6236 verses = 7 of The Key + 6229 of The Message
The Key has 7 verses, 29 words, 139 letters) all are primes, with prime digit sums (7=7, 2+9=11, 1+3+9=13) and amazingly concatnating them left-to-right (729139) and right-to-left (139297) also primes with primes digit sums (7+2+9+1+3+9=31)
The rest can be found at www.heliwave.com pr www.primalogy.com.
Make sure not to miss the 355 days of chapter The Merciful that map to the leap Hijri year 1433AH = 2012 
I suspect the Hijir year becomes leap evey PI years a “PI in the Sky” if you like
Ali Adams
God > infinity
Great article very informative and helpful, the only think I could see needing some furthre explainging this quote
“faster than finding the closest answer from the outset”
What is outset?
@GW: Ah, I just meant “rather than finding the closest answer immediately, from the very beginning”.