I have always been a little confused about pi. I have gone all the way up into partial differentiable equations (I hope I spelled that right) so I understand a little bit about pi, but what I wonder is is it essential in the mathematical world. It’s kind of hard for me to put into words, but what I am wondering is how was it calculated and how did the person/persons know it was correct and helped in calculating the true dimensions of certain objects and equations. Thanks.
Archimedes discovered that the ratio of a circle’s circumference to its diameter is constant, and named the constant of proportion [symbol]p[/symbol]. He calculated a rough value, other people did more accurate calculations, and so on and so forth.
As for calculating pi, there are various ways. One way I’m familiar with off the top of my head is:
pi = 4 - 4/3 + 4/5 - 4/7 + 4/9 - …
You can derive this by finding the power series expansion for arctangent(x).
Using this formula, you can calculate pi to any desired accuracy. However, this series for pi converges slowly; I’m sure there are much more efficient ways to calculate pi, but that’s not something I have a lot of experience with.
Well, back before the days of scientific calculators with preprogrammed constants, we would use the mnemonic ratio 355/113 (note the pairs of odd digits), which is accurate to 0.3 parts per million… better than remembering six digits of pi.
Or, if you’re really not concerned with accuracy, you can just go with the Bible’s value of 3. Three is good enough for the Lord’s work.
As for how the ancients determined its value, they just used the definition. The ratio of circumference to radius. The ancients were quite resourceful in their graphical analytical methods… since it’s all they had. Along the same lines that the generation of engineers that built WWII radars were a hell of a lot better at using slide rules than just about every engineer alive today.
Every time I need pi accurate to 7 decimal places I simply recite "May I have a large container of coffee."
What does that stand for?
Count the letters.
3 1 4 1 5 9 2 6
…except that the number after the 6 is 5, so it should be rounded up to 7. But “May I have a large container of sausage?” doesn’t have quite the same ring to it.
You can also compute [symbol]p[/symbol] by generating random numbers. Repeatedly generate pairs of random numbers, both between 0 and 1. Count how often the sum of their squares is less than 1. Divide by the total number of pairs you generated. This ratio is the ratio of the area of a quarter circle of radius one to the area of a square with sides of length 1. (If you don’t see why, think of each pair of numbers as an (x,y) coordinate. The ratio of the areas is the same as the probability that a random point in the square lies in the quarter circle.) In other words, it’s [symbol]p[/symbol] / 4. Then just multiply by 4 to get [symbol]p[/symbol].
It’s no where near the most effecient way to do it, but I think it’s kind of clever.
May I have a large container of coffee, buddy.
That made my brain hurt. 
The fact that such a constant as pi existed was known long before Archimedes, who lived at the time of the Punic Wars. The Ancient Egyptians and Ancient Babylonians both attempted to calculate pi by the method of inscribing a regular polygon inside a circle and then adding up the lengths of the sides. The more sides the ploygon had, the better the estimate they got. And before the invention of the calculator the mideival Chinese performed such a calculation with a 3000-sided polygon and got a very good result. See this book for further details.
As for how we know that it is the correct value to use in calculating surface areas and volumes, it’s all in the multivariable calculus. Certain integrals can only be solved my means of trignometric substitution, a method that frequently gives results in terms of pi.
Every time I need pi accurate to 300 decimal places I simply realize I’ve memorized that many 
When come back, bring pi. 
For an equation that kinda ties things together, consider:
e[sup]i[symbol]p[/symbol][/sup] = -1
I haven’t read The Joy of Pi. I have, however, read A History of Pi by Beckman and enjoyed it quite a bit. Pi has been calculated lots and lots of different ways throughout history. For example, if you drop a needle on a grid you can calculate pi by the proportion of times the needle crosses a line, or something like that. Very good book, IMO.
I was off by eight on how many replies there would be before this happened.
Trigonal Planar: :dubious: So, what’s the 247th value?
My dad loves to joke about the time he was in grad school at Yale, and he was required to memorize Π to 100 places. :eek:
You think your math homework was bad? 
Adam
For those who are into this, it is possible to calculate any digit of Pi (without having to calculate the previous digits) using the “BBP formula.” Take a look here . Truly an amazing and fascinating number.
May I have a large container of coffee buddy. Now bring Patricia something totally indulgent for it has actually been plenty of months sans she had complete fun!
Damn, couldn’t think of a four letter word for “since” thus I have the french word sans in there, which only sorta makes since. By the way, the sentence ends there because that is how many digits I know… 3.141592653589793238462643383