I have one last thing that I want to cackle about before finishing with pi: exactly what number should we use for pi?
Pi is an irrational number, but it’s just not practical to use an infinite amount of decimal places when using pi in an equation, right? So, how many decimal places should we use? This depends, in part, on how accurate you want your answer to be.
Okay, so pi equals about 3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510….. but it’s just silly to think about all these numbers in terms of measurement because most of us simply do not have the means to measure anything this accurately. So, what number should we use? To answer this I used Microsoft Excel to help me show you a picture (!) of a comparison of approximations of pi. (I’m really excited about this.)
Ta-Dah! Now, take a look at this! It looks like using 3 as an approximation of pi can really be off the mark. Going out one digit is better, and it’s close to the value that the Babylonians came up with (3 1/8) two thousand years ago, but it’s still not so good. Now, 3.14 looks like it’s close to the rest of the representations of pi, which become increasing indistinguishable from each other, and it’s a value that I’m satisfied with. But for anyone who is not so easily satisfied we get to zoom in on this picture for a closer look.
I have to tell you that when I created these images I didn’t know how the various approximations of pi would look compared to each other. You can see here that 3.14 looks like it’s less than 3.141, and 3.141 looks smaller than 3.1415 and maybe 3.1415 looks smaller than 3.14159 but after that, even though there are differences between the subsequent values of pi, the differences are so small that, to the senses and for the purposes of us mere mortals, these differences are indistinguishable from each other.
So there you have it. A real look at pi. So, what’s so important about this number anyway? There are books on this subject, but, briefly, pi shows up predominantly in two places. One is standardized tests. Every student who takes standardized tests, which include the SATs and ACTs will absolutely 100% be asked a question that will test their understanding of pi. Is this a practical reason to understand it? I dunno, but that’s the way it is. Will you ever use it for any practical reason? The only time I have ever used it was when I needed to roll out a clay coil that was long enough to become the foot of a bowl I was making. The bottom of my bowl was about 6 inches across so I knew that my coil needed to be nearly 19 inches long plus about another inch for overlap.
I think that the next time pi shows up in a meaningful way is in Calculus class. There’s a really cool thing that pi does with calculus formulas:
If you ever wondered what calculus does, well one of the things it does is provide a formula using pi which can calculate the surface area of an egg-shell, which of course, is not a perfect circle. And other use of Pi in calculus is that it is used in yet another formula which can calculate the volume of an egg cup, which, again, is not a regular sort of shape. Now. just think, if you can figure out the surface area of an egg you can figure out how much material you will need if you ever want to build irregularly shaped buildings like Santiago Calatrava’s.
For more big eggs like this take a look at http://artmagonline.wordpress.com/2011/12/15/on-calatrava
And if you want to know how much liquid it takes to fill up your bathtub, your swimming pool or your oil tanker, calculus and pi (paired with the right calculus formula) comes in handy again.
Now, it’s a good thing that Chinese New Year is coming up, because I’m feeling inclined to abandon pi and start writing about accordion style bindings…
Addendum: I just came across a great post on pi in the Scientific American Blog. The articles reports that NASA uses 15 or 15 digits in their Pi calculations, in the comments reader suggested that the cube root of 31 is an excellent pi approximation. Read for yourself http://blogs.scientificamerican.com/observations/2012/07/21/how-much-pi-do-you-need/