Vincent Bouchard, associate professor and associate chair (undergraduate) in the Department of Mathematical & Statistical Sciences, talks Pi.
March 14 (also known as 3/14) is Pi Day, an annual celebration of the mathematical constant pi. Also written as π, pi represents the ratio of a circle’s circumference to its diameter, which is approximately 3.14. In the world of mathematics, pi is everywhere--and it’s very important.
Interested in learning more? Hear from Vincent Bouchard, associate professor and associate chair (undergraduate) in the Department of Mathematical & Statistical Sciences.
So, what exactly is pi?
Pi is, of course, a greek letter. But in mathematics, the symbol pi is the ratio of the circumference of a circle to its diameter. Its definition in terms of the geometry of a circle is rather elementary, but the constant pi appears all over mathematics.
Pi is an irrational number, which means that it cannot be written as a ratio of two integers. It also follows that if you write pi in its decimal representation, it has an infinite number of digits. Pi is irrational, so one could keep calculating more and more decimals of pi, forever! In fact, using cloud computing technology mathematicians have managed to calculate pi's two-quadrillionth digit!
Why is pi so important in mathematics?
Pi is an important mathematical constant because it is rather ubiquitous. It appears just about everywhere in mathematics, from the simple geometry of a circle, to more advanced mathematics such as contour integration in complex analysis, Fourier analysis, spectral theory, etcetera. And since it appears in so many different contexts in mathematics, this means that it also plays an important role in science. For instance, the period of a simple pendulum is always proportional to pi.
Pi also appears in Euler's identity: e^(Pi i) = -1, where i is the imaginary unit (satisfying i^2=-1) and e is the base of the natural logarithm. This equation provides a deep connection between some of the most fundamental numbers in mathematics: in fact, for many mathematicians, this is one of the most beautiful equation in mathematics
What is Buffon's needle experiment?
This is a really fun experiment. The idea is to experimentally "measure" the value of pi by throwing toothpicks on a sheet of paper. Isn't it cool?
The experiment goes as follows. Pick a toothpick. On a sheet of paper, draw horizontal straight lines, vertically separated by precisely the length of the toothpick. Then throw your toothpick on the sheet of paper, and take note of whether the toothpick crosses a horizontal line. Throw it again. And again. You need to keep track of the number of tosses and the number of "crosses" (that is, the number of times the toothpick crossed a line). Now, the remarkable claim is that if you take twice the number of tosses and divide by the number of crosses, you should get the value of pi. Amazing! For instance, if after 200 tosses, you obtained 125 crosses, then 2*200/125 = 3.2, which is not too far off from pi =3.14159... !
Note that this a probabilistic experiment, so to get an accurate measure of pi, you need to throw a lot of toothpicks (and by a lot, I mean a lot!). But try it. It works!