Whew! Busy week for me, hence the missing post or two. Traditionally Saturdays on this blog tend to be non-physics fare – either links, commentary on something nonscientific, or whatever. Today I think it will be the celebration of Pi Day!
Pi Day: As you know from the SB main page, today is pi day. 3/14, representing the famous number 3.14159265359…
You first meet is as the ratio of the circumference of a circle to its diameter. As you go on in math, you meet it pretty much everywhere else too. Seriously everywhere.
There’s also a Pi Approximation Day, on 7/22. Which of course only makes sense if you write your dates in the 22/7 format. 22/7 is approximately pi, because 22/7 = 3.14286…
Let’s look a little more closely at that concept. To approximate a number as a fraction, we’re saying there are two numbers a and b such that a/b is close to our number. You would start with a low number for b and work your way up with the desired approximation. Why don’t we start with b = 1? In that case, clearly 3/1 = 3 is our best approximation.
Now it turns out that as we try higher values of b, we won’t be able to find a closer approximation to pi until we hit b = 7. Then we’ll find that a = 22, leading to the famous 22/7 approximation. But we can take b higher: can we (say) find some number a such that a/13 or is a better approximation? Nope. b = 13 (which I picked as an example for no particular reason) won’t do it. In fact 22/7 is a very good approximation. There’s no a that will give a better approximation for any b until we reach b = 106.
With a little effort, we find that the next rational approximation to pi that’s better than 22/7 is 333/106. And further it will turn out that the next approximation to improve upon that doesn’t have to make its own b much higher: 355/113.
So what are all these approximations buying us compared to the actual value of pi? Let’s see:
Approximation: 3/1 = 3
Percent Error: 4.5
Approximation: 22/7 = 3.14286…
Percent Error: 0.04
Approximation: 333/106 = 3.14151…
Percent Error: 0.0026
Approximation: 355/113 = 3.1415929…
Percent Error: 0.0000085
And just for kicks, the next one after that:
Approximation: 103993/33102 = 3.14159265301…
Percent Error: 0.000000018
Going that high kind of defeats the entire point of a rational approximation, which is to make the math easier. And it’s worth pointing out that in some sense 22/7 and 355/113 are especially good approximations because you have to make a quite sizable jump in b before you can find a better approximation. If you’d like to take a look at the machinery of this calculation, check out the theory of continued fractions. It’s a beautiful subject but not one that can really be done justice in a single Pi Day news post.
Enjoy, and go eat a pie!