Wednesday, September 10, 2008

Wednesday Math, Vol. 38: Pi, or the Circle and the Rope Stretchers

People have known about pi for a long time. It's an irrational number, which means that you never have a pattern in the decimal digits that repeats itself forever. It's also a transcendental number, a special kind of irrational number, which means it isn't the square root or cube root or n-th root of any rational number. I've heard of people who have memorized the first few hundred digits. I never saw the point. When I was a kid, we were told to round it to 3.1416; I learned that the first five places after the decimal were 3.14159, so that's what I used. I never went any farther than that in memorizing digits of pi.

The easiest way to represent pi as a concept is as the ratio of the circumference of a circle to the diameter. In ancient Egypt, some careful calculations were done that gave a very good rational approximation using a method called rope stretching.

Let's say we have a circle that is one cubit wide. Wikipedia says that there were a lot of measurements that called themselves a cubit over the millennia, but let's say this cubit is 22 inches long, or 560 mm. The rope stretcher had a long length of rope, and also an official length of rope that was exactly one cubit. If he wrapped the long rope around the circle and stretched it tight then cut it, he would have a length of rope for the circumference. He could mark the cubit length against the circumference length, and he could see that the longer piece of rope was three times the cubit with some leftover. What he would do then is cut a small piece of rope the length of the leftover, and see how many times it would go into the cubit. He would find that seven times the leftover is almost exactly the cubit. This means that pi is almost 3 + 1/7, or 22/7 if we write it as an improper fraction. This is a pretty good approximation, but the "almost exactly" is important.

If our rope stretcher had good eyes, he would see that a cubit is slightly longer than the leftover times seven, and the new smaller leftover is about .03 of an inch or .7 of a millimeter, small but not invisible. That new leftover goes into the original leftover 16 times, so this means 3 + 1/(7 + 1/16) is a better approximation. This isn't exact, but how much it misses by is not visible to the naked eye. There is evidence the ancient Egyptians used this better approximation. This is called a continued fraction, and to write it as an improper fraction, it becomes 355/113. This is a much better approximation.

How much better? Let's say we had a circle where the diameter was a mile. If we say the circumference of the circle is 22/7 miles, we are off by 6 feet, 8 inches. If we say it is 355/113 miles, we are off by the thickness of a sheet of bond paper.

Of course your calculator has a much better approximation inside it if it has a button labeled pi, but it is interesting to see how folks did math in the dark era before 1973.

Or interesting to me, at least.


Anonymous said...

I feel constrained to thank you for this post, but I have no idea why. LOL. I love your political junk, and will be adding you to the next blogroll update. Also you get a mention in my Friday wrap up, coming ummmm, Friday. :)

ken said...

You gonna do e and i too, and then wrap them all up together with our friend -1?

Matty Boy said...

afeatheradrift: Thanks! Always glad to have a new bloggy addition.

Ken: You know me too well. e and i are in the on deck circle and strecthing in the dugout as we speak.

Anonymous said...

There's a great Burns & Allen TV episode where a woman in college is staying at their house. Gracie asks her what she's studying. She replies "Math". Gracie wants to hear some math so the woman says "Pi r squared". Gracie gets an annoyed look on her face and responds "No, pie are round!"