Sunday, April 24, 2011

Sunday Numbers 2.0, Vol. 8:
The approximations of pi.

This is an update and correction of a post from late 2008. I forgive any reader who thinks this is the first time I wrote about this.

The number pi is the ratio of the circumference of a circle to its diameter. It has been considered a useful number since at least the time of the ancient Egyptians. We now know it's an irrational number, which means we can't write it exactly as a/b, and there is a proof that it is a transcendental number, which means it's not the square root of 10 or the fifth root of 306 or any root of a rational number.

These proofs take some damn tricky math. Best to take my word for this stuff.

We've known for a while that pi is pretty close to 3 1/7 or 22/7. It's a little high, but only a little. For example, if the diameter of the circle was a mile, 3 1/7 of a mile overshoots pi miles by about than six feet eight inches. If the diameter was a kilometer, the overage is about one meter and 26 centimeters.

The ancient Egyptians could have found this approximation by the method they called rope stretching. Let's say they made a circle whose diameter was a cubit, the length from the tip of the middle finger to the elbow. In the English system, we approximate this to 22 inches and in metric, we could use 56 centimeters.

They would have taken a rope of cubit length and cut another piece of rope the length of the circumference. They easily could mark the longer rope to see it was a little more than three times the length of the cubit rope. They would then take the excess rope and see how many times it would go into the cubit length. Seven copies of the shorter length fit with a little left over, about a fifth of an inch or half a centimeter. This is a small length, but clearly visible to the naked eye, even if your eyes are as bad as mine at short distances.

The Egyptians would have taken the smaller part and checked to see how many times it would fit into the first remnant. The correct answer is 15.99659..., which means to the naked eye it looks like it goes in sixteen times.

So now we have the numbers 3, 7 and 16, in that order. What good would this have done the ancient Egyptians?

The answer is called continued fractions. Instead of saying pi is close to 3 1/7, we will say it's closer to 3 1/(7 + 1/16).

We change the mixed number 7 1/16 into the improper fraction 113/16.

1/(113/16) = 16/113, so out new approximation is 3 16/113, usually written as 355/113.

Like 22/7, 355/113 is not exactly pi but it's awfully damn close. Now if we have a circle whose diameter is a mile or a kilometer, the amount of difference between the true circumference of pi times the diameter and 355/113 times the diameter is about the thickness of a piece of bond paper, less than 1/50 of an inch or half a millimeter.

Mathematicians, precision loving nerds that we are, have taken the continued fraction representation of pi way past this already pretty damn good approximation. This picture makes it look like the numbers on the list are as follows:

3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 2, 1.

This is just a partial list. If a continued fraction ends, the number it represents must be rational, and we know pi isn't.

Happy Easter to everybody, and I hope the only time you have to think about pi for the rest of the day is if it has an e at the end.


ken said...

Whoops, typo. Irrational means you can't write it as a/b

Matty Boy said...

Thank, Mr. Editor. I owe you a beer.