Project Work For Additional Mathematic 2009 Part 1

Part 1 Question B History of Pi The oldest value we know for π comes from the Babylonians. (Man, but those guys were impressive mathematicians; almost any time you look at the history of fundamental numbers and math, you find the Babylonians in the roots.) They tended to work in ratios and the approximation that they used 25/8s (3.125), which is not a terribly bad approximation. Especially when you realize when they came up with this approximation: 1900BC! The next best approximation came from Egypt, around the time of Pharaoh Amenemhat in the mid 17th century BC, where it had been refined to 256/81 (3.1605). Which isn't such a great step forward; it's actually a hair farther from the true value of π than the Babylonian approximation. We don't see any real progress until we get to the Greek. Archimedes (yes, that Archimedes) worked out a better approximation. He used a really neat trick. He worked out how to compute the perimeter of a 96-sided polygon; and then worked out the perimeter of the largest 96-gon that could be drawn inside the circle; and the smallest 96gon that the circle could be drawn inside. Here's a quick diagram using octagons to give you a clearer idea of what he did:

That gives you an approximation of π as the average of 223/71 and 22/7 - or 3.14185. And next, we find progress in India, where the mathematician Madhava worked out a power series definition of π, which allowed him to compute π to 13 decimal places. 13 decimal places, computing a power series completely by hand! Astounding! Even better, during the same century, when this work made its way to the great Persian Arabic mathematicians, they worked it out to 9 digits in base-60 (base-60 was in inheritance from the Babylonians). 9 digits in base 60 is roughly 16 digits in decimal! And finally, we get back to Europe; in the 17th century, van Ceulen used the power series to work out 35 decimal places of π. Alas, the publication of it was on his tombstone.

Then we get to the 19th century, when William Rutherford calculated 208 decimal places of π. The real pity of that is that he made an error in the 153rd digit, and so only the first 152 digits were correct. (Can you imagine the amount of time he wasted?) That was pretty much it until the first computers came along, and once that happened, the fun went out of trying to calculate it, since any bozo could write a program to do it. There's a website that will let you look at its computation of the first 2 hundred million digits of π. The name of π came from Euler (he of the great equation, eiπ + 1 = 0). It's an abbreviation for perimeter in Greek. There's also one bit of urban myth about π that is, alas, not true. The story goes that some state in the American Midwest (Indiana, Iowa, Ohio, Illinois in various versions) passed a law that π=3. Didn't happen.. What is π? Pretty much everyone is familiar with what π is. Take a circle on a plane. Measure the distance around the outside of it, which is called the circumference. Divide that by the diameter of the circle. That's π.

It also shows up in almost anything else involving measurements of circles and angles, from things like the sin function to the area of a circle to the volume of a sphere. Where it gets interesting is when you start to ask about how to compute it. You get the relatively obvious things - like equations based on integrals to calculate the area of a circle. But then, you get the surprising ones. After all, π is a fundamental geometric number; it comes from the circumference of a circle. So why in the world is the radius of a circle related to an infinite sum of the reciprocals of odd numbers? It is.

π/4 is the sum of the infinite series 1/1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 ...., or more formally:

Or how about this? π/2 =

What about a bit of probability? Pick any integer at random. What's the probability that neither it nor any of its factors is a perfect square? 6/π2. How about a connection between circles and prime numbers? Take the set of all of the prime numbers P, then the product of all factors (1-1/p2) is.. 6/π2. What's the average number of ways to write an integer as the sum of two perfect squares? π/4.