A piem is a little poem where the length of each word is a digit of π. There's also a funny little trick for memorizing π, called a piem (argh!). What's the average number of ways to write an integer as the sum of two perfect squares? π/4. How about a 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/p 2) is. What's the probability that neither it nor any of its factors is a perfect square? 6/π 2.
What about a bit of probability? Pick any integer at random. So why in the world is the radius of a circle related to an infinite sum of the reciprocals of odd numbers? It is. After all, π is a fundamental geometric number it comes from the circumference of a circle. You get the relatively obvious things - like equations based on integrals to calculate the area of a circle.
#246 in babylonian numerals how to
Where it gets interesting is when you start to ask about how to compute it. 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. Divide that by the diameter of the circle. Measure the distance around the outside of it, which is called the circumference. Pretty much everyone is familiar with what π is. The story goes that some state in the American midwest (Indiana, Iowa, Ohio, Illinois in various versions) passed a law that π=3. There's also one bit of urban myth about π that is, alas, not true. It's an abbreviation for *perimeter* in Greek. The *name* of π came from Euler (he of the great equation, e iπ + 1 = 0). There's a website that will let you look at its computation of. 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. (Can you imagine the amount of time he wasted?) The real pity of that is that he made an error in the 153rd digit, and so only the first 152 digits were correct. Then we get to the 19th century, when William Rutherford calculuated *208* decimal places of π. Alas, the publication of it was on his tombstone.
#246 in babylonian numerals series
9 digits in base 60 is roughly 16 digits in decimal!Īnd finally, we get back to Europe in the 17th century, van Ceulen used the power series to work out 35 decimal places of π. *13* decimal places, computing a power series completely by hand! Astounding!Įven 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). That gives you an approximation of π as the average of 223/71 and 22/7 - or 3.14185.Īnd 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.
Here's a quick diagram using octagons to give you a clearer idea of what he did: 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 96-gon that the circle could be drawn inside. Archimedes (yes, *that* Archimedes) worked out a better approximation. We don't see any real progress until we get to the Greek. Which isn't such a great step forward it's actually a hair *farther* from the true value of π than the Babylonian approximation. 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). Especially when you realize *when* they came up with this approximation: 1900BC! (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. The oldest value we know for π comes from the Babylonians. How can you talk about interesting numbers without bringing up π?
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