Proportionality

Proportionality Part 8

= 3.141 592 653 5... • Pi has excited mathematicians for thousands of years. Why all the interest? • Pi arises in may fields of mathematics, for example: • In area of a circle • In functions and graphs • In analytic trigonometry • In complex numbers • In differential and integral calculus • In Fourier series

Where does Pi come from? • Take your circular objects and measure the diameter and the circumference and fill in the chart below. • Circular object circumference diameter ratio of circum to diam_____________ ______ ______ ______________ _____________ ______ ______ ______________ _____________ ______ ______ ______________ _____________ ______ ______ ______________ _____________ ______ ______ ______________ • It is the ratio of the circumference to the diameter of a circle

Why ? • The Greek letter is used since it is the first letter of the Greek word for "perimeter", meaning "distance around the outside". If we have a circle with diameter 1 unit, then the perimeter will be pi units. • = 3.141 592 653 589 793 238 462 643 383 279 502 884 197 169...

Pre-History of Pi • The Bible contains two references which give a value of 3 for pi. • I Kings 7:23 states, "Also he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits in height thereof; and a line thirty cubits did compass it round about." This implies that pi = C/d = 30/10= 3 • The Babylonians gave an estimate of pi as 3+ 1/8 = 3.125, while the Egyptians gave = 3.1605 in the Rhind papyrus and 22/7 elsewhere. • http://mathworld.wolfram.com/Pi.html

Some better approximations • As far back as the 3rd century B.C.E., Archimedes found that pi was between 223/71 = 3.140 845 07... and 22/7 = 3.142 857 143... • By the 5th century, the Chinese had the respectable value (correct to 6 decimal places): 355/113 = 3.141 592 92... • By the 17th century, the use of infinite series meant that we could get more and more accurate values of pi, as long as we had plenty of time (or computing power) to calculate more decimal places.For example, Leibniz (of calculus fame) developed the following expansion for pi, (but it wasn't very good):

Pi and Computers • One of the first things that computer engineers do when they are testing a new computer chip (or programming language or operating system) is to calculate the value of pi to check that everything is working correctly. • For example, just after World War II, the huge ENIAC computer managed to compute pi to just over 2,000 places. • The most recent value was achieved by the Japanese in 2002, with 1.2 trillion decimal places.

Pi and Art • How's this for a great piece of pi art in downtown Seattle?

In Art and Music….. • Pi appeared in Alfred Hitchcock's 1966 film Torn Curtain, including in one scene where Paul Newman draws a symbol in the dirt with his foot at the door of a farmhouse. In this film, the symbol for pi is the pass-sign of an underground East German network that smuggles fugitives to the West. • The 1998 film Pi is a movie about a mathematician who is slowly going insane searching for a pattern to the Stock Market. Unfortunately, the film has essentially nothing to do with real mathematics. • 314159, the first six digits of Pi, is the combination to Ellie's office safe in the novel Contact by Carl Sagan. • The 2005 album Aerial features a song called "Pi" in which the first digits of pi are interspersed (unfortunately incorrectly) with lyrics.

Suduku anyone?

What does Pi sound like? • http://www.avoision.com/experiments/pi10k/index.php • This website allows you to assign notes to each of the digits 0 through 9. Then it plays the notes as they appear in Pi? • Do you think you can pick the notes so that the result is a nice melody? (Why or why not?)

And you can't beat this: Mmmmmmmm………… pi

And of course for those who have nothing better to do….

Proportionality