student activity 11.4 So what is this thing called pi? the value 22 7 Many primary schools students are taught that pi ( ) has . Indeed some students reach high school believing that is exactly 22 or exactly 3.142. 7 Hopefully, you know better by now. It is well known that the Egyptians and Babylonians understood that there was something special about the ratio of the circumference of a circle to its diameter, but they didn’t know its value to any real degree of accuracy. The modern symbol of was invented in the early 1700’s by William Jones, who decided “Pi (rather than some other Greek letter like Alpha or Omega) was chosen as the letter to represent the number 3.141592...because the letter in Greek, pronounced like our letter 'p', stands for 'perimeter'.” It was discovered that pi was a little over three through simple measurements using string (as you may have done in primary school). The Babylonians did slightly better with a value of 3 1 but the 8 Egyptians had a less accurate version of 4 ( 8 )2 9 that worked out to approximately 3.160484. If nothing else, this was not exactly a value that was easy to use in calculations! What value does your calculator approximate for ? ................................... 1. What is the percentage error for these values for ? a) Egyptian? .............................................................................. % b) Babylonian? .............................................................................. % c) Section 11.4 22 7 ? .............................................................................. % Page 229 As computers gained power we were able to calculate the value of to any number of decimal places required. By 1995, the Chudnovsky brothers in New York, had computed to 2 billion decimal places using a home made super computer. Purely for your amusement, the first 1000 decimal places of are given below. = 3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986 280348253421170679821480865132823066470938446095505822317253594081284811174502841027019 385211055596446229489549303819644288109756659334461284756482337867831652712019091456485 669234603486104543266482133936072602491412737245870066063155881748815209209628292540917 153643678925903600113305305488204665213841469519415116094330572703657595919530921861173 819326117931051185480744623799627495673518857527248912279381830119491298336733624406566 430860213949463952247371907021798609437027705392171762931767523846748184676694051320005 681271452635608277857713427577896091736371787214684409012249534301465495853710507922796 892589235420199561121290219608640344181598136297747713099605187072113499999983729780499 510597317328160963185950244594553469083026425223082533446850352619311881710100031378387 528865875332083814206171776691473035982534904287554687311595628638823537875937519577818 577805321712268066130019278766111959092164201989… Over the centuries of mathematical development one way to have your name remembered has been to find a way to calculate more accurately or quickly! For example, a monk called Gregory became famous for finding Gregory’s Series: 1 1 1 1 1 1 1 = 1 + + + +… 3 5 7 9 11 13 15 4 John Wallis (1616 – 1703) discovered Wallis’ Product which gave as: 2 2 4 4 6 6 ... = 2 1 3 3 5 5 7 ... The problem with both of these is that they converge very slowly to , requiring that you evaluate many thousands of terms just to obtain a value accurate to four or five decimal places! 2. Calculate the decimal value of Gregory’s Series up to the number of terms shown above. Multiply by four to find the approximation of and calculate the percentage error. 3. Calculate the decimal value of Wallis’ Product up to 20 x 20 on the top line, remembering that the values on the bottom are slightly out of step with those on the top. Multiply by two to find the approximation of and calculate the percentage error. Section 11.4 Page 230 4. If your graphics calculator has a seq uen ce tool then use it to evaluate these sequences using the iterative rules below. In each case the first two terms have been supplied, as some calculators require them. The final multiplication by four and two respectively has been incorporated into the formulae. The formula for Wallis’ Product evaluates two terms of the product per term in the sequence. Gregory’s Series: T(1) = 4 * 1 T(2 ) = 4 * (1 1/3 ) T( N ) = T( N 1)+(1)^( N 1) * 4/(2 * N 1) Wallis’ Product: T(1) = 2 * 2 2 /1 * 1/3 T(2 ) = 2 * 2 2 /1 * 42 /3 2 * 1/ 5 T( N ) = T( N 1) * (2 * N )2 /(2 * N 1) * 1/(2 * N +1) a) Use these formulae to calculate the 10th, 100th and 1000th terms for each sequence. b) Comment on the accuracy of these values. c) Comment on the manner in which they converge on the true value of . Are they approaching from above, from below or in some other manner? Section 11.4 Page 231 5. If you have basic programming skills it is quite easy to write a program for your graphics calculator or computer that will calculate to any degree of accuracy your machine will handle. For a calculator it’s generally around 12 or 13 decimal places. The program below uses a method called the Brent-Salamin Algorithm. The calculations in each iteration are very simple and it quickly converges on the true value for . Explaining why it works is beyond the level of mathematics in this worksheet! A=1 {Load initial values for A,B,R and S} B = SQRT(2) / 2 R=1 S = 0.5 {Precision of the result?} INPUT “How many decimal places?” D DO M = (A+B) / 2 N = SQRT (A * B) 2 C = (M - B) R=2*R S=S–C*R 2 P=2*M /S IF ABS(C) < 10^-D THEN {Accurate enough yet?} PRINT “PI = ”, P ELSE A=M {M and N become the new values of A and B} B=N END UNTIL ABS (C) < 10^-D {Repeat until accurate enough} If you have the facilities to use this program then use it to calculate to as many decimal places as your machine’s accuracy allows. Section 11.4 Page 232 6. Archimedes (287 BC – approximately 212 BC) was one of the most famous investigators into the value of . As you did in the earlier sections of this worksheet, Archimedes also approximated a circle using polygons. He used an inscribed and a circumscribed polygon, as shown right, to obtain values for which were slightly smaller and slightly larger than the true value. rad ius To make things difficult, Archimedes didn’t have a modern calculator and sin, cos and tan had not been discovered, nor was there any real understanding of square roots. In fact, use of the modern decimal system had not even begun so all calculations had to be completed using fractions and ratios. So what did Archimedes do? He invented a series of very clever techniques that allowed him to: • find fractional approximations to square roots, and • find the perimeter of a polygon with twice as many sides if he knew the perimeter of an existing one. In other words, if he knew the perimeter of a six sided polygon then his rule would allow him find the perimeter of a 12 sided polygon relatively easily. Archimedes obtained the enclosing range for of 3 10 < < 3 10 71 70 This is equivalent to the decimal range of 3.140845… < < 3.142857… So where did the value 22 originally come from? It came from Archimedes’ value of 7 3 10 or 3 1 or 70 7 22 7 . Now use the Polygonal paths explorer (see section 11.6). This will allow you to use inscribed and circumscribed circles to find enclosing ranges for of increasing accuracy. How many sides must the polygons have before the resulting enclosing range of values for is accurate to at least, a) 1 decimal place? Answer: .............................................................. b) 2 decimal places? Answer: .............................................................. c) 3 decimal places? Answer: .............................................................. d) 4 decimal places? Answer: .............................................................. compared to your calculator value? Section 11.4 Page 233 Section 11.4 Page 234