Life of Pi





Preface¡¦
What day is March 14th? Most of the people will say ¡°White day.¡± The day when boys give candies to girls whom they have crushed on. However, outside Korea, many people know March 14th as Pi day! In Europe and US, Pi Day has already become universal. Especially in San Francisco, USA, they celebrate the birth of Pi(¥ð) very grandly at March 14th 1:59. In year 2000, a math club in Pohang University of Science and Technology, first hosted the celebration of Pi Day. It¡¯s a little hope of the mathematicians and scientists for people to remember Pi Day more than White Day made by merchants who wants to sell candy^^.


What is Pi?

Before going on and on about the life of pi, let¡¯s find out what pi is. The word ¡®pi¡¯ was first adopted from the word ¡®periphery¡¯ meaning the circumference. Pi is the ratio of the circumference to the diameter of a circle which is known to be a constant. We usually use the approximation 3.14. However, the number value ¡®pi¡¯ is not 3.14 but a transcendent number meaning there isn¡¯t an end to it like you can see the column beside the article. This article will be about how mathematicians or scientists were able to get the value of pi through the centuries before computer was invented and after the creation.


From the Bible
There is a verse in the Bible where it gives us the information of how they calculated the constant pi: And he made a molten sea, ten cubits from the one brim to the other: it was round all about, and his height was five cubits: and a line of thirty cubits did compass it about. (I King 7, 23) There is the same verse in II Chronicle 4, 2. The interesting fact here is that the number pi equals 3. This value isn¡¯t that accurate even in the view of that time since the Egyptians and the Mesopotamians used much more accurate values like 25/8=3.125 and ¡î10=3.162. The important fact about this value is that they are obtained through measurement and not by a specific reason or theory.



The First Theorectical Approach


First, consider a circle with the radius of 1. Then inscribe a regular polygon of 3x2n-1 sides, with the semi perimeter as bn. Also circumscribe a regular polygon of 3 x 2n-1 sides, with semi perimeter an. If n=2, it will look like the picture on the right. Then, use trigonometric notation which will give the solution:
an=Ktan(¥ð/K), bn=Ksin(¥ð/K)
will equal each semi perimeters since the radius of the circle is 1 and K=3 x 2n-1. With the same process, we will get:
an+1=2Ktan(¥ð/2K), bn+1= 2Ksin(¥ð/2K)
It is not so difficult to get to this conclusion:
(1/an + 1/bn) = 2/an+1 ------ ¨ç
an+1bn = (bn+1)2 ------ ¨è
Archimedes¡¯ conclusion of the calculation of ¡®pi¡¯ is actually b6 < ¥ð < a6. Here¡¯s something surprising. Archimedes was before the time to have the advantage to use an algebraic and trigonometric notation which means he had to derive ¨ç and ¨è only using his geometrical knowledge. In addition, he also did not know the decimal notation of numbers, meaning he had no way to calculated a6 and b6. So, he was a amazing thinker to imagine and calculate such constant. It¡¯s a wonder how he went so far as n=96!!!!!




Using Arctangent Infinite Series
In the 1600s, the discovery of arctangent infinite series (the inverse function of tangent function) brought another big step to the calculation of pi. It was first used to compute the value of pi by two different mathematicians, who worked independently, Leibniz and Gregory. This is the arctangent series used to calculate pi:
If we expand the equation above to an arbitrary number, it¡¯s possible to get any arbitrary precision. What Leibniz and Gregory found was the term to find an equation to find the term of pi. From the diagram on the right, the following equation can be derived: sin(¥ð/4) = cos(¥ð/4) = r/r¡î2 = 1/¡î2 tan(¥ð/4) = sin(¥ð/4)/cos(¥ð/4) = (1/¡î2)/ (1/¡î2) = 1 Inverse the tangent to arctangent which will be like: tan(¥ð/4) = 1 -> 1= ¥ð/4 Then we put in the result into the firs formula which will then end like this: From the diagram on the right, the following equation can be derived: sin(¥ð/4) = cos(¥ð/4) = r/r¡î2 = 1/¡î2 tan(¥ð/4) = sin(¥ð/4)/cos(¥ð/4) = (1/¡î2)/ (1/¡î2) = 1 Inverse the tangent to arctangent which will be like: tan(¥ð/4) = 1 -> 1= ¥ð/4 Then we put in the result into the firs formula which will then end like this: This is the famous Gregory/Leibniz series, the first infinite series ever discovered by pi. However, this series is practically useless because we have to calculate 1.0x1051 repeated series in order to get 100 accurate decimal points of pi.


Who is Leibniz?
Leibniz occupies a large place in both the history of philosophy and the history of mathematics. He invented calculus, which we generally use instead of Newton¡¯s because it¡¯s much easier. In philosophy, he is one of the three great 17th century rationalists. Leibniz also made major contributions to physics and technology, and anticipated notions in almost all scientific areas.He was such a genius that even today there are so many notes that he left that cannot be interpreted. What a man!


Machin¡¯s Variations
In 1702, John Machin made another record in calculating pi. Machin increased the rate of convergence of Gregory/Leibniz variation which made him possible to get 100 decimal places of pi. Machin¡¯s variation was so effective that even today we use this formula to program computers to compute the pi digits. This is his how he got his variation: tan¥â = 1/5 ¡æ then use double-angle formula which gives tan2¥â = 2 tan¥â/1-tan2¥â = 5/12 ¡æ use the double angle formula again tan4¥â = 2 tan2¥â/1-tan2 2¥â = 120/119 Note that tan4¥â only has 1/119 more than 1 which is arctan of ¥ð/4. So, Machin decided to calculate this small difference: tan(4¥â- ¥ð/4) = (tan4¥â-1)/(1+ tan4¥â) = 1/239 Therefore: arctan(1/239) = 4¥â- ¥ð/4 Now, substitue the value ¥â with its arctan value: arctan(1/239) = 4(arctan1/5)- ¥ð/4 Finally solve the equation for ¥ð/4 which will bring the conclusion:


Inverse Trigonometric Function
In the passage above, it keeps using the value ¡®arctangent¡¯ or ¡®arctan¡¯. Then what is this thing that sounds something similar to tangent? Arctangent is the inverse function of tangent function. We call the whole set of arcsin, arcos, arctan etc. as inverse trigonometric function. Like we¡¯ve learned in class, to find the inverse function all we have to do is to switch x and y. The graph on the right is the graph of arctangent.


Mistake of Shanks
William Shanks was an English amateur mathematician who calculated up to 707 digits using Machin¡¯s formula which unfortunately was only correct up to 528th digit. This was noticed by De Morgan (do you remember De Morgan¡¯s Law in sets?). Very soon after Shanks¡¯ calculation, he found out that a statistical error in Shanks¡¯ result. There was a shortage of 7¡¯s statistically. De Morgan mentioned this factor in his book, Budget of Paradoxes and it remained as a mystery until 1945 when Ferguson found out the reason. He found out that the 528th digit was mistaken meaning all the other calculation was wrong. So in 1949 a computer was used to calculate pi to 2000 places which resulted the predicted amount of 7¡¯s. And all the other digits passed the statistical tests of randomness.


34080 times of Needle throwing?!
Calculating and finding the value of pi became a silly and needless thing to do after Lindemann proved that pi was a transcendental number meaning that pi is not the solution of any polynomial equation with integer coefficients. As a result, computing pi with computers became a tool to measure the capacity of the computer rather than the mean of curiosity. In the 20th century, those kind of pi formulas disappeared and it was replaced by statistical interpretation. The method to calculate pi using statistics or by throwing needles is called the Buffon¡¯s needle experiment.
Like the picture on the left, throw needle with the length of §¤ (§¤ The value of pi calculated by Lazzerini(1901) was amazingly accurate. It¡¯s probably because of the effort of his 34080 throws! The result he got was this: ¥ð = 355/133 = 3.1415929 This is again surprising because the result was exactly the same as the one Zu Chongzi found before hundreds of years.


To close¡¦
Life of Pi is not as simple as many people think. It¡¯s basically because so many researches have been done by many curious researchers. Pi was always the challenge and the interest of the mathematicians until the proof that pi was actually a transcendental number. Like you can see from the article, mathematicians gained the same value of pi to which Zu Chongzi have gained way before all those theories were found. Interesting? No? Oh, Well; A constant that we use everyday right now had went through a long period of polishing to be the very ¥ð we are using right now. Of course, we wouldn¡¯t need to use all those infintively long digits of pi in our life but it would be the ¿¹ÀÇ for Pi to at least think of it at least for one minute a year, March 14th 1:59. ^^