### Mathematicians & The Number PI

Canadian mathematician brothers Jonathan M. Borwein and Peter B. Borwein have as one of their interests the mathematical theory behind the calculation of Pi (and other mathematical computations) accurate to many digits. In this case, many means billions of digits.

Born in St. Andrews, Scotland and educated in Canada since the early 1960's, the Borwein brothers have had distinguished careers in Mathematics. Jonathan Borwein is currently at Dalhousie University; Peter Borwein is at Simon Fraser University. In 1993, Jonathan Borwein founded the Centre for Experimental and Constructive Mathematics at Simon Fraser University in order to marry the advanced capabilities of computers to branches of mathematical research.

Although much mathematical research usually takes a form that most people cannot easily understand, one of things that has captured the imagination of many people throughout history has been the calculation of the exact value of Pi. Here is a brief history of Pi. Thanks are due to the Borweins for information from their lectures.

### How Was PI Calculated?

The ancient Greek, Archimedes was the first to find a method for calculating Pi accurately using successive approximations with polygons of sides 6, 12, 24, 48, and 96 sides. The method he used relied on numerical calculation and reasoning, not measurement.  His method was the first (and the most enduring) for getting closer and closer approximations to the value of Pi. As well, his method can be extended indefinitely to get better and better results. Until the calculus, his was the best method of calculating Pi.   This method has been used by mathematicians from different parts of the world to calculate Pi accurately:

250 BC   Archimedes      2 digits of accuracy
700's    Tsu Ch'ung Chi   7 digits of accuracy
1500's   Al Kashi            14 digits of accuracy (using base 60 notation!)

Then, in 1579, the mathematician Viete, in finding a different means of calculating Pi, introduced a new sort of notation for forming a sequence of products that get closer and closer to a particular number, a limit.  This gave rise to some important pieces of modern mathematics.

With the discovery of calculus in the 1600's by Leibniz and Newton, some new formulas for Pi were derived. Many summation formulas that get closer and closer to the number Pi the more additions you do, were found.
Euler gave this formula:

Using calculus, Newton cleverly devised this specific formula from a more general case:

then confessed that he used idle time to calculate Pi to 15 digits.

Much naive work on Pi was devoted to the notion that it might be possible to find a repeating pattern of digits in its expansion.  If it repeats, of course, that means that it is a rational number.  Some rational approximations to Pi are:

but Pi was finally shown to be irrational in the late 1700's by Lambert and Legendre.

Pi was shown to be not only irrational but also transcendental (think "harder" or "even more irrational") in 1882 by Lindemann.  What transcendental means is that it can't be the solution of a polynomial equation with rational coefficients. What Lindemann proved was what the Greeks had "known" two millennia earlier.

Patterns in Pi are not easily discovered but modern mathematicians now think that it possible that every possible sequence of digits can be found somewhere in the expansion of Pi.  For example, the first occurrence of the digit sequence 0123456789 appears starting at position 17,387,594,880 of Pi's decimal expansion.

In the early 1900's Ramanujan produced this formula for Pi:

This formula was recently discovered from the notes of Ramanujan by Peter Borwein. It turns out that this formula adds correct digits at a great rate. Modified versions of it were used recently to calculate Pi to billions of digits.

### Pi Calculation Challenge

The first 70 digits of Pi:

3.141592653589793238462643383279502884197169399375105820974944592307816

Challenge: Use your calculator to calculate these approximations to Pi.  Compare your results to the actual value of Pi. How many digits are correct?

The first term of Ramanujan's Formula:

Newton:

Euler:

Tsu Ch'ung Chi:

Which formula is closest?

### Problem

Why can't you use Ramanujan's formula on your calculator to calculate more than one term (below is Ramanujan's formula with three terms in it)?

### More Interesting Facts about PI

The number of digits of accuracy used in calculations by computer are what limits the accuracy of the result, just as for your calculator. Normally, the calculation of the digits at position, say 100, depends on the accurate calculation of all of the previous 99 digits. So if you want to calculate a billion digits of Pi, your calculating machine will have to be able to keep that accuracy throughout its set of calculations. In the last 10 years Peter Borwein, using the experimental facilities of the CECM, discovered a most unusual formula for the digits of Pi. This formula:

easily calculates the binary and hexadecimal representations of Pi. (Binary or base 2, uses just the digits 0 and 1 for counting rather than the ten digits we use in the decimal system. Hexadecimal, base 16, uses 16 "digits".)

What is most important about this formula is that the digits in binary at any position can be calculated without knowing any of the digits before. Recently, this has been used to calculate the quadrillionth binary digit of Pi.

Mathematicians have used calculations of Pi to advance their subject which then indirectly have benefited the sciences. Its also true that some of these advances have gone immediately in other beneficial directions. Means of performing rapid multiplication (Fast Fourier Transforms), perfected in doing high precision arithmetic on computers, are used in processing medical imaging devices. Pattern searching in the digits of Pi parallels discovering patterns in the human genome. Pi calculations are often used to test the correctness of some computer components and were used in the last twenty years to find, test, and fix some computer chips.

Do we know everything about Pi now? Not likely.

For example, although nobody thinks the expression (Pi + e) is rational, we don't yet know how to prove that it is transcendental or irrational.