By Emma Laws, Cathedral Librarian
Friday 14 March is International Pi Day. It makes better sense in America where the month is typically written before the day, as in 3/14 or 3.14. Pi, or π, has a special place in mathematics – it expresses the ratio of a circle’s circumference to its diameter. You may have happy memories of calculating the area or circumference of a circle at school.
Pi is an irrational number – i.e. it has an infinite number of non-repeating decimal places. This never-ending number is usually approximated to 3.14159 but 3.14 is good enough for me. Akira Haraguchi, a retired Japanese engineer, holds the unofficial world record for reciting over 110,000 decimal places of Pi from memory. According to the Japan Times, Haraguchi describes Pi as “the religion of the universe” and as a quest for an “eternal truth”. His words bring to mind the major mathematicians of the 17th century who described mathematics as the language of God.
There are many formulas that attempt to express the exact value of Pi. One of these formulas is known as the Wallis Product and was devised by the mathematician and clergyman, John Wallis (1616-1703) in his Arithmetica Infinitorum, published in 1656. You can read an online copy on archive.org but it’s in Latin and not for the fainthearted.
Wallis’s formula expresses Pi as an infinite multiplication (product) of fractions. Each fraction is made of two even numbers on the top (numerator) and two odd numbers on the bottom (denominator), arranged in a repeating pattern:
- The first fraction is 2×2 divided by 1×3
- The second fraction is 4×4 divided by 3×5
- The third fraction is 6×6 divided by 5×7
- The fourth fraction is 8×8 divided by 7×9
and so on, forever.
Or, you can write it like this:
Each step in this multiplication brings the result closer to π / 2. Just multiply your answer by 2 and you have an approximate value of π. The more pairs of fractions you multiply together the more accurate your answer will be – or the closer you will be to an exact value of Pi. In his Treatise of Algebra (1685), Wallis describes similar calculations as “Infinite Series… or continual Approximations”. (The Exeter Cathedral Library has the first edition of this important work and, thankfully, it is in English).
I’m afraid, this is as much as I can get my head round. I rather like the story of how John Wallis performed calculations in his head to pass the time during sleepless nights. One night he calculated the square root of a 53-digit number. The answer was a 27-digit number, and he recalled it perfectly the next morning. His contemporaries were so impressed they wrote a paper about him in the Philosophical Transactions of the Royal Society in 1685.
Wallis was appointed to the Savilian Chair of Geometry at Oxford in 1649. He held this post for 54 years and made contributions to trigonometry, calculus and geometry. He was ordained in 1650 and published widely on theology, philosophy and logic. He was even proficient at French, Greek and Hebrew as well as Latin. You could say he was infinitely talented.