Simple Simon met a Pi Man
Going round and round
Said Simple Simon to the Pi Man
“Tell me what you’ve found.”
The Pi Man said “I went around,
But now I’m at a loss.”
Said Simple Simon to the Pi Man,
“Why not cut across.”
The Pi Man did as Simon said
And went the whole way through
And when he factored round o’er ‘cross,
He found a number true
The Pi Man tried another orb
and near as he could see
The ratio of round to ‘cross
Was always close to three.
The Pi Man knew not what he had wrought. His number, which turned out to be greater than three by about 5%, is probably the most interesting in mathematics, cropping up as it does in unexpected places. Pi is a mysterious number, not even classified properly until the 19th century. Let’s celebrate this Pi Day by looking at the various aspects of the ratio of the circumference of a circle to its diameter — a veritable potpourri of π.
What Kind Of Number Is π?
Numbers are classified in various ways into many useful categories: primes, integers, fractions, decimals, powers of nine, etc. Into what category or categories does π fall?
The ancient Greeks of Euclid’s time (approximately 300 B.C.) suspected, but could not prove, that the ratio of the circumference to the diameter of a circle was always the same. With a bias towards the tidy, they thought, as did the Pi Man, that it was the integer three. Proof that it was a constant — and a remarkably accurate estimate of pi — were derived around 250 B.C. by Archimedes. Archimedes’ method was to inscribe and circumscribe polygons in and outside a circle. The value of the circumference would lie between the areas of the two polygons. After calculating this number for polygons with 96 sides, Archimedes arrived at limits of 3.141 and 3.143 for pi.
The question then arose whether π could be precisely described by a fraction. Numbers that can be so designated are called rational; those that cannot are called irrational. This has nothing to do with the sensibility of the numbers. The words are based on the root ‘ratio.’ Numbers that can be expressed as a ratio of integers are rational. 22/7 approximates π to three decimal places.
Irrational numbers can be thought of as numbers that slip through the cracks between the rationals — but that would be metaphorically wrong. In fact, there are many more irrational numbers than rational ones. It’s as if we thought we had a few cracks in our roof through which a couple of drops of rain leaked, only to find that there was really little roof at all and it was mostly holes.
Still, though most numbers are irrational, it is devilishly difficult to prove that an individual number is irrational. And so, as the value of pi became ever more refined, there was little doubt that it was irrational — but it was a long time until Johann Heinrich Lambert proved it in 1761.
To make matters worse, there is a subset of the irrational numbers known as transcendental numbers. Although most irrationals are transcendental, it is even harder to prove that any individual number is transcendental.
So it wasn’t until 1882 that Ferdinand von Lindemann proved that π is transcendental.
How Accurate Does Pi Have To Be?
Modern calculating machines (computers) have made it possible to determine pi to unimaginable accuracy — well beyond a trillion decimal places.
This game of extending the value of pi to trillions of decimal places is overkill in the extreme. Using pi to an accuracy of 40 decimal places is sufficient to calculate the circumference of the universe to an accuracy of the diameter of a hydrogen atom.
The Great Leonhard Euler And π
Leonhard Euler popularized π as the designation for the well-known ratio of the diameter of a circle to its circumference. Euler also found a remarkable relationship between π and the imaginary numbers (multiples of the square root of negative one). Euler’s formula is perhaps the most aesthetically pleasing relationship in all of mathematics: e^iπ + 1 = 0. The formula relates Euler’s number, e, the basis of natural logarithms and continually compounded interest rates, to the ubiquitous π, the square root of minus one (designated i), and the two simplest ‘numbers,’ one and zero.
What Does It Mean?
Euler’s formula is derived from and leads to, deep mathematical relationships. One of its uses is in modeling wave functions, which is beyond the scope of this article. Let’s just rejoice on this Pi Day that our heroic number is involved in this beautiful, simple, and elegant mathematical truth.