Monday, April 20, 2015

Pi -- Part Two

Pi is irrational, meaning it cannot be expressed as the ratio of two whole numbers. There is no way to write it down exactly: Its decimal expansion continues endlessly without ever settling into a repeating pattern. No less an authority than Pythagoras repudiated the existence of such numbers, declaring them incompatible with an intelligently designed universe.

I was taught in school the simple fraction 22/7 as an approximation for pi. It only has 3 digits, yet it is a bit closer to the correct value than the apparently equivalent 3.14 in digits.

Twenty-two-sevenths works out to 3.142857… compared to the actual pi of 3.14159…. Another approximation from history is 256/81= 3.16049… and 339/108 = 3.1888…. The Chinese used 3927/1250 = 3.1416 exactly … about the closest of any of these attempts. Whether these ancients thought these were approximations or simply the best they could come up with isn’t clear.

The ancient Greeks had a puzzle that was never solved. It is called “squaring the circle.” That means to create a square using just the geometric tools of a compass and a straight edge that has the same area as a given circle. Modern mathematicians know that this is impossible since we now understand that pi is transcendental which means it that is not algebraic — that is, it is not a root of a non-zero polynomial equation with rational coefficients. The most prominent transcendental numbers are π and e, the base of natural logarithms. Further, it can be proven that any shape created with only a compass and straight edge are the algebraic numbers and doesn't include a value like pi.

So the ancient search for a method to “square a circle” is now known to be impossible. Sometimes in math and science it is as important to know what is impossible as it is to know how to calculate something.

It’s fair to ask: Why do mathematicians care so much about π? Is it some kind of weird circle fixation? Hardly. The beauty of π, in part, is that it puts infinity within reach. Even young children get this. The digits of π never end and never show a pattern. They go on forever, seemingly at random — except that they can’t possibly be random, because they embody the order inherent in a perfect circle. This tension between order and randomness is one of the most tantalizing aspects of π.

And yet π, being the ratio of a circle’s Circumference to its Diameter, is manifested all around us. For instance, the meandering length of a gently sloping river between source and mouth approaches, on average, π times its straight-line distance. Pi reminds us that the universe is what it is, that it doesn’t subscribe to our ideas of mathematical convenience.

Pi also opens a window into a more uncharted universe, the one consisting of transcendental numbers, which exclude such common irrationals as square and cube roots. Pi is one of the few transcendentals we ever encounter. One may suspect that such numbers would be quite rare, but actually, the opposite is true. Out of the totality of numbers, almost all are transcendental. Pi reveals how limited human knowledge is, how there exist teeming realms we might never explore.

A short explanation of rational and irrational numbers. Start with the integers … the whole numbers. (We’ll ignore zero and negative values.) Rational numbers are any number that can be formed from a ratio of integers or whole numbers. That is, a fraction of integers like 22/7. The ancient Greeks thought that was all there is. That’s why they called them “rational.” Then some guy figured out that the square root of 2 can’t be rational. He proved that, if it was a ratio or fraction of whole numbers then the numbers were not even, nor were they odd. Since all integers must be one or the other, there could not be such a ratio. That made such an impact on the Pythagoreans (a club that the guy was in and famous for their theorem or formula) threw him out of the boat … literally … they drowned him … or so the story goes.

Makes a good story whether true or not. One aspect of irrational numbers is that they can not be represented by a finite decimal expansion such as 1/2 = 0.5 or a repeating decimal such as 1/3 = 0.3333… (or 1/11 = 0.090909…). Transcendental numbers are even more complicated. They can’t even be represented by any root of a regular polynomial equation … but I promised to keep this simple, so I’ll stop there.

The combination of utility and mystery makes π a perfect symbol for all of mathematics. Surely the ancients, had they understood π better, would have worshipped it, just as they did the moon and the sun. They would have praised pi’s immutability: Pi = 3.14159... is one of the few absolutes that remain, unchangeable in a world of temporary existence.

Or is it absolute? The ratio of circumference to diameter might not be as fixed as we think. To understand why, imagine a circle drawn on the surface of a sphere. Its diameter, as measured along the bulging surface, will be greater than if the same circle is traced out on a flat sheet of paper. This observation might have been of only academic interest except for our inability, so far, to definitively determine whether the geometry of our universe is flat. If there is even a little curvature, then the value of π, as defined by this ratio, is not what we think. Thanks to Einstein we now have an absolute speed of light, but π might not be fixed.

Yes π, on cue, reminds us that it is an abstraction, like all else in mathematics. The perfect flat circle is impossible to realize in practice. An area calculated using π will never exactly match the same area measured physically. This is to be expected whenever we approximate reality using the idealizations of math.

To this day, some think that 22/7 or 3.1416 are the exact value of pi. Perhaps they fear the unknown of the unknowable and are as traumatized as Pythagoras by the idea of a non-fractional universe. The Indiana State General Assembly once proposed establishing pi as something like 3.2 to make commerce easier, but the law didn’t pass because Professor C. A. Waldo of Purdue University talked them out of it.

Maybe life would be simpler if pi = 3.2, but I would argue that life could not exist if π has such a simple value. There is no reason to be afraid. I’ve learned that it’s only when we try to stretch our minds around mathematics’ enigmas that true understanding can set in.

That’s the beauty of it!

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