Pi Day: The logic of irrational numbers…
Pi is the ratio of the circumference of a circle to its diameter. In other words, it is how many times the distance across a circle takes to make the distance around the circle. Regardless of the size of the circle, this ratio is always pi – about 3.14 – times. This number is also shorthand for March 14 – which is why the date is celebrated as “Pi Day.”
Pi is an important number because it appears often in mathematics and physics formulas. It is particularly fascinating because it is irrational – which means it cannot be expressed as a simple fraction. That also means that when we do a decimal expansion of pi, it never ends and it never repeats. Rounded to eight places, Pi is 3.14159265. But it has been calculated to millions of places (what else are you going to do with your super computer on Pi Day?) and there is no end in sight.
A few other important irrational numbers include “Euler's number” e (a common mathematical constant of approximately 2.7182 that is the base of natural logarithms), the square root of two, and the “golden ratio” φ (two amounts are in the golden ratio – about 1.618 – if their ratio is the same as the ratio of their sum to the larger of the two quantities: a/b = [a+b]/a).
Numbers are even or odd, right? Not irrational numbers. And it was this enigma that helped prove irrational numbers existed. And there was resistance. The followers of Pythagoras, for example, were certain that all numbers could be expressed as the ratio of integers. While it is probably not true they drowned one of their own – Hippasus – for proving irrational numbers were real, that such stories exist reveals that principles of math can be important to how people understand the world.
Why do irrational numbers exist? To start, let us think about the math system we now use most commonly – the base 10 digital system (once you reach 10, you add a new place and start over with one). What we need to remember is that we use this system because humans have 10 fingers to count on: in English, each finger is a “digit” – hence “digital” logic. The metric system defines the world in digital terms: water freezes at zero degrees and boils at 100; there are 10 millimeters in a centimeter and so on. Other math systems are based on dividing and fractions. The gallon, for example, is divided into four quarts, those into pints and those into cups. There are eight ounces in a cup and 16 cups in a gallon. Base eight or 16 might seem weird until you think of them in terms of dividing in half and then half again – 16, 8, 4, 2, 1. Inches, for example, are divided and then divided again and again. And this is something we can often do without measuring. Think of cutting a pizza: half, then half, then half again = eight even slices.
One surprisingly old math system is the “sexagesimal” system for describing time: We all use it today and it was invented by the Sumerians over 4,000 years ago! Each hour is divided into 60 minutes and each minute into 60 seconds. We keep it around because it works – it’s easy to divide 60 into two, three, four, five or six equal parts.
If we think of pi (again, the ratio of the circumference of a circle to its diameter) or the square root of two (the diagonal length across a perfect square where each side is one unit), it is easy to see these irrational numbers have real values. So, the problem lies with the kind of math we use to describe any given situation. For daily life, digital math is more than sufficient, so we tend to think of it as perfect. But once we’re doing serious math or physics, it is both easier and far more accurate to use the square root of two (√2) or pi (π) rather than their digital approximations.
In everyday English, “irrational” means emotionally unreasonable. It comes from the Latin, “ir” = “no” and “ratio,” so… “no ratio” – and that is the precise definition of an irrational number: It cannot be defined by a simple fraction, which is a ratio.
Numbers, whether rational or irrational, integer or containing fractions, allow us to describe the universe and allow its mysteries to be comprehended. For all our technology, we require the tools of both elementary and complex mathematics to allow us to grasp the fundamentals of the cosmos. From everyday electrical current, to the furthest reaches of outer space, pi gives us the intellectual constructs to push the bounds of knowledge ever further.
Words by Daniel Kany