In today’s geek-positive culture, it’s cool to be able to post knowingly about 3/14 being Pi Day, and to use it as an excuse to eat pie. (I like pie.)
But being me, I see Pi Day as a chance to make a point about the reality of irrationality—at least in its literal, mathematical sense, but to some extent, in a wider sense.
If you’re like me, pi (or π) was one of the first irrational numbers you ever encountered, probably in a geometry class—the ratio of the circumference of a circle to its diameter. “Irrational” means that this number cannot be expressed with a ratio of integers—for instance, by 22/7, which is often used as a close approximation. What that also means is that the numbers to the right of the decimal place when pi is represented as a number—for example, in 3.14159265…–do not repeat in a permanently repeating pattern. The digits are infinite. Probably the most common irrational numbers commonly encountered are the square roots of any number that is not the square of a rational number—for instance, the square root of 2. π, however, is also a transcendental number, meaning it is not the root of any polynomial with rational coefficients (unlike irrational square roots).
π is thus a gateway into higher mathematics, where we start to realize that there are different kinds of numbers beyond those we encounter in basic arithmetic Yet π itself is not a difficult number—what it represents is very simple, and key to understanding any function that is based on circles and curves.
Irrational numbers, therefore, are highly rational in the language sense of the word. It’s easy to understand π. We’ve all seen circles. We can see and measure their circumference and diameter, and will know that at least one of these is not going to land on an easy-to-express number. Your 12” pie will have a circumference of 37.6991118-ish inches. π, as a constant, is also neutral to whatever unit of measure we choose to use.
Where things actually get weird are with negative numbers At first, they’re understandable in real terms—having -20 dollars means that someone owes you (or you owe someone) 20 dollars. You can also conceive of them in two or three dimensional space. It’s when you start to think of them in the abstract that things start to get harder to understand. Taking the square root of -1 gets you the imaginary unit i. You can form more imaginary numbers by multiplying i by a real number. (A real number is any number that can be represented as a place along an axis. Both rational and irrational numbers are real.) So yes, you can have an imaginary number represented by πi. Initially, the term imaginary number was a derisive one, but as their mathematics were worked out, the imaginary became real (in the literary sense of the word “real.”). Incidentally, zero is both imaginary and real. And then you have complex numbers, which are the sum of a real number and an imaginary one. Technically, since zero is both imaginary and real, real numbers are also complex numbers.
Now if you’re like me, that last part started to get too complex to really grasp (pun absolutely intended). With imaginary numbers, we cross over into more abstract concepts, although complex numbers are definitely used in areas of engineering and physics. But the thing to remember is that all of our numbers are just a language of sort—a way of classifying what we see in nature. I like this explanation by M B Drennan:
“Don’t forget that maths is an invention, if you like the rules of a game by which we play. Maths is NOT a science, it does not represent reality (or even attempt to) – it is a system which merely WORKS. It is designed for convenience. Since the roots of maths predate the complexity that led in much later centuries to imaginary numbers, it is no surprise that later elaborations for functions originally unforeseen proved problematic. Further: do numbers have a use in the “real” world? Well, yes, but apparently crucial (and well-established!) continuations, like algebra, have only THEORETICAL or abstract uses: which is, in effect, the use imaginary numbers have. Imaginary numbers run contra to common sense on a basic level, but you must accept them as a system, and then they make sense: remember that nothing makes 2+2=4 except the fact that we SAY SO. Same with imaginary numbers. The discomfort you feel is the awkwardness not between reality and the i series but between the (deceptively named) “real” series and its i counterpart. And if by “use” you mean application, I am sure plenty of mathematicians and physicists and engineers will swear to their importance.”
So, back to π. As this article mentions, the story of pi is also the story of the “taming of infinity.” As Steven Strogatz elaborates, “pi represents a mathematical limit: an aspiration toward the perfect curve, steady progress toward the unreachable star. It exists, clear as night, with no end in sight.” And this is what appeals most to me. A circle drawn with a compass exists. We can see it, we can adjust the radius so that it is a simple number, like 4”, and we will know that its circumference will be 8 π”, which also looks to be a simple number when written thus. It is not the actual circumference but our ability to express the number that reflects that measurement that invokes the infinite. We are used to looking out at the sky to conceptualize infinity, but the infinite also comprises the infinitesimal. And the irrational turns out to be completely rational, simple and elegant. Mystery is to be had at the farthest edges of the universe, but also at the subatomic level, where the weird and wonderful and hard to comprehend both hold sway.
Meanwhile, I think I’ll have some pie. Or at least, I will continue to think about it.
Very nice post and amusing with that. I was not aware that there is a Pi-Day, I was more familiar with D-Day. However I like the idea. And devoting a post to Pi and imaginary numbers certainly woke my brain up while standing on the platform and waiting for the morning train! 😄
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