The 14th of March is Pi Day. It’s the day in March when a math lover might eat a slice of pie to celebrate the division of a pie’s circumference by its diameter. I don’t know how many do that, actually, but the United States House of Representatives supported the designation of Pi Day back in 2009, and there’s a web site (https://www.piday.org) where you can buy Pi Day t-shirts and have all sorts of mathematical fun. If I wanted you to quit reading and go there I’d have made that address into a link. Keep reading.
If you recall any math at all from your high-school classes you may remember that pi, the lower case Greek letter π, stands for the number you get when you divide the circumference of a circle by its diameter. And when you carry out that division you get 3.14 or March 14th. If you carry out the division a couple of decimal places further, you get 3.1415, so pi day was an especially big deal in 2015 when 3.14.15 or 3/14/15 came around. I have the feeling I’ve already lost half the readers who read the first sentence of this paragraph.
Some numbers come up so frequently that it’s convenient to give them a name, and 3.1415 comes up very, very often and it saves time to write π instead of the number. Maybe this is the time to point out that 3.14159 is the way the number begins, but it doesn’t ever end. It goes on and on forever, never repeating and never ending.
Another endless number that comes up a lot is called e, and that begins with 2.718. That name doesn’t have the cachet of a Greek letter but, after all, e isn’t as famous as pi. And probably the reason e isn’t as famous as pi is because high-school math usually doesn’t get to e. But this is a digression and I think the only reason it’s mentioned here is because I re-discovered e one day while trying to figure out the magic of compound interest on $100.00.
In mathspeak, pi is an irrational number — not to imply that 3.14159 is a particularly wacky numeral, but simply that you cannot get it by dividing a number by another number. Or, to speak mathematically, it isn’t the ratio of two numbers and, hence, it’s irrational. So, it’s not equal to 22/7, which many students pick up in high school.
You can calculate pi by drawing a regular hexagon inside and outside a circle, and you can use the sides of the hexagon as the base of triangles whose vertex is at the center of the circle. It’s simple enough to calculate the areas of the triangles and hence the hexagons, and you know that the area of the circle is bigger than the inside hexagon and smaller than the outside hexagon. Archimedes did that and kept doubling the number of sides until he had a 96-sided polygon. Then he was able to show that pi was bigger than 223/71 and smaller than 22/7. And that’s where 22/7 comes from.
Our pi is transcendental. That sounds more unusual and ecstatic than it is. Again, in mathspeak, a transcendental number is not algebraic, which is to say that it’s not a root of a non-zero polynomial equation with rational coefficients. That should satisfy you. Most real numbers are transcendental, but if we start discussing what we mean by that previous sentence we won’t enjoy much of the day.
At this point I’m going to stop writing pi and instead use π, because this is about math, not literature. And as we finish I’ll return to what we first learned about dividing the circumference of a circle by its diameter, namely that the answer goes on forever. Maybe a better way of grasping why or how π is infinite is to think of dividing the diameter into little equal pieces with nothing left over, or dividing the circumference into little equal pieces with nothing left over. You could do that; it’s not impossible at all. What’s interesting is that if you were to do it, you’d never find a size of equal little pieces that that would work on the circumference and on the diameter, too, with nothing left over. If that puzzles and fascinates you, you might have a head for mathematics. Or not.