Using playful mathematics to illuminate practical problems in thermal system design.

When I was in elementary school, around the age of 10, I was introduced to a mathematical concept that is extremely simple and yet appears in all sorts of sophisticated, complex mathematics. This is, of course, the number known as π, or pi.

What we learned was that the ratio of the perimeter of a circle to the distance across its middle was the same no matter the size of the circle, and that this ratio is called “pi.” We were also told (not entirely accurately as it turns out) that π was 22 divided by 7. This is quite handy for mental arithmetic. To multiply by π just double your number, multiply by 10 (add a zero at the back), add 10% (add the number before the zero was tacked on) and then, the hard part, divide by seven. As I got older I was told that π wasn’t actually 22/7, it was 3.14—another half-truth. Later we learned that in reality π is transcendent, which sounded very hippy-dippy, but in other words it means it goes on forever without ever repeating itself (as far as we know). Both of the school versions are approximations.

Oddly, the elementary school fraction 22/7 is a more accurate representation of π than 3.14 because the fraction in decimal form is 3.142857… and the true value of π to six decimals is 3.141593. This introduces one of my other favorite numbers, the decimal representation of 1/7 which is 0.142857142857142857…. There’s not space here to explain all the wonderful things about this number, but I’ll just say that 1/17 is even better.

The Greek letter, π, short for “periphery,” was first used to denote the ratio by the mathematician William Jones in the early 18th century, but engineers and scientists had been aware of the importance of the ratio for thousands of years before Jones named it, using a variety of approximations. The Chinese used 355/113 which is accurate to seven decimal places and the Indians used √10, which isn’t really accurate at all—it is 3.16227766… which is 10,000 times worse than the Chinese estimate. However it does give a handy hint that π2 is close to 10 (within 1.5%)—another useful shortcut for mental arithmetic. The second century mathematician, Ptolemy, came in between, with 3, 8 minutes and 30 seconds, or 3.141666….

Of course if you start your understanding of π by thinking about circles, as we all did in school, it is easy to get confused when it starts popping up in all sorts of other places, such as Euler’s identity: the beautifully simple equation eπi+1=0. This shouldn’t be a surprise since Euler was investigating the exponential function—the one where the rate of increase at any point equals the value of the function at that point. The thing about diameters and circumferences is just a special case of Euler’s more general work which proved that the exponential function is periodic in the complex plane, but that’s probably beyond most fifth graders.

I will be writing more about the teaching of math in school in a coming issue. Meanwhile, if you want to remember more digits of π then the following sentence, attributed to the English physicist, Sir James Jeans, is useful. Jeans said “How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics.”

Sir James liked a drink, alcoholic of course, with his slice of pi.