Kitchen Math: How to Eat π Pies
Methodically slice up four pies to reach the irrational number pi.
Tune in to Science Friday on March 14, 2015, for a Pi Day Celebration.
Every child knows how to make their pie last: Eat half, then eat half of what’s left, then eat half of that…and so on. The part that’s left over will get smaller and smaller until eventually you can’t even see it any more, and the total amount of pie eaten will get very, very close to one, although it will take forever.
What if you want to eat not one pie, but π pies? Here’s how you do it. Start with four whole pies on plates. Remove a third from each of them. Then, out of each third, cut a slice the size of a fifth of a whole pie. Put those fifths back with the pies on the plates.
So far, you have this much pie on the plates:
Now keep going. Remove a slice the size of a seventh of a whole pie from each of the pies on the plates. Then cut a sliver the size of a ninth of a whole pie from each of those slices, and put the slivers back with the pies on the plates. Then remove an eleventh. Put back a thirteenth. Remove a fifteenth. Put back a seventeenth. The process you’re carrying out is this:
The amount of pie left on the plates will eventually have to stabilize, because at each stage you move a little less than at the previous stage, so your total amount of pie will have to approach something—and that thing happens to be the irrational number π. Now, enjoy your pi(e)!
It’s a fascinating fact about numbers that you can add up an infinite sequence of rational numbers (fractions where the top and bottom are whole numbers) and get an irrational number (a number that can’t be expressed as a fraction). This doesn’t sound like it makes any sense, because every time you add a fraction on, you get another fraction…but when you start adding up infinite numbers of things, weird stuff can start happening.
It takes several chapters of an advanced calculus book to get to the point of understanding why the process described above really gives pi. But if you’re into baking, there’s an oddly easier way to get an approximation for pi using kitchen math. Many bakers know that you can use a recipe for a 9-inch round cake to make an 8-inch square cake instead. Considering the formula πr^2 (used to calculate the area of a circle), this baker’s trick suggests that π4.5^2 =8×8=64.
If we use this to approximate pi, we get:
…which is closer than the approximation we got above using fancy math, and definitely good enough for cake!
Dr. Eugenia Cheng is an Honorary Fellow in the School of Mathematics and Statistics at the University of Sheffield, U.K., and is Scientist in Residence at the School of the Art Institute of Chicago. Her latest book is Beyond Infinity: An Expedition to the Outer Limits of Mathematics (Basic Books, 2017).