0! = 1 Because I Said So

I remember sitting on the deck of my parents’ house in suburban Philadelphia back in high school an explaining factorials to my friend Paul. I don’t know why I was explaining them to him, since Paul always had way more mathematical chops than me. (As an undergrad he did the Great Books program at St. John’s College in Annapolis, whose math component involved doing proofs about geometric solids using contemporary Euclidean techniques which Paul ate right up but would have given me, a geometrophobe, fits.) “A factorial,” I said, “is a certain kind of multiplication. When you write an exclamation point after a number–an exclamation point because the numbers get really big, really fast!–that means you multiply the number times its predecessor, times its predecessor, and so on down to 1.” Paul grasped the idea immediately. He saw, for instance, that the final multiplication by 1 was a bit of a formality, and asked what the factorial of zero was, because like I said he had mathematical chops and knew the right questions to ask.

“That’s the interesting part. The factorial of zero is one.”

“That’s stupid,” said Paul. “Because zero times any number is zero.”

“Well,” I sputtered. I wasn’t expecting this. “That’s the way it’s defined.”

“Then that definition doesn’t make any sense!”

A bit of background: back in high school the math thing was a sideline, and the main preoccupation of Paul, me, and our circle of friends was reading Lester Bangs’ Psychotic Reactions and Carburetor Dung and using it as an example of how to hold very specific aesthetic opinions very adamantly, so a strong reaction to a thing that didn’t fit wasn’t entirely out of order. The problem was that I couldn’t justify the fact that 0! = 1. I fell back lamely on Appeal To Authority, and hope I didn’t put Paul off factorials for life.

Here, many years later, is the answer I should have given. Of course you could define 0!=0. You can define anything any way you want. Ink is cheap. The limiting factor is the utility of your definition. Factorials, it turns out, are useful when you want to count the number of ways a thing can happen. Pat, Kim, and Sam stand in line for a movie. Maybe Pat is first, then Kim, then Sam. Or maybe Kim is first, then Sam, then Pat. The number of possible configurations is 3! = 3 x 2 x 1 = 6. All manner of sophisticated statistics are built on top of this basic accounting. Now ask, how many ways are there to arrange no things? Granted, this has an annoying Zen koan-like ring to it, but a reasonable answer is: one. “Reasonable” here has two components. First, it’s not unreasonable. When no one shows up to the movie there’s just an empty sidewalk with sad little puddles reflecting blinking marquee lights, and that’s a single thing. Second, when you start using factorials as building blocks for more complex tallies you find yourself writing n! terms, some of which end up in the denominators of fractions, so you either have to define n! so that it doesn’t ever equal zero up front or insert caveats into every equation where that might hurt you. Ink may be cheap, but the mental effort it takes to parse extraneous notation is not.

Mathematics is often a refuge from utility. In theory, once we have specified our initial axioms, the rest is just formal manipulation. Furthermore those axioms usually express such head-slappingly obvious concepts that it’s difficult to imagine things being any other way, and so take on a formal quality of their own. But the definition 0! = 1 does not naturally arise from the business of multiplication, nor does it express some self-evident extra-mathematical concept. Instead it is a finesse we pull to make things easier later on. Mathematics is a real world activity, and demands a certain canniness. As in the rest of life, it helps to be scanning the road ahead.

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