Let me start by saying that when I was in college, I majored in English. I proudly wear the sweatshirt my husband bought for me that reads “English Major. You Do The Math.” Nevertheless, I did have really good math teachers in high school, and in my first year of college I found myself taking two semesters of calculus as well as two semesters of symbolic logic, so I’m not a complete doofus when it comes to things mathematical, and when my son, a high school senior, started talking to me recently about factorials, I felt well up for the conversation. Now to be quite honest, I don’t remember ever having studied factorials before. In fact, I couldn’t recall ever having heard of the little buggers in my life. Apparently they have something to do with statistics (which is why I missed out on them), and my son had been watching some mathemagician playing around with them online and wanted to tell me about what he had seen. It’s really a pretty simple concept: a factorial is any whole number multiplied by every whole number less than it all the way down to 1. So 5 factorial is 5 x 4 x 3 x 2 x 1. I can’t imagine why anyone would need to make such a calculation, but I am willing to concede that factorials must have some usefulness somewhere or they would not have been invented. In my mind, factorials are in the same mathematical category as limits and cosines and cube roots–mysterious calculations that only really make sense to the mathematically elite, much as the semicolon and the pronoun “whom” must appear to those uninitiated in the secrets of English grammar. The symbol used to express a factorial is an exclamation point, and, as an English major, this is where I hit my first serious problem with factorials.
“Wait a minute,” I said to my son. “You mean to tell me that 5 factorial is written 5! ?”
“Yeah, it means 5 x 4 x 3–”
“I know what it means,” I replied haughtily. “I’m talking about the exclamation point. Why the exclamation point? There’s nothing exciting about factorials, is there? Or am I missing something?”
“No, Mom.” (A roll of the eyes.) “That’s just the way they do it.” (The word “convention” is apparently beyond his adolescent ken.)
I object to this convention. I object strongly! I had always assumed that if the people of the math world were going to appropriate a mark of punctuation from the real world, they would at least have the courtesy to change the name of said mark. Thus, the little dot that most of us know as a period becomes, in the world of mathematics, a decimal point, or just a “point”. When a period does a floating mid-air magic act, it means multiplication and isn’t called anything at all. The lowly hyphen becomes a minus sign. All of this is fine with me. Punctuation takes on a new identity in a new world. But alas! Not so in the case of factorials and the exclamation point; since it is not re-named, this stoic mark of punctuation retains all of its real-world personality traits even while existing in the world of numbers. By nature, the exclamation point is connotatively both emphatic and ambiguous. It indicates some kind of strong emotion, either positive or negative. But let’s face it: there is nothing strongly emotional about factorials, necessary as I assume they must be, and the use of the exclamation point is therefore eminently inappropriate. So I say this to the world of mathematics: The use of the exclamation point to indicate factorials is outright theft from the world of words. It is morally wrong. The poor exclamation point is totally out of its element, and is likely on the verge of suicide. A NEW SYMBOL IS DESPERATELY NEEDED. I humbly submit the following solution.
(At this point in my writing, my son walked through the room. I mentioned that I was working on a piece about factorials, and asked him if he would like to read what I had written so far. He read for a few seconds and looked up, shocked. “Mom,” he said, “this isn’t about factorials. It’s about punctuation!” “Well I’m not finished yet,” I explained, “and I was really upset about those exclamation points.” He rolled his eyes yet again. “So you’re OK with numbers and letters being together in algebra, but when there’s punctuation involved, it’s like, ‘Oh my God, hold the phones!'” And he stalked out of the room. He just doesn’t get it.)
To return. I humbly submit the following solution: the ampersand. This is the fairly intricate symbol that lives above the number 7 on the standard QWERTY keyboard. It looks like this: &. It’s not technically a mark of punctuation, so I’ve no professional objection to its being used in the world of mathematics. And it means “and,” so it’s really the perfect symbol for factorials since multiplication is just a quicker way of doing addition anyway. Now I know that there will be at least one immediate and strong objection to the ampersand: it’s nearly impossible to make one by hand. Written attempts at the ampersand come out looking like either miniature creations of abstract art or, if you’re really lucky, the symbol used for the G clef. And even such scribbles take time, more time than the average high-schooler has while copying last night’s math homework from his buddy during homeroom. But wait! According to Wikipedia, “in everyday handwriting, the ampersand is simplified in design as a lowercase epsilon or a backwards numeral 3 superimposed by a vertical line.” PERFECT! Whether you want to call it a lowercase epsilon or a backwards 3, it’s the same shape with a vertical line through it. The math world is filled with both numbers and Greek letters; the handwritten ampersand will feel right at home. Now the opposite problem remains–how to type it–but that has a simple solution. Let the “handwritten” ampersand symbol live atop the 7 on the QWERTY keyboard, while the old-style fancy-pants ampersand (henceforth to be known as the fancypantersand) can be added to some special font that you can only find by clicking on the question mark icon on your computer. It will still be there; it will just be harder to get to. I admit that, over time, the fancypantersand may well become obsolete, but I believe that the handwritten version will serve us just fine, and after all it’s a small price to pay for the inappropriate (and I must say immoral) use of the exclamation point.
Alright, now I can get back to factorials and the ideas that I had originally intended to express. A few days after our initial conversation, my son casually asked this question: “What do you think 0 factorial is?” We had just finished dinner and were clearing the table, so I wasn’t able to give the question the proper amount of thought, or I would have answered differently (more on that later). As it was, I replied, “0 factorial? It would have to be 0.” He smiled triumphantly. “Nope. It’s 1, and I can prove it.” “NO, sir,” said I with an air of finality, “0 factorial is 0.” And I continued loading the dishwasher, making a mental note to email Joe’s guidance counselor because someone had clearly led the poor child astray. Later that evening Joe came to me with pencil and paper. “Watch this.” And he proceeded to show me this formula: (Much as it pains me, I will continue to use the exclamation point to indicate factorials, though I am confident that it will soon be replaced by the handwritten ampersand.)
For any whole number n,
n! = (n +1)! n + 1
So, for instance,
1,272! = 1,273! 1,273
Well, that one I would just have to take on faith because I was not about to sit and work out all that multiplication. It occurred to me to wonder who came up with that formula in the first place. Someone with way too much time on his hands, no doubt. Anyway, it was easier to see with smaller numbers. I could agree that 4! = 5!/5, or that 3! = 4!/4. Big deal. The kicker came when Joe said,
“So if you follow the pattern, then 0! = 1!/1 which equals 1. So 0! = 1.”
“Whoa there, cowboy. Patterns be damned. Factorials only go down to 1. There are no zeros involved in factorials. Zero factorial doesn’t exist!”
Now, for those of you playing along at home, this was obviously a huge error on my part because just hours before I had stated quite unequivocally that 0! = 0, clearly implying that zero factorial does in fact exist. Luckily, my son failed to notice or care about this egregious logical contradiction, and I just forged ahead.
“Look, zero’s not even really a number. It’s just a place holder.” (I’ve no idea whether this is true or not.) “That’s why you can’t divide by zero. It just doesn’t make sense. Sure, you can say, ’21 divided by 7′ and that makes sense, but you can’t say . . .” here I cast my eyes around the room and said the first thing they landed on “you can’t say ’21 divided by chair.’ That doesn’t make any sense at all. And coincidentally, ‘chair’ is a place holder too. It’s a place to hold your butt.”
But he was not diverted by my diversion.
“Look, Mom,” showing me a video on his phone, “there’s a college math professor online saying that zero factorial equals one, and–”
“Oh, honey, you know better than that! I don’t care if there are a hundred college professors online saying the same thing! It’s like any given statement made on FOX news–just because you say a thing where lots of people can hear it, that doesn’t make it true!”
“Fine.” His face was set with a fierce determination, and I could tell that some final coup de grace was coming. He reached for his calculator. “Look.”
The calculator did in fact have a little button for factorials, and when he entered “0,” instead of the error message that I expected, the screen showed “1.” It seemed that the entire math world, as well as Texas Instruments, were conspiratorially allied against me. I had no choice but to admit a grudging and ill-humored defeat.
Still, I couldn’t let it go. Hours after the boy had gone to bed, I consulted my old friend, The Merriam-Webster Dictionary. There were two entries for the word “factorial.” The first entry filled me with hope: “the product of all the positive integers from 1 to n.” Ha! No zero! I win! But with the second entry, that hope was dashed like seagull shit on an ocean jetty: “the quantity 0! is arbitrarily defined as equal to 1.” I comforted myself with the notion that my defeat was only arbitrary.
The question remains: Why should I get so worked up about factorials in the first place? Factorials have absolutely nothing–and I mean NOTHING–to do with my life. Until two weeks ago, I didn’t even know that they existed. So why would I expend so much mental and manual energy on their behalf? I think I can hazard a pretty good guess. I retired from teaching on full disability at the age of 50. I spend most of my days alone, giving me lots of time to think. When my husband and son and I engage in “intellectual” discussions (which I’m proud to say we frequently do), and we talk about heady topics like history or politics or science or art, we share information and exchange views, but we seldom argue, not the way my son and I argued about factorials. That was an intellectual exercise of the highest degree. That argument stuck in my mind, prompted further thought, led me to research (at least a little), and demanded expression. That is the kind of argument that I hope for you in college, Joe. May they be many, and may you have joy of them.