# Factorials

Isabel finds a curmudgeonly 1842 quotation from Augustus de Morgan, about the way we write factorials:

Among the worst of barabarisms is that of introducing symbols which are quite new in mathematical, but perfectly understood in common, language. Writers have borrowed from the Germans the abbreviation n! to signify 1.2.3.(n – 1).n, which gives their pages the appearance of expressing surprise and admiration that 2, 3, 4, &c. should be found in mathematical results.

You know, I’d always thought the formula for “n choose k” was a little, well, enthusiastic:

$$\left(\begin{array}{c} n \\ k \end{array}\right) = \frac{n!}{k!(n-k)!}.$$

n! k! n minus k!! You gotta believe me, guys!”

Still, though, if I saw de Morgan’s way of writing the factorial of n, I’d read it as

$$1 \cdot 2 \cdot 3 \cdot (n-1) \cdot n,$$

which is only the factorial of n when n is 5. I guess there’s little point in pleasing the dead. . . .

## 11 thoughts on “Factorials”

1. Sorry, de Morgan, but math is a glutton for symbols. As new ideas and operations come in, sometimes you need a new notation to express an idea. The sheer amount of crazy symbols you can use in LaTeX speaks volumes about what people are trying to say without words.

Hopefully deM got over it one day. Everyone else certainly did.

2. gg says:

If de Morgan really wanted something to complain about, he should have complained about using \nu to represent frequency. Since it often appears hand-in-hand with velocity v, and italic v looks in many fonts like \nu, it’s a source of confusion. That’s one of my only notation gripes…

3. I’ve never really reconciled myself to writing $$\sin^{-1}$$ for the inverse sine and $$\sin^2$$ for the square of the sine. If the first were your standard, then surely the second would mean “apply the sine twice.” But we don’t do that: raising trig functions to powers is far more common.

4. Eric says:

â€œn! k! n minus k!! You gotta believe me, guys!â€

Speaking of somewhat ridiculous notation, when I first read this I almost thought you were saying there was a double factorial in the denominator, and I was worried. As for the sines, I have no issue with sin-1 for the inverse sine, since -1 seems to be pretty standard notation for the inverse of anything. Sin2x instead of sin(x)2 is just convenient shorthand (yes, I’m actually lazy enough to not want to have to include the parentheses); it probably wouldn’t be nearly as acceptable if sin(sin(x)) were at all common.

5. Blake: that’s why I make the point to my students that though I’ll accept $$\sin^2(x)$$ I will always write $$\sin(x)^2$$ and always include the parens. It’s also a good time to point out that there is a difference between the reciprocal and the inverse, not that they’ll remember.

gg: you mean $$v$$ and $$\nu$$ (hopes that the renderer works)? How can you confuse those? The former is so much more… feminine, for lack of a better word.

6. Bleh. Blake, can you take the LaTeX above and make it go? And how can I invoke it on my own in the future. It’s not the same as on WordPress hosted weblogs.

7. gg says:

John: LaTeX makes a pretty clear distinction, as long as you know what you’re looking for. I guess I was thinking more about students unfamiliar with /nu trying to decipher a professor’s chicken-scratch!

8. My LaTeXRender plugin delimits formulas with $$ and $$ instead of $latex and $. I haven’t gotten around to adding the more common WordPress way, yet.

9. And in trying to add that alternate syntax, PHP seems to be giving me trouble. Feh.

10. This is slightly off-topic, but in Google Reader, all you can see of the formulas is the black dithering on the edges, so you probably shouldn’t make the background transparent.

11. Hmmm, OK, I’ll fiddle around with that.