6×4 or 4×6?

Every so often, a question that comes up that looks incredibly trivial, but - no matter how much each side protests that the preference doesn’t really matter - sets down clear divides in the maths community.

My podcasting partner in crime-fighting @reflectivemaths (Dave Gale in real life) stumble upon such a question the other day: does $4\times 6$ mean four groups of six or six groups of four?

Of course it doesn’t sodding matter

One of the first things you learn about maths is that it doesn’t matter which way you multiply things: like addition, multiplication commutes on the real numbers ¹ - you’re within your rights to see $4\times 6$ as either four groups of six, six groups of four, or even both at the same time (a position I suspect most mathematicians instinctively hold).

But still: almost everyone comes down on one side or the other.

Dave’s wrong-headed thinking

Oh, poor @reflectivemaths and his muddleheaded simpleness. Dave believes, bless his little plastic pocket protector, that $4\times 6$ means six groups of four, reasoning wrongly that you’d say it as “4 multiplied by 6”, which means you start with four things and then create six groups of the same size. It’s a seductive argument, and one that’s hard to find proper fault with, which is obviously why I’ve resorted to the ad-homs before I put the case for my side of the story.

Instead of saying Dave’s argument is wrong, I’m going to show that mine is right.

Argument 1: Nanny Pat’s times tables

“Twice six,” my grandmother probably chanted in school, “is twelve.” Not, you’ll note, “six twice.” Similarly, “Thrice six is eighteen, four times six is 24.”

The notation $4\times 6$, which I’d say as “four times six” as that’s how I learnt my times tables, too, clearly means four groups of six - the version of it where you say “twice” or “thrice” gives it away - that’s the number of groups.

Argument 2: Algebraic convention

$3x$ - or three times $x$ - is almost impossible to think of as anything other than three groups, each with $x$ in. The algebraic convention is that the number of things always goes before the variable: you’d never write $x3$, not least because you’d get it mixed up with $x^3$ unless your handwriting was splendid.

Argument 3: Linguistic convention

Literally the only example I can think of, in either of the two languages I can speak well, or either of the three languages I can speak awfully, of the number of groups following the contents of the group is Old King Cole’s “fiddlers three” - which, in itself, is only there to make a pretty poor rhyme. In almost any other situation, saying the number of things after the things themselves means the ordinal: Toy Story 3 is the third Toy Story movie, not the set of three movies.

Even in pop groups like the Dave Clark 5, it’s not that there are five Dave Clarks, just five people let by Dave Clark.

Argument 4: $n$ times

Here’s the clincher:

$\underset{\overbrace{x+x+\cdots +x+x}}{ntimes}$

This is not $x$ times $n$, it’s $n$ times $x$.

Argument 5: @reflectivemaths is a poopyhead

There, I said it.

Probably best to close with an ad-hom as well, just for the sake of good form.

In conclusion

On this point of utter triviality, @reflectivemaths is clearly - clearly in the wrong. It’s ok, though, it doesn’t necessarily make him a bad person.

Footnotes:

1. and the complex numbers, since you ask - but not the quaternions or matrices, or non-Abelian groups. But I digress.