The Correct Usage of Multiplication Symbols (and also, other types of products!)

This post was written as part of my submission to MIT's Blogger Applications for 2024

What is something totally inconsequential that you have a strong opinion about?

No, there isn't a best symbol to represent multiplication. If there was a best symbol, it definitely would not be the center dot. Different occasions call for different symbols, and it's crucial that you use them for the right occasion, or you might seriously offend your residential mathematician.

I'll be typing my examples in LaTeX, a math processing language, because my writing software (Obsidian) has LaTeX integration. However, Listed.to doesn't, which is kind of lame. I'm planning on moving to another site that has that kind of integration, or maybe even making my own website to be honest. Anyways, you can tell that it's in LaTeX by the dollar signs surrounding the code.

The center dot ($\cdot$) - As a budding student mathematician makes their first step into abstraction, they run into a common problem: x's and the traditional times symbol look too much alike. Part of their rite of passage into the realm of algebra involves forsaking the cross for the center dot.

I have one big problem with the center dot: normally, with a pen or a whiteboard marker, it suffices to just press the point of your writing utensil into the page, producing a perfect dot. However, if you were using a pencil or chalk, which rely on friction to leave marks, you'd have to draw a tiny circle. That's not dotting!! I want to be able to boop the page and leave a dot there. The final nail in the coffin is that pencil and chalk is exactly what one should be doing their math with!

Thus, I claim that the best and only permissible time one should use the center dot is when you're multiplying a long (>3) string of variables together, so it doesn't look like $abcd$ but rather $a \cdot b \cdot c \cdot d$ to accentuate that they're being multiplied. Even then, I don't think it's necessary. I mean, a lot of the time, when I'm writing the product of two variables, I just put them next to each other, instead of writing the dot in the center. There really is no need for this small, lackluster notation.

There are other uses for the center dot that I'm aware of. The one that I have in mind is to denote group action [https://en.wikipedia.org/wiki/Group_action], for denoting how an element of a group acts on an element of a set. Let's not also forget the dot product [https://en.wikipedia.org/wiki/Dot_product], which tells us how much a vector lies on another vector.

The cross ($\times$) - Ahhh, the cross, what we all grew up using, and what we should still be using to denote multiplication in a vast majority of cases, especially when we are only multiplying numbers and not variables. I'm very, very fond of this symbol, and I still use it fairly often, especially when I'm multiplying things.

"But it looks exactly like a variable x," I hear a cdot supporter whine. No, it doesn't! I write my x's in cursive, with tons of extra curliness, which helps me easily distinguish between my crosses (which are written with very straight lines, are significantly smaller than my x's, and do not rest on the bottom of the line, but rather are in the center) and my x's. That being said, I still avoid using crosses for multiplication of variables. Like I said earlier, I prefer omitting the multiplication symbol.

Plus (pun unintended), it's also the standard convention for writing scientific notation [https://en.wikipedia.org/wiki/Scientific_notation]. Another common occurrence is the cross product [https://en.wikipedia.org/wiki/Cross_product]. I think it's fairly interesting that we are allowed to use both the dot and the cross for multiplication since the dot and cross product agree for 1 by 1 matrices (note how one would usually denote it as $1 \times 1$ matrices, with the cross symbol. Another sign (pun intended) of cross's greatness).

Other usages of the cross: the Cartesian product [https://en.wikipedia.org/wiki/Cartesian_product], where A cross B are all the ordered pairs (a, b) with the first coordinate a in A and the second coordinate b in B. It's in a way the natural way of juxtoposing a copy of A and a copy of B together. For example, the real plane comprises of two copies of the real line: $\mathbb R² = \mathbb R \times \mathbb R$. That's also why it's called the Cartesian plane.

The asterisk (*) - Remember what I said about there being no best symbol? Well there definitely is a worst, and that's the asterisk. The only reason it's used is because whoever came up with the keys on the keyboard decided that the better multiplication operators didn't deserve to be included like how + is, and so we're stuck with using the asterisk. The only time you use it is when you're coding, I suppose. It's just yucky.

The centered asterisk is used to denote the free product [https://en.wikipedia.org/wiki/Free_product]. It's like the cross product, combining two sets together naturally, but with minimal structure. I don't even like the free product anyways. The asterisk is just bad.

Bonus: No symbol - My preferred choice of denoting multiplication is not denoting multiplication. It's when the aura of the multiplication is so strong that it doesn't even need to make its presence known visually.

Bonus: Tensor product ($\otimes$) - It's the cross with a circle around it, like an O. With a times in it. Hence the LaTeX name otimes. I still don't understand what the tensor product does, except it just somehow puts together two modules (vector spaces over rings) together somehow. I wanted to give this one a shoutout because it's the thing that I've been trying to understand for such a long time but still can't. It's okay, I'll get it one day.

My credentials: I'm a math major :3