If, for a moment, you conceptualize of multiplication on non-negative whole numbers as repeated addition, then this is the algorithm:

  procedure product(multiplier, multiplicand)
      acc := 0
      for i = 1 to multiplier
          acc := acc + multiplicand 
      end
      return acc
  end

Swapping the arguments is a different computation. But after thinking about it, you realize that you get the same answer all the same. That's the point of

> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

The point with bringing up dimensional analysis is that the above algorithm doesn't work because what does it mean to do `for i = 1 to 3 chairs/row`? You might think of it like

  procedure product(multiplier, multiplicand)
      acc := "0" # an "absolute" zero that cooperates with any dimension
      each single_multiplier in multiplier
          acc := acc + (multiplicand * single_multiplier)
      end
      return acc
  end

But then what is `(multiplicand * single_multiplier)` ?

This was exactly my thought, I don't understand why those two word exist since it's commutative. Perhaps all this confusion is purely semantic.

Multiplication over the reals is commutative. Matrix multiplication of non-square matrices isn't. Multiplication in a Ring isn't necessarily commutative. Other algebraic structures also have non-commutative multiplication.

One could argue that these things aren't "multiplication" even if they are "products" since they don't satisfy all the properties of multiplication over the reals. But it is common to call the use of the product operation "multiplication", at least in cases where there's only one product operation to use. EG Geometric Algebra has Inner, Outer, and Geometric products, so calling them "multiplication" seems less common IME.

I really don't get the point you're making. If we're going to pull out random examples, monoids aren't guaranteed to be abelian; strings and concatenation form a monoid that's not abelian.

If you have enough mathematical sophistication to conceptualize a non-commutative ring, you're well past the point where naming conventions are even remotely an issue.

The original article was contrasting addition and multiplication on the basis that addends are called the same while factors are supposed to be called differently, which not only makes no sense (it's just a naming convention), but it also breaks down when you have more than two factors: what is the "c" in a x b x c called? Or we're talking about non-associative operations now?

My point is only that the commutative property is not inherent to all multiplication operations, so there can be a distinction between the operands. It's not necessarily a useful distinction, and in the usual use of multiplication it's utterly useless and only adds confusion.

But matrix multiplication is taught in high school (and usually promptly forgotten), it's not particularly advanced math.

Personally I'm of the opinion that the terminology is muddled. There's no need to distinguish the operands of a multiplication over any of the usual domains (reals, rationals, integers, etc). And when you reach the point where it does become important there's generally more than one product operation and we should stop calling it multiplication. "Matrix multiplication" is a bad term. You also typically wouldn't name the operands, since as you note there can be more than two!

Sure, my counterpoint was just that the same reasoning technically applies to addition as well, but as you note "matrix multiplication" is taught in high school, while my example wouldn't come up.

I agree not calling it multiplication would probably help, perhaps something like "linear transformation composition" might encourage students to keep it separated from real multiplication, I jsut found the argument in the original article kind of ridiculous to be honest.

Yea, math tends to reuse terms and notation across disciplines in ways that adds confusion rather than clarity. It’s much better to think of infinity for example as multiple independent concepts than assume it’s all the same idea.

They exist to distinguish the element being operated on (the LHS) and what it is operated on by (the RHS).

Total technicality, but I could see myself using the term multiplicand/multiplier in my code if I had to implement e.g. a stack-based parser for arithmetic expressions.

> to distinguish the element being operated on (the LHS)

I'm not sure what you mean, they are both operated on, it is a binary operator and commutative, there is literally no difference.

I agree that it's a generally pointless distinctions, although it might be useful in some cases such as number systems where the multiplication isn't commutative, or for a particular implementation where the distinction matters.

After all if you were to code a multiplication that was implemented naively as a series of additions it'd be generally much faster todo 2x1000 than 1000x2.

To return to TFA I think the author is talking from a pedagogical standpoint, that teaching that multiplication is just a bunch of additions under a trench coat is not the best way to go. I'm not sure that I agree personally.

In particular this bit regarding multiplier/multiplicand makes zero sense to me:

>Different names indicate a difference in function. The multiplier and the multiplicand are not conceptually interchangeable. It is true that multiplication is commutative, but (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

Of course 3 rows and 2 rows aren't the same, but what does it have to do with the order of the multiplication? Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.

Suppose your friend Alice arranges your wedding. You ask her to arrange the lawn chairs in an arrangement of 2 rows and 3 columns. But she misinterprets your request as 3 rows and 2 columns. Oops. Now a particular family can't all sit in a single row without rearranging the chairs.

If all you care about is the total number of chairs, the order of operands is irrelevant. but if you care about the structure, "2 x 3" may encode information that "6" does not.

I address that point specifically in my comment. What you say is that the units of the operands matter, which I agree, but the order doesn't. 2rows x 3chairs and 3chairs x 2rows is the same thing.

Except we are talking about scalar multiplication, that was the premise, not matrices.

The point isn't "matrix multiplication != scalar multiplication". The point is "the process of evaluating a reducible expression inherently discards information about the original expression", which is a fact about evalution rather than any specific operator. The fact that "the information discarded is of little consequence to the compressed result" is a quirk specific to scalar multiplication. Thus, the commutative property distracts from explicitly modeling the "AST" so that the student understands what multiplication represents under the hood, beyond the rote memorization of scalar multiplication tables.

Perhaps an analogous situation would be: Suppose a teacher wanted to introduce the notion of limits to a calculus curriculum. "That makes zero sense. The only things a student needs to know are the shortcuts for each parent function, e.g. that (d/dx x^2) reduces to (2x) via handwavey magic." But what if an engineer needs to integrate over an arbitrary curve? Can students solve the problem without being comfortable with Riemann Sums? Maybe 1st-year calc students should rederive the shortcuts from scratch? "Except we're talking about a math course, not an engineering course."

> In particular this bit regarding multiplier/multiplicand makes zero sense to me.

> Isn't 2 rows x 3 chairs the same thing as 3 chairs x 2 rows? It's a bizarre argument.

It's bizarre to simias (and you, I assume) because y'all can't imagine performing the operation without thunking. (Don't get me wrong, I think "repeated addition" is the best method. I'm just attempting to explain the opposite perspective so that it feels less bizarre.)

It distinguishes these two rectangles.

  ***  **
  ***  **
       **

If we are just looking at numbers with no units attached there is no difference. As soon as there are units, a difference arises.

I will collect 12 apples 5 times. The 12 apples are the multiplicand and the 5 times is the multiplier.

The answer retains the units from the multiplicand.

Still don't see the difference.

Density times volume is mass. Volume times density is mass. Distance times force is work/energy...

"5 times I will connect 12 apples"

You're correct. The GP in incorrect about their dimensional analysis. 5 apples 12 times yields apples because it's apples times a dimensionless scalar (count). Newtons times meters is always Newton-meters, never Newtons or meters. Units are never magically dropped in dimensional analysis.

Not all multiplication is commutative, e.g. matrices.

> (2 rows × 3 chairs/row) is not the same as (3 rows × 2 chairs/row), even though both sets contain 6 chairs.

The author has it all backwards. Basically that you can turn this arrangement by 90 degrees (and turning the chairs, if you do not modell them by points by 90 degrees) is the reason why multiplication is commutative. It is non trivial that 5+5+5+...5 (100 times) is the same as 100+100+100+100+100