Agreed. Multiplier and multiplicand are different words but the commutative property says their values can swap equivalently, so . . . what was the author's point again?
procedure product(multiplier, multiplicand)
acc := 0
for i = 1 to multiplier
acc := acc + multiplicand
end
return acc
end
> (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
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.
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 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.
Density times volume is mass. Volume times density is mass. Distance times force is work/energy...
"5 times I will connect 12 apples"
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.
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.
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.)
*** **
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**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.
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?
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!
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.
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