What’s Wrong with “Multiplication Is Repeated Addition”? (2008)

Post by Garlef » Wed, Mar 03, 2021, 2:33 PM UTC

prionassembly wrote: post #64

Garlef wrote: post #53

Earlier quoted context omitted.

No (At least not yet: open research question). But neither can "repeated multiplication". How would you multiply 2 with itself "a half" times?

Operationally, unsure. But if there was an answer, it would be something that multiplied with itself twice would have be the same as 2 multiplied with itself once. I think it was Gerolamo Cardano who in a book of worked exercises came up with a problem like "obtain two numbers that add to 10 and multiply to 40" and arrived at the solution had to involve square roots of -1. Then he goes "this makes no sense, but let's…

Post by sudosysgen » Wed, Mar 03, 2021, 2:46 PM UTC

parenthesis wrote: post #41

rustybolt wrote: post #27

These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. You can definitely see multiplication as repeated addition. It's even a useful thing to do, how else would you define multiplication? This is the whole idea that made mathematics great. You define more complicated operations in terms of familiar operations. Then, if you…

> These articles bring back memories of teachers telling me I was wrong, because I didn't use the quadratic formula, but completed the square instead. That's hilarious, since the quadratic formula is just what you get by solving ax^2 + bx + c = 0 by completing the square. In fact, I have a high school textbook that makes that derivation.

Post by fifilura » Wed, Mar 03, 2021, 2:51 PM UTC

D13Fd wrote: post #42

I've been teaching my 5 year old multiplication and division for the last couple of weeks. I'm at a loss how you would teach it without explaining that it is repeated addition. For example, the other day I asked her how many fingers and toes the three kids at the table had, and she came up with "20 fingers and toes each times 3 kids means there are 60 fingers and toes." I think the units are intuitive in most cases a…

Post by mantap » Wed, Mar 03, 2021, 2:52 PM UTC

prionassembly wrote: post #50

y7 wrote: post #39

I think the author's argument becomes more clear when you consider multiplication in rings other than the integers, for example (square) matrices. The product of two matrices A*B does not correspond to repeated addition, and it is not commutative (A*B does not equal B*A in general). I can see that having an engrained belief that multiplication is defined via addition becomes problematic at some point when learning ab…

The better line of thought here might be "A*B is not multiplication, it's function composition". It literally looks nothing like multiplication when carried out manually.

Post by np_tedious » Wed, Mar 03, 2021, 2:55 PM UTC

anbende wrote: post #74

tomxor wrote: post #69

Earlier quoted context omitted.

> 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.

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.