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

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

#61

Post by yakubin »

gugagore wrote: post #56
yakubin wrote: post #37

Earlier quoted context omitted.

Not algorithms. There will be infinite addition involved, and algorithms are finite. Thinking of multiplication as repeated addition also won't explain anything about it. It's a separate operation. Deal with it. For similar reasons, you can't calculate x-th power of a number, when x is irrational, by decomposing it into exponentiation and roots. This metaphor is just training wheels. At some point you should lose it.…

Algorithms are not finite. E.g. Newton's method.

First sentence on Wikipedia[1]:

> In mathematics and computer science, an algorithm (/ˈælɡərɪðəm/ (About this soundlisten)) is a finite sequence of well-defined, computer-implementable instructions, typically to solve a class of problems or to perform a computation.

Newton's method is finite too. You perform finitely many iterations. It doesn't calculate roots. It calculates their approximations.

[1]: https://en.wikipedia.org/wiki/Algorithm>

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

#62

Post by sumtechguy »

tromp wrote: post #5

In some settings, "repeated addition" is the only sensible way to think of multiplication. Such as multiplying an elliptic curve point with a scalar.

Multiplication is a shorthand for 'repeated addition'. Much like power is short hand for 'repeated multiplication'. Even with imaginary numbers. You can create an in infinite series of additions that looks a lot like the imaginary number.

The argument they are making is kind of strange. But not 'wrong' per se. If you mix 2 types of number systems you get a different number system out of it. It is a similar argument I make about fractions. It is basically physically impossible to have a fraction of some things. If I have a car and divide it in half. I no longer have a car. I have two of something else. They are clearly not a car because they can not act like a car anymore.

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

#63

Post by YeGoblynQueenne »

AzzieElbab wrote: post #31
Garlef wrote: post #13

My favourite way of thinking about these operations comes from category theory: * addition: the cardinality of the disjoint union of two sets * multiplication: the cardinality of the direct product of two sets * exponentiation: a^b is the cardinality of the set of maps from B to A where B is a set of cardinality b and A is a set of cardinality a

That is exactly how I explained multiplication to my 5 years old, right after getting him into programming by playing with profunctor optics

You waited 5 years to teach programming to your 5 year old?

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

#64

Post by prionassembly »

Garlef wrote: post #53
threatofrain wrote: post #43

Earlier quoted context omitted.

Can this perspective accommodate fractional exponents, like 2^(0.5)?

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 calculate nevertheless..."

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

#65

Post by prionassembly »

eigenket wrote: post #55
prionassembly wrote: post #50

Earlier quoted context omitted.

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.

It looks exactly like scalar multiplication when your matrices are diagonal.

Then: all matrices look like the identity matrix if applied to a suitably rescaled eigenvector.

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

#66

Post by tgv »

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…

You start by teaching rings, duh.

But seriously, her first objection is

> To define multiplication as repeated addition is to make multiplication a sub-species of addition.

To me, that means the author is wrong from the get-go. Not all cases of multiplication might be easily seen as repeated addition, but to get hung up on a deliberate linguistic misunderstanding? And to add insult to injury, she finishes by proposing to see multiplication as the answer to "how many or how much OF the unit", and shows a diagram with _repeating_ units.

Oh, never mind, the answer is at the end: she's got books to sell, and probably tries to improve her sales by appealing to parents with children that can't multiply 83 by 17 when they're 8.

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

#67

Post by eigenket »

prionassembly wrote: post #65
eigenket wrote: post #55

Earlier quoted context omitted.

It looks exactly like scalar multiplication when your matrices are diagonal.

Then: all matrices look like the identity matrix if applied to a suitably rescaled eigenvector.

We're talking about matrix/matrix multiplication here, not matrix/vector, right?

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

#68

Post by ThePadawan »

tomxor wrote: post #40
drewcoo wrote: post #34

Earlier quoted context omitted.

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?

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.

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.

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

#69

Post by tomxor »

ThePadawan wrote: post #68
tomxor wrote: post #40

Earlier quoted context omitted.

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.

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.

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

#70

Post by AzzieElbab »

Garlef wrote: post #51
AzzieElbab wrote: post #31

Earlier quoted context omitted.

That is exactly how I explained multiplication to my 5 years old, right after getting him into programming by playing with profunctor optics

I learned adding from counting disjoint unions of dots. I learned multiplication from counting squares of dots. But agreed: I learned about exponentiation first via repeated multiplication.

That is very cool. I prefer geometric/visual approaches. Like move that many steps along the line, figure out area of a rectangle and so on
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