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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)

#101

Post by midjji »

Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

Not defending the article, but how would you compute: \pi*\pi using repeated addition?

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

#102

Post by scythe »

The nicest formal definition of multiplication is "the operation that distributes over addition". When defining operations on abelian groups that might be called "multiplication", it's the distributive property that makes the term useful, and brings us into the categories of ring, semiring, [weird math prefix here]-ring, etc.

But the thing is: repeated addition does distribute over addition! So there is really a very natural extension from "multiplication is repeated addition" to "multiplication is any operation that preserves the nicest property of repeated addition, which is distribution".

I hasten to add that as a GTA I have had untold numbers of students who apparently did not learn the distributive property correctly -- likewise, the biggest difficulty my students seem to have with dimensional analysis in practice is that they have trouble dividing fractions symbolically. Also, substitution (replacing an expression with a letter) continues to trip students up: e.g. when pointing out that, say, newtons per coulomb is the same as volts per meter. (Students are no doubt tired of hearing me yak about how math expressions are a form of communication...)

Sometimes, I do think we need to teach students to "manipulate expressions" rather than just "solve problems", but then again, don't we do that already?

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

#103

Post by Jtsummers »

eru wrote: post #92
sudosysgen wrote: post #73

Earlier quoted context omitted.

We did that derivation in class, actually! I was so lucky to have a great math teacher in high school.

That's not the norm?

No, it's not. It depends a bit on your teachers but also on which "level" of math you're in (in the US). Lower level but still algebra/geometry classes tend to teach facts, not derivations from foundational concepts. Those are the classes aimed at non-Honors and maybe non-College Prep students (2 of the 3 typical "tracks" students end up in the US, names may vary by state and decade).

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

#104

Post by spacedcowboy »

Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

I have a PhD in physics and more maths qualifications than I can shake a stick at; to me, multiplication is repeated addition.

I’m not sure what the teacher is trying to do here, but I do think the outcome of what they’re trying to do is far more complicated than the simple “multiplication is repeated addition”.

I also happen to have an 8-year-old going through third grade right now, and when we were talking through his homework, it was quite clear that using simple concepts he already knew (addition & subtraction) to explain slightly more complex things that he was learning (multiplication and division) was really useful to him. As I recall it being to me.

[aside] I think the maths schedule is more advanced now than it was in my day anyway - he only did multiplication and division this year, but he also did algebra and simultaneous linear equations now, as in:

a + b + 8 = 24

a - b = 4

“Solve for a and b”

Pretty sure I only did that in senior school (11 and up), not at age 8. No powers as yet (presumably they’ll come after the multiplication/division stuff), so no quadratic formula, but still...

[/aside]

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

#105

Post by cjfd »

Ah yes, there is Keith Devlin involved.... I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods while at the same time being taken seriously.

You know, most, if not all, mathematical objects of significance can be defined in many, many different ways. In fact, this is the hallmark of an mathematical object of significance: it keeps popping up in many context and can therefore be defined in any or all of these contexts if one enjoys doing so. There are far ranging context to define an object and less far ranging. I am quite sure that in grade school it is quite unhelpful to look for the furthest ranging context in which one could define multiplication. E.g, let us start in grade 1 with the definition of an algebra and derive everything from that.

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

#106

Post by marcosdumay »

eigenket wrote: post #97
SAI_Peregrinus wrote: post #84

Earlier quoted context omitted.

Multiplication of the Surreals is a recursive operation using sums (addition and subtraction on the left and right sets). Since the Reals are a strict subfield of the Surreals one can define multiplication of the reals using only the same recursive formula and restricting both operands to be Reals.

Its probably worth emphasising that the recursion you need to "construct" the Surreals is infinite, in other words this does not give a reasonable algorithm to (for example) add two real numbers, you need S_omega in order to have even all rational numbers. The construction is rather involved but if we're only interested in the reals for now you can think of it as defining a real number by a set of rational numbers, i…

Of course, once you decide to iterate over uncountable sets, infinity starts to appear all the time.

This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.

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

#107

Post by Retric »

SAI_Peregrinus wrote: post #76
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.

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…

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.

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

#108

Post by yakubin »

gugagore wrote: post #94
yakubin wrote: post #61

Earlier quoted context omitted.

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.wiki…

If you use a termination condition that has to do with convergence of iterates instead of a number of iterations (often the case), then you generally don't know beforehand the length of the finite sequences. Maybe you know a bound, but in general you might not even have that. In an important sense, it only becomes a finite algorithm. It isn't one. You cannot write the finite sequence of instructions down. It's got lo…

> To your point about approximations vs not, if you have an algorithm that, for any desired approximation accuracy can compute the square root to that accuracy in a finite number of steps, then that process is as much "the square root" as anything involving the real numbers.

Not really, since approximations, no matter how accurate, don't preserve algebraic properties. You only get to know what it's bigger/smaller than.

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

#109

Post by horsawlarway »

midjji wrote: post #101
Agingcoder wrote: post #9

I don't understand. The author is mixing up physics (dimensional analysis) and maths, and trying to give the multiplicand a special role (I didn't even know there was a distinction between the two terms - to me they are both factors). This might be true in the physical world, but in the world of numbers, I think the distinction is irrelevant. Furthermore, being able to compute/define multiplication through repeated a…

Not defending the article, but how would you compute: \pi*\pi using repeated addition?

pi + pi + pi + (.141592... of pi) ~= 9.8696

You need the concept of a ratio, so arguably I'm using multiplication to define multiplication, but you're sort of cheating by asking about a fractional number.

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

#110

Post by alisonkisk »

marcosdumay wrote: post #106
eigenket wrote: post #97

Earlier quoted context omitted.

Its probably worth emphasising that the recursion you need to "construct" the Surreals is infinite, in other words this does not give a reasonable algorithm to (for example) add two real numbers, you need S_omega in order to have even all rational numbers. The construction is rather involved but if we're only interested in the reals for now you can think of it as defining a real number by a set of rational numbers, i…

Of course, once you decide to iterate over uncountable sets, infinity starts to appear all the time. This isn't the only case of infinite calculations that can't be computed in practice but that mathematics use all the time anyway; and it reflects quite well the fact that multiplying irrational numbers isn't something that one can do practice. There is no problem with it.

Multiplying algebraics is trivial. A rectangle with sides sqrt(3) and sqrt(2) has area sqrt(6), which can be approximated by a decimal if needed.

There are countable/computable/constructable subsets of the reals where multiplication has a finite algorithm and is it repeated addition.

One example is the algebraics, as well as extensions the including a few special constants like pi. These are the subsets of the reals most commonly used for math and science. So in a wide range of problems areas, multiplication is not just repeated addition.

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