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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)

#341

Post by ahepp »

roenxi wrote: post #308
majormajor wrote: post #213

Earlier quoted context omitted.

The continuing series of qualifications "it's kinda like this, for what you're talking about" gets pretty rough there, even when explaining, say, Kubernetes to adults! It can be a confidence crusher. There's also the flip side response which is always asking a lot of questions about "well then what are the other sorts of numbers" and eventually getting shut down "we're not talking about that now" which comes back to…

> It can be a confidence crusher. If confidence gets priority over truth in explanations, then society will churn out people who are confidently wrong. This is a bad idea even if everybody does that and even if it is the traditional approach. If people were honest that they don't know something then the world at large would be a lot nicer to live in.

I think this is backwards. The public at large are pretty honest that they don't know math. The world would be a much better place if they were able to do practical algebra, even if the higher truths of abstract mathematics never enter their minds.

I really don't see people being confidently wrong about abstract mathematics as an issue. I certainly don't know the rigorous definitions of an integral, but I can apply the concepts of calculus to everyday life perfectly fine. People who care about rigorous math can do rigorous math, and I'm glad if they teach me an intuitive understanding that lets me live a happy, productive life.

Personally, I find layers of abstraction necessary for learning. Maybe there are people who don't, I suspect they would have to be prodigies though. Tell me how to add fractions practically, then teach me the principles when I need to know them. Framing that as a lie seems wrong to me, I'd call it "bounded knowledge".

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

#342

Post by kortilla »

pessimizer wrote: post #313
pdonis wrote: post #299

Earlier quoted context omitted.

> Explaining the structure of an atom without having to explain atomic orbitals and standing waves. My high school chemistry teacher had no problem explaining this to me, when teaching the periodic table of the elements, without telling any lies and without going into the details of the quantum mechanics involved. The Pauli exclusion principle and a general statement that the details of the quantum mechanics were out…

Why would you explain classical mechanics at all? It's an inaccurate simplification.

Classical mechanics covers the vast majority of the interactions in day to day life people will have. There is a reason it’s “classical mechanics”.

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

#343

Post by dragontamer »

thaumasiotes wrote: post #338
dragontamer wrote: post #198

Earlier quoted context omitted.

> At any rate we have to impose that n·0=0, which can't be writen cleverly as "repeated addition" and worked up backwards. n * 0 = n * (1 + (-1)) = n + (-n) = 0. ----- The 0-element in a Galois Field works identically btw. In GF(5), the 0 element is 5 (5 mod 5 == 0). n * (5) == n * 0 == n * (1 + (-1)) == n * (1 + 4) == 0 mod5. For example, if we take "n" == 2, 2 * 5 == 10 mod 5 == 0. 2 * (1 + 4) == 2 + 8 == 2 + 3 (mo…

> n * 0 = n * (1 + (-1)) = n + (-n) = 0 You don't think it's easier to say "n·5 = n·(5+0) = n·5 + n·0"?

The (1 + (-1)) thing was very precisely written.

All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is:

    [ 0 0
      0 0 ]
And 1 is:

    [ 1 0
      0 1 ]
Because A * 1 == A, A * 0 == [0 0; 0 0]... etc. etc. In general, you cannot really assume much more than "0 exists" and "1 exists" when working with Rings (at least, if you want your proof to extend out to all possible ring algebras). 0 and 1 may look like something you wildly don't expect... they're abstract labels that are kind of undefined aside from being additive-identity or multiplicative-identity, respectively.

--------------

As such, the concept of "5" does not necessarily exist in all possible Ring-systems. "5" exists in GF(5) for example, but not really in GF(3). Case in point, what does "5" mean in 2x2 Matrix Linear Algebra over GF(2)?

--------

"-A" is called the additive inverse of A, which also exists in all rings. A - A = 0.

In GF(5), -1 is 4 for example. In 2x2 Linear Algebra, -1 is [-1 0; 0 -1]. In Real Numbers, -1 is... well... -1.

----------

Anyway, the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings. (And then later, I did an example in the GF(5) system as a specific example).

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

#344

Post by thaumasiotes »

shadowofneptune wrote: post #135
Jtsummers wrote: post #103

Earlier quoted context omitted.

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).

The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.

> Instead of learning that you can use shapes to do useful calculations like square roots

Huh?

You can easily draw shapes that conceptually represent square roots, but how do you get from that to calculating the square root? You'd need an infinitely-graded ruler.

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

#345

Post by dragonwriter »

macspoofing wrote: post #337
cjfd wrote: post #105

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 def…

>I am sure he enjoys it a lot to be in a position where he can write pedantic falsehoods Are you sure it's pedantic? As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again -…

> As a matter of practically, neither people, nor computers actually compute multiplication in such a way. Even children, though they may be exposed to the 'multiplication is repeated addition' concept as an introduction to multiplication, are quickly ushered past this and it is never brought up again.

The standard manual method of multiplication of large numbers relies on leveraging heavily:

(1) the fact that my multiplication is equivalent to repeated addition for nonnegative integers,

(2) the fact that shifting digit positions are equivalent to multiplication by the base (usually 10) or it's multiplicative inverse, depending on direction, and

(3) memorization of multiplication tables for single digits in the base.

So, no, I don't think the “multiplication is repeated addition” thing is something people are exposed to and then never use.

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

#346

Post by thaumasiotes »

redshirtrob wrote: post #168
eru wrote: post #92

Earlier quoted context omitted.

That's not the norm?

I don't think so. I was on the Calculus track in high school so we derived it in...Pre-Calculus. Prior to that the quadratic formula was something that seemed to be handed down from on high. We used it in Algebra II and maybe even before that, but I had no idea where it came from. It was a mind-opening experience when we derived it in class one day. Our teacher didn't ruin the surprise. She just said, let's complete…

> Our teacher didn't ruin the surprise. She just said, let's complete the square on a general quadratic equation.

How is this not ruining the surprise? The only possible outcomes of doing that are that (1) you make a mistake; or (2) you get a formula for solving quadratic equations. Quadratic equations have the same solutions regardless of your methodology, so there's only one formula you can get.

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

#347

Post by sudosysgen »

thaumasiotes wrote: post #344
shadowofneptune wrote: post #135

Earlier quoted context omitted.

The way Geometry is taught in the US is awful. Instead of learning that you can use shapes to do useful calculations like square roots, you slog through postulates and theorums without any sense of why you have to do them. Rarely is what is learnt in geometry ever used in later high school courses, save for trigonometry. I hope it is different in other countries.

> Instead of learning that you can use shapes to do useful calculations like square roots Huh? You can easily draw shapes that conceptually represent square roots, but how do you get from that to calculating the square root? You'd need an infinitely-graded ruler.

All calculation is inherently imperfect. But if you want to have something of length sqrt(2) you can.

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

#348

Post by thaumasiotes »

sudosysgen wrote: post #347
thaumasiotes wrote: post #344

Earlier quoted context omitted.

> Instead of learning that you can use shapes to do useful calculations like square roots Huh? You can easily draw shapes that conceptually represent square roots, but how do you get from that to calculating the square root? You'd need an infinitely-graded ruler.

All calculation is inherently imperfect. But if you want to have something of length sqrt(2) you can.

I'd say the opposite; the drawing is guaranteed to be imperfect, but the calculation won't be.

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

#349

Post by codethief »

alentist wrote: post #334
aaplok wrote: post #289

Earlier quoted context omitted.

Multiplication is literally repeated addition, as in "taking a number multiple times". It is in the name. The fact that the operation is so useful that it has been generalised to the point where that original meaning is eventually lost through more and more abstractions doesn't invalidate that, because with all the generalisation and abstraction, multiplication as repeated addition still works, and any generalisation…

Define π times e in terms of repeated addition. I wouldn't say multiplication is literally repeated addition. I'd say it reduces to repeated addition when the multiplier is a natural number.

Maybe it is not literally addition but it certainly reduces to addition in all cases: https://news.ycombinator.com/item?id=26329855

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

#350

Post by thaumasiotes »

dragontamer wrote: post #343
thaumasiotes wrote: post #338

Earlier quoted context omitted.

> n * 0 = n * (1 + (-1)) = n + (-n) = 0 You don't think it's easier to say "n·5 = n·(5+0) = n·5 + n·0"?

The (1 + (-1)) thing was very precisely written. All rings have 0 and 1 as elements. 0 is the additive identity. 1 is the multiplicative identity. 0 and 1 are NOT necessarily numbers. In Linear Algebra of 2x2 matricies, 0 is: [ 0 0 0 0 ] And 1 is: [ 1 0 0 1 ] Because A * 1 == A, A * 0 == [0 0; 0 0]... etc. etc. In general, you cannot really assume much more than "0 exists" and "1 exists" when working with Rings (at l…

> As such, the concept of "5" does not necessarily exist in all possible Ring-systems.

Don't think of 5 as a quantity; think of it as a variable name. The proof only depends on the concept of addition and an additive identity (and distribution of multiplication over addition, which you're using anyway); no property of 5 appeared.

> the A * 0 == A * (1 + (-1)) == A - A == 0 thing is built up from fundamental portions of Ring theory. As such, the proof I constructed at first applies to all rings.

To repeat myself:

  n·5 = n·(5+0)         [definition of 0]
  n·(5+0) = n·5 + n·0   [multiplication is distributive over addition]
  n·0 = 0               [definition of 0]
Every step in that proof is a direct application of one of the ring axioms; it doesn't matter what 5 is.
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