Is Multiplication Really Just Repeated Addition?

[apeiron]

You have now divided the original line perfectly into n equal parts.

Thanks for that. But are you saying it is a geometric representation of either inverse multiplication or repeated subtraction as arithmetic operations?

Rather I think what it shows is a process in which "divide by x" is handled by creating a model of x outside the numberline and then morphing it to fit between two points on the numberline.

So a "whole" is constructed by additive steps, but then the whole is shrunk to fit. Which is a continuous transformation rather than as a series of discrete steps.

The other three operations are constructing a whole from the parts (additive actions). Whereas division starts with the whole and asks for a reduction to a set of parts.

So we can start by trying repeated subtraction with an example like 7/3. We can subtract twice then get down to having to divide the remainder 1 into 3 parts. We are now dealing with a "whole" unit - and subtraction depends on working with multiples of this unit. As does addition and multiplication.

The answer is to shrink the numberline in scale - morph the 1 to 10 to create an internal decimal division of .1 to 1. Then pick up with the subtraction at this new scale. And morph again if we need to get into hundredths or thousandths.

So division does seem deeply different in this light. The other three operations are straightforwardly constructive - operations that are discretely additive. But division involves the extra step of a morphing of a constructed co-ordinate space. Something quite different in nature is required to go from the whole to its parts. Even if in the end it is no big deal because you have multiplication as a "look-up table" of inverse operations and decimals as a standard way to fractalise the dimensionality of the numberline. (Base 10 is just a choice, not something derived axiomatically, is it? God created the integers but not the decimals? )

 

[Studiot]

But are you saying it is a geometric representation of either inverse multiplication or repeated subtraction as arithmetic operations?

I didn't say anything of the sort.

I said before that I answered a specific question you made, asking for a proceedure to "divide a given random number into x equal parts" and thought you also implied that you did not think this could be done.

As it happens this proceedure was documented centuries before we had anything like modern arithmetic so it preceeded such theory and cannot therefore be said to be derived from it in any way.

I am rather suprised you have not heard of it since in another thread you claimed a classical education within the British/Irish system.

I fully agree with Hurkyl that there are also arithmetic algorithms fully developed to handle the question. Obviously these came later in the hsitory of mathematics.


go well

 

[apeiron]

I said before that I answered a specific question you made, asking for a proceedure to "divide a given random number into x equal parts" and thought you also implied that you did not think this could be done.

Then you misunderstood the question. It was about the OP point that "division seems different" and so about the precise nature of that difference in number theory.

I am rather suprised you have not heard of it since in another thread you claimed a classical education within the British/Irish system.

They were teaching new maths by the time I came along I guess.

 

[Studiot]

I can only understand what is written.

If I gave you any randomly chosen real number and asked you to split it into x equal portions, and you had no access to multiplication tables or other forms of prior knowledge, how in terms of a mathematical operation would you proceed?

Conforms exactly to the question I answered and later paraphrased.

 

[apeiron]

I can only understand what is written.

Conforms exactly to the question I answered and later paraphrased.

OK, I understand. You can't answer the larger question that was posed. You are not interested in how a procedure works, only that it works.

It was about the OP point that "division seems different" and so about the precise nature of that difference in number theory.

 

[Studiot]

You can't answer the larger question that was posed.

You do seem to like putting (incorrect) words into the mouths of others.

I have been following this thread since near its inception, and even posted way back although my comment at that time has not been addressed.

 

[disregardthat]

:sigh: If only the answer were so simple as long division.

As should be clear, the issue is the precision. The answer for simple constructive operations is always going to be exact. But for division, answers are only going to be effective. You have to introduce a cut-off on the number of decimal places as a further pragmatic choice.

This is purely due to the choice of representation by decimals. Division is a wholly constructive operation, but it is simply the case that not all rational and real numbers can be written as a finite decimal expansion in base 10. There isn't anything imprecise about the result of an operation that iteratively gives you the base 10 digits of a given real or rational number, it gives you exactly what you want.

You talked about the necessity to introduce limits when it comes to real numbers - as a sort of flaw, but it is in this sense real numbers are defined (or can be defined) - as limits.

It seems like the prime number factorisation problem. You have to guess repeatedly to crack the answer. There is no simple iterative operation to employ.

Actually, there are simple constructive methods to give you the full prime factorization of any given integer. It is not necessary to guess at any point. Even the most naive (ineffective) ones will not involve any guesswork.

 

[apeiron]

This is purely due to the choice of representation by decimals. Division is a wholly constructive operation, but it is simply the case that not all rational and real numbers can be written as a finite decimal expansion in base 10. There isn't anything imprecise about the result of an operation that iteratively gives you the base 10 digits of a given real or rational number, it gives you exactly what you want.

You talked about the necessity to introduce limits when it comes to real numbers - as a sort of flaw, but it is in this sense real numbers are defined (or can be defined) - as limits.

Actually, there are simple constructive methods to give you the full prime factorization of any given integer. It is not necessary to guess at any point. Even the most naive (ineffective) ones will not involve any guesswork.

Yes, division can be a wholly constructive operation (namely, repeated subtraction) but only because a further "natural" step has been taken in breaking the symmetry of the number line by choosing a base 10 numbering system.

So it seems to me that an extra geometrical argument has been introduced at this point. Whereas the numberline is a linear additive concept, we are now laying over the top of it a geometric expansion which gives us "counting in orders of magnitude and decimal scale".

Now of course I am sure people will say they see no issue here because all the points on the numberline exist. So 1.3, or pi, are just as natural as entities as 1,2,3.

But that is what I am musing about. Some extra constraint appears needed to break the naive symmetry of the numberline. The challenge was to connect something that is essentially discrete (a string of points) with what also had to be essentially continuous (a line) and breaking the scale of counting in this way, using a base as a further constraint, seems like the way it has been done.

So anyway, the answer for me now goes clearly beyond the original question about the nature of division and is clearly part of all the conversations about irrational numbers and infinities.

The numberline is founded on the notion of "one-ness". And that is a symmetric or single-scale concept. But as soon as you introduce an asymmetry, a symmetry-breaking constraint - such as any base system starting even from base 2 - then there is something new. A connection is forged between the original point-like discreteness and the continuity implied by a numberline. Scale is broken geometrically over all scales. Allowing then measurement down to the "finest grain".

 

[Hurkyl]

Yes, division can be a wholly constructive operation (namely, repeated subtraction)

I can't guess what you mean from this description.

but only because a further "natural" step has been taken in breaking the symmetry of the number line by choosing a base 10 numbering system.

I can't guess what you mean from this description.

Whereas the numberline is a linear additive concept, we are now laying over the top of it a geometric expansion which gives us "counting in orders of magnitude and decimal scale".

I can't guess what you mean from this description.

The challenge was to connect something that is essentially discrete (a string of points) with what also had to be essentially continuous (a line) and breaking the scale of counting in this way, using a base as a further constraint, seems like the way it has been done.

I can't guess what you mean from this description.

So anyway, the answer for me now goes clearly beyond the original question about the nature of division and is clearly part of all the conversations about irrational numbers and infiities.

I can't guess how you draw that conclusion from this description.

The numberline is founded on the notion of "one-ness". And that is a symmetric or single-scale concept. But as soon as you introduce an asymmetry, a symmetry-breaking constraint - such as any base system starting even from base 2 - then there is something new. A connection is forged between the original point-like discreteness and the continuity implied by a numberline. Scale is broken geometrically over all scales. Allowing then measurement down to the "finest grain".

I can't guess what you mean from this description.


Try using math instead of prose.



Incidentally, the attached diagram depicts a rather simple purely geometrical construction of division.

Segments AD and AG were constructed to be unit length.
Segments BD and CE were constructed to be parallel.
Segments BF and CG were constructed to be parallel.

length(AE) = length(AC) / length(AB)
length(AF) = length(AB) / length(AC)

I can't be sure if this is relevant to whatever you're thinking, though.

 

[apeiron]

Try using math instead of prose.

Try being helpful. And if you don't wish to be, simply don't respond.

 

[disregardthat]

Yes, division can be a wholly constructive operation (namely, repeated subtraction) but only because a further "natural" step has been taken in breaking the symmetry of the number line by choosing a base 10 numbering system.

You should explain what you mean by "breaking the symmetry" of the number line. Real numbers are not defined as "what makes up the number line" if that is what you are driving at.

So it seems to me that an extra geometrical argument has been introduced at this point. Whereas the numberline is a linear additive concept, we are now laying over the top of it a geometric expansion which gives us "counting in orders of magnitude and decimal scale".

The number line is a "linear additive concept"? What does that mean? The definition of real numbers is not referring to the number line. The number line is used as an analogy or intuition of real numbers, as it e.g. captures the geometric interpretation of the intermediate value theorem.

Now of course I am sure people will say they see no issue here because all the points on the numberline exist. So 1.3, or pi, are just as natural as entities as 1,2,3.

Why would there be an issue? You should explain that first.

But that is what I am musing about. Some extra constraint appears needed to break the naive symmetry of the numberline. The challenge was to connect something that is essentially discrete (a string of points) with what also had to be essentially continuous (a line) and breaking the scale of counting in this way, using a base as a further constraint, seems like the way it has been done.

What symmetries are you talking about, and how are they broken? A base representation is not a constraint (or what do you mean by that?), it is just what it is called: a representation.

The numberline is founded on the notion of "one-ness". And that is a symmetric or single-scale concept. But as soon as you introduce an asymmetry, a symmetry-breaking constraint - such as any base system starting even from base 2 - then there is something new. A connection is forged between the original point-like discreteness and the continuity implied by a numberline. Scale is broken geometrically over all scales. Allowing then measurement down to the "finest grain".

You should describe what you mean mathematically. "One-ness" is meaningless as a mathematical "foundational notion" as you put it, unless you describe mathematically what you mean by it.

 

[Hurkyl]

Try being helpful. And if you don't wish to be, simply don't respond.

It really is good advice. It's way too easy to be a never-ending fount of nonsense if you never even try to connect your thoughts back to mathematics.

It would also make my job easier and more fun -- it would mean fewer threads I have to step in and moderate, and more threads that I might enjoy helping someone learn something or formulate their own ideas.

(For the sake of clarity -- yes, this thread is nearing the point where I would lock it. On another day I might have locked it already)

 

[apeiron]

...more threads that I might enjoy helping someone learn something or formulate their own ideas.

You say that, but you aren't giving me the slightest help in connecting my prose to your maths.

From the start, I simply pointed out the bit of the OP that intrigued me and asked about how that would be handled. I was expecting someone to say, that is just the xyz conjecture, or whatever, a very standard issue you can go google to discover more.

The only response slightly like that was Studiot's, and it took several goes and a few insults to discover something that indeed gave me a better insight.

If I could package up my own thoughts in terms that you would consider properly mathematical, of course I would. But I still feel the gist was clear enough.

And when I did make further effort to explain myself more clearly that your response - "I can't guess what you mean from this description." - just appears childish and mean spirited.

It's your choice to lock the thread. Just don't insult me anymore by pretending that you have made an effort, whereas I have not.

 

[brydustin]

Of course there will always be much said on something so ridiculous as a post like this... everyone wants to give their two cents and anyone can. With that said...
Multiplication IS repeated addition for children because they are children.

 

[Hurkyl]

Multiplication IS repeated addition for children because they are children.

I don't think you give children enough credit -- they can form abstract concepts ideas too. In fact, I understand they're generally better at it than grown-ups.

Of course, if you drill a child into mentally substituting "repeated addition" whenever he sees a multiplication symbol, that would make it very difficult for him to mentally form the notion of multiplication being an operation in its own right.

 

[coolul007]

The correct answer is: multiplication defines area. It can be an area Pi x e, or .4 x .005

 

[TylerH]

The correct answer is: multiplication defines area. It can be an area Pi x e, or .4 x .005

How do you know area (of a rectangle in R2) doesn't define mulitplication?

 

[Robert1986]

The correct answer is: multiplication defines area. It can be an area Pi x e, or .4 x .005

OK, multiplication defines area. Big whoop! Read the title of the thread; read the posts in the thread. We're not discussing what multiplication defines; we're discussing what defines multiplication. If you mean that multiplication is defined as area, then you are very wrong. At best, your "definition" works when you multiply two real numbers. When you do three, you are (in your terms) talking about volume. Then, what about when you multiply complex numbers? Or, what if you are not multiplying numbers at all? What if you are dealing with matricies? Or composition of functions?

 

[Stephen Tashi]

Try being helpful. And if you don't wish to be, simply don't respond.

If you think you can become the thread controller, you're dreaming.

 

[coolul007]

OK, multiplication defines area. Big whoop! Read the title of the thread; read the posts in the thread. We're not discussing what multiplication defines; we're discussing what defines multiplication. If you mean that multiplication is defined as area, then you are very wrong. At best, your "definition" works when you multiply two real numbers. When you do three, you are (in your terms) talking about volume. Then, what about when you multiply complex numbers? Or, what if you are not multiplying numbers at all? What if you are dealing with matricies? Or composition of functions?

Well maybe, I should have said area/volume/? defines multiplication. Functions define a result, geometrically, whether in hyperspace or Euclidean planes. Matrices define multidimensional space. All are consistent with the definition.

 

[Hurkyl]

How about just saying "multiplication and area are related by the fact that the product of the lengths of two sides of a rectangle is equal to the area of the rectangle", rather than dogmatically trying to assert one defines the other?

 

[coolul007]

How about just saying "multiplication and area are related by the fact that the product of the lengths of two sides of a rectangle is equal to the area of the rectangle", rather than dogmatically trying to assert one defines the other?

I wasn't being dogmatic, I was trying t answer the question/discussion with a consistent definition that would work for the original poster. Repeated addition is taught as it is a convenient way to describe to a 7 year old what is taking place. We don't always teach things in the correct way, so that we don't cause inconsistencies down the road. I am a fan of teaching primes and prime factorization at an early age. That will ease most elementary problems that children struggle with, I.E. fractions/rationals, however, that won't happen as our elementary educational system is not enlightened enough, but I digress.

 

[TylerH]

Does the origin of a function have any bearing on what we say its definition is? If it does, I think we could say that multiplication is defined as repeated addition in N, with extensions for everything else. I doubt the cavedudes were thinking pi * e when they invented/discovered multiplication.

 

[Stephen Tashi]

Neo-Platonisitic discussions certainly become contentious! When threads are posted along the lines of "Is .9999 =1?", "Are infinitismials nonzero?", "Is sqrt(-1) an actual number?" they do well as light mathematical chit-chat. Everyone gets to express their own personal intuitions. I think it's interesting to put the real mathematical definitions of things aside and hear about how people imagine things.

However, there are always some participants that take the topic seriously and insist that they are the ones who see the THE TRUTH. Formal mathematical definitions may be dull, but after reading attempts at discussing math as serious personal philosophy, I begin to appreciate them.

 

[Hurkyl]

I think it's interesting to put the real mathematical definitions of things aside and hear about how people imagine things.

I actually view this as a sort of trick question. I've always viewed one of the greatest strengths of mathematics is the ability to seamlessly flow back and forth between 'pictures'.

If someone answers the question "how do you imagine multiplication" with something like "I imagine it as talking about area!", that's a bad thing -- they've limited themselves to one particular 'picture'.

This person will probably have success at applying multiplication to area problems, and have some success applying area to multiplication problems.

However, this person will also face unnecessary difficulty in applying multiplication to problems that aren't related to area, or converting multiplication problems into other sorts of problems.

 

[coolul007]

I actually view this as a sort of trick question. I've always viewed one of the greatest strengths of mathematics is the ability to seamlessly flow back and forth between 'pictures'.

If someone answers the question "how do you imagine multiplication" with something like "I imagine it as talking about area!", that's a bad thing -- they've limited themselves to one particular 'picture'.

This person will probably have success at applying multiplication to area problems, and have some success applying area to multiplication problems.

However, this person will also face unnecessary difficulty in applying multiplication to problems that aren't related to area, or converting multiplication problems into other sorts of problems.

What I find interesting, many of you have tried to turn this into a personal issue instead of a mathematical one. It is not about people it is whether the 'definition" fits the problem. If it does use it, if it doesn't, find a better one.

 

[Stephen Tashi]

I actually view this as a sort of trick question. I've always viewed one of the greatest strengths of mathematics is the ability to seamlessly flow back and forth between 'pictures'.

If someone answers the question "how do you imagine multiplication" with something like "I imagine it as talking about area!", that's a bad thing -- they've limited themselves to one particular 'picture'.

I agree that having a variety of pictures is a good thing. I disagree that there are any serious consequences from the way people answer questions like this. For one thing, the way that people imagine things probably isn't as consistent as the way they answer questions about their imagination. I can't imagine this type of thread damaging the career of any budding mathematicians. If they have the talent to do math, they will enjoy these discussions in the way that people enjoy discussions about "What's your favorite beer?" or "Who's your favorite author?".

 

[hillzagold]

Won't someone think of the children?

Ahem. The author in the column from the initial post had to make 5 separate entries explaining to emailers who disagreed with him. I myself initially disagreed with him and believed that multiplication was repeated addition and exponentiation was repeated multiplication. When I had to do 2x3.3, I broke it up into 2 + 2 + 2 + .3x2, and went my whole life. But reading Devlin's column made my realize that I was defining multiplication as repeated addition, and was defining this instance of repeated addition using multiplication! I was pretty confused until Devlin explained a key difference being multiplied values can have different units, such as kilowatts and hours, and I could finally accept multiplication as it's own distinct operation.

As one poster previously mentioned, seeing multiplication and exponentiation as repeated addition made understanding i²=-1 completely impossible for me. It's presumably still impossible for at least 95% of America. But some posters argue that it's necessary to that children can master multiplication at all. May I propose we try and make a pros and cons list?Pros:
This is the status quo, at it is what almost the entire country, including many of the teachers, honestly believes.
It's apparently easier to learn and understand, to those simple 1st grader minds. (?)
It can be retaught properly later, maybe in middle school or high school or even college. (?)

Cons:
It is mathematically flawed, according to any mathematician you can find who speaks your language.
It makes more advanced concepts, from fractions up to complex numbers and beyond, more confusing.(?)
It actually can't be retaught properly later, with a dependable success rate. (?)

Would anyone like to contribute to this list in any way?

 

[coolul007]

I think of the children, being taught one thing and then having to "unlearn" it later in favor of a better concept later, is the confusing part. And what if the better concept never comes. The state of lower education is not in favor of the rigor of mathematics, but some "touchy feely" form to make the children have a false sense of confidence. Mathematics is a system that does not have a lot of "stand alone" concepts. (I will get a large reaction for that last statement) I tutor high school students, the part most of them are missing is the foundation for understanding the concepts being taught to them. The modern trend is also to rename a lot of concepts, some good some bad, fractions, rational numbers, etc. Vocabulary and English should be taught in that class not mathematics.

 

[Studiot]

Won't someone think of the children?

Motivation is all.

Of course this applies to all ages, not only children.

It is easy to demotivate.
Just teach a list of arbitrary definitions and statements, really designed to keep pupils quiet and sitting up straight.

No go down the pub and ask a darts player who has just scored 347 and who flunked school, what he need to finish and I guarantee he will know more quickly than any 5 maths professors put together.