So, what is multiplication?

In summary, the conversation discusses the concept of multiplication and whether it is truly repeated addition. The author of the article argues that multiplication is a distinct operation from addition and should not be taught as repeated addition, while others believe it is a useful way to introduce the concept to children. The author also brings up examples of how the idea of repeated addition becomes problematic when dealing with rational and real numbers. Ultimately, the conversation highlights the different perspectives on how to define and teach multiplication.
  • #1
JyN
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2
I just read an article (http://www.maa.org/devlin/devlin_06_08.html [Broken]) saying that multiplication is not repeated addition.

I am a first year engineering student, and i am very interested in mathematics. The picture i have had of multiplication for ~12 years or something is definitely of repeated addition. Wikipedia also seems to give this definition. http://en.wikipedia.org/wiki/Multiplication

I read that there are 4 elementary operations for arithmetic. Addition, subtraction, multiplication, division. I notice that subtraction is the inverse of addition, and division the inverse of multiplication. So, if multiplication was indeed repeated addition, there would only be two elementary operations. Addition, and subtraction. And since subtraction is inverse addition, that would mean that division is repeated subtraction, and it certainly isn't.

As a side note: I actually remember seeing it like this when i was very young and first learning about arithmetic. And, because i saw multiplication as repeated addition, it seemed to me that division was really not like the others. Although i just accepted this and havn't thought about it since.

tl;dr If multiplication isn't repeated addition, what is it? Do you think i am missing something based on my above explanation?
 
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  • #2
Hmm, I think what he's getting it is the following: multiply 0,1 and 0,1 with each other. The answer is 0,01. But this is not repeated addition since you can't add up 0,1, for 0,1 times.

Of course, in higher mathematics, addition and multiplication can be totally different things. The multiplication of matrices can not be seen as a repeated addition of matrices. And there are many other kind of examples.

What the author of the article wants, is that teachers explain to their students that there are 2 kinds of operations on the integers: addition and multiplication. And that these operations have nothing to do with each other.

While this is certainly true for higher mathematics, I think that the auther is barking up the wrong tree. I think it's very useful for children to look at multiplication as repeated addition. And I wouldn't want my child to be taught otherwise. Of course, a few years later, we can say that this is not quite true, but I think that it is very useful to think of multiplication as repeated addition.

So, while he technically right, I don't agree with him...
 
  • #3
try to multiply e by pi. e from exponential 1 and pi = 3.14...you'd see it's not a repeated addition at all.
 
  • #4
JyN said:
I just read an article (http://www.maa.org/devlin/devlin_06_08.html [Broken]) saying that multiplication is not repeated addition.
Multiplication is bilinear. That means (a+b)c = ac + bc and a(b+c) = ab + ac. In other words, the distributive property holds.


In the very special case that "a" can be written as repeated addition of a multiplicative unit:
a = 1 + 1 + 1 + ... + 1​
then "ab" can be written as repeated addition of b:
ab = (1 + 1 + ... + 1)b = 1b + 1b + ... + 1b = b + b + ... + b​
 
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  • #5
Multiplication is certainly not repeated addition. Asking what is multiplication, the correct answer will be what do we do when we multiply? So the author is entirely correct, both technically and pedagogically, the method of multiplying is entirely different from the method of adding. It is a matter of fact that we do not ordinarily calculate 4*71 the same way as 71+71+71+71. If we did the matter would be different. Even though the methods are interchangeable, we simply do not call 71+71+71+71 multiplication!

To ask what multiplication is if it is not addition is like asking what addition is if it is not multiplication. What type of answer do you expect? Multiplication is the activity or method of multiplying.
 
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  • #6
I've always just considered repeated addition a special case for the natural numbers, that can easily be extended to rational numbers. It's not a contradiction, just as division, as an extension to Cancellation Law,
isn't a contradiction, just an extension.

This is similar to a question I've wondered about for a while. How would one rigorously define addition and multiplication?
 
  • #7
I agree with micromass.

When you start teaching children about math you don't go straight to the reals, you start with specific examples involving integers.

When you deal with integers, there is in fact a direct relationship between multiplication and addition.

Even with the reals though you can break up a number into its "integer" and "fractional" part with distributivity and treat the multiplication with the "integer" parts and the "fractional" parts. By the time the students have gone through learning the rationals, the same kind of idea that was introduced in primary school can be used in high school.
 
  • #8
I sympathize with the author. It is not right to consider multiplication as an application of addition, the method of multiplication is simply not repeated addition. Young students ought not be taught that. These are two quite different operations with interesting relations which can be presented in various interesting and in my opinion more instructive ways.

And it is a valid point that this notion of repeated addition is queer when we get to rational numbers, and completely abolished when it comes to real numbers (or at best the analogy is in a twisted form). The method of multiplication on the other hand is generalized to rational and real numbers in a much more natural way. Why should we consider 2.3*4.2 repeated addition and how does it help us ?

There are other reasons for why the notion of repeated addition is not appropriate. What happens in an application when you are calculating the area of, say, a rectangle? Say it is 3 m long, and 2 m wide. To calculate the area, we have 3 m * 2 m = 6 m^2. What is added here? 2 m^2 + 2 m^2 +2 m^2 ? This certainly does not generalize neatly when it comes to a rectangle of length 3.3 m, and width 2.5 m. How is it intuitive that we should think of calculating area as repeated multiplication?

The differences are much more important than the similarities.

TylerH said:
This is similar to a question I've wondered about for a while. How would one rigorously define addition and multiplication?

This is the seed of the problem. What exactly is not rigorous with multiplication and addition?
 
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  • #9
Multiplication is, at heart, repeated addition. Your 1st grade teacher wasn't lying.

>Why should we consider 2.3*4.2 repeated addition and how does it help us?

If you ask me to pay you $2.3 times 4.2, then the "multiplication as repeated addition" metaphor shows me clearly you are asking to be repeatedly paid $2.3 more than 4 times but fewer than 5 times. The total amount is 2.3+2.3+2.3+... where the number of iterations is 4.2. I can continue perfectly thinking of the multiplication as addition as long as I admit the idea of a fractional iteration.

Another example: construct a rectangle 2.3 inches wide by 4.2 inches long. Then repeatedly add 1x1
squares on top of the rectangle until the rectangle is exactly filled up. You are also allowed to insert pieces of squares. How many repeatedly added squares did it take to fill the rectangle? I believe I can solve the problem to arbitrarily high accuracy while performing only additions, subtractions, and comparisons.

The concept of "multiplication as repeated addition" applies in every case, depending on how flexibly you can think. For example, what is "i"? It is the number, that when repeatedly added i times, yields -1. If this sounds awkward, it's because "i" is awkward, not because the basic idea of multiplication has somehow changed.

There are of course other ways of viewing multiplication. In the complex plane, it can operate like a rotation. But I maintain that the "multiplication as repeated addition" metaphor can remain valid in almost any scenario and, if it helps you think, shouldn't be discarded.

>This is similar to a question I've wondered about for a while. How would one rigorously define addition and multiplication?

You don't. They exist purely as concepts in the human mind. Regarded as an element of natural science, the human brain produces patterns, but cannot produce true or false ideas any more than the stomach can produce true or false acid.
 
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  • #10
Goongyae said:
if it helps you think, shouldn't be discarded.
While focusing on an analogy (possibly torturing it in the process) may help you think about something initially, it eventually becomes an obstruction, preventing you from developing an understanding of that thing in its own right.

I'm pretty sure you're past the point where it's an obstruction when you are trying to think of adding something i times. :wink:
 
  • #11
In the spirit of some of the previous posts I wonder if it would be helpful to think of multiplication as the natural generalization of repeated addition to non-integer numbers much in the same way that the gamma function interpolates non-natural numbers for the factorial function.

So that it is in some sense an artifact of human intuition that the manifestation of multiplication as repeated addition in the natural numbers carries such weight with us. If we grew up learning math from the perspective of the real numbers the fact that multiplication and repeated addition happen to coincide for integer values might be seen as simply a natural by-product of the 'cleanliness' of the integers.

I suppose in this sense the process seems almost empirical in that we are constantly exploring more and more exotic sets of numbers and thus having to constantly refine our definition of multiplication to account for the new data.

So what is multiplication? It seems to me to be one of those base facts that if not considered self-evident cannot be adequately explained. I think the author in the MAA article has it essentially correct, our notions of scaling and counting, central to meaningful sensory experience, seem to be at the core of our intuition for these two concepts.
 
  • #12
micromass said:
Hmm, I think what he's getting it is the following: multiply 0,1 and 0,1 with each other. The answer is 0,01. But this is not repeated addition since you can't add up 0,1, for 0,1 times.

Of course, in higher mathematics, addition and multiplication can be totally different things. The multiplication of matrices can not be seen as a repeated addition of matrices. And there are many other kind of examples.

What the author of the article wants, is that teachers explain to their students that there are 2 kinds of operations on the integers: addition and multiplication. And that these operations have nothing to do with each other.

While this is certainly true for higher mathematics, I think that the auther is barking up the wrong tree. I think it's very useful for children to look at multiplication as repeated addition. And I wouldn't want my child to be taught otherwise. Of course, a few years later, we can say that this is not quite true, but I think that it is very useful to think of multiplication as repeated addition.

So, while he technically right, I don't agree with him...

As a personal story, I recall that I didn't know that I wanted to major in Math until sometime during my first Calculus course. When I started taking my second Calc course, I was really upset that the professor nor the book bothered to prove things as rigorously as possible. For example, we still relied on that "a function is continuous at a if you can draw it without picking up your pencil near a." Clearly, this isn't a very rigorous definition, in fact, speaking as a more mature math student (that is, more mature in math than your average Calc II student) it is a really silly definition.


Stuff like this really irritated me, but now I see that these sorts of "definitions" are actually pretty good to teach a Calc 1,2,3 student. First of all, without having several months/weeks of an analysis course, it is really impossible to define continuous. So, the only option is to start first year college students in an analysis course, but I don't think this would work for most people. So, students get crappy, though very intuitive definitions of mathematical ideas. This way, we are not overloaded with tons of definitions. We get to develop an intuition for stuff, and THEN we get to learn the material more rigorously.


The same goes for multiplication. Start kids out by telling them that multiplication is repeated addition (though, I, too was taught that division was repeated subtraction and I thought that was insanely stupid). Don't mention rationals or reals until they have a good intuitive understanding of the integers.


And, if you look at the way most maths were developed, we see that they usually started out describing real objects, and then got more abstract. If I understand correctly, the foundations of calculus weren't laid until well after other areas of calculus had been laid.
 
  • #13
Robert, the issue here is not rigor. There isn't anything non-rigorous about multiplication if it isn't insisted on that it's "really just" addition at work all along, so I don't at all see the analogy with the pencil-definition in calculus.
 
  • #14
Jarle said:
Robert, the issue here is not rigor. There isn't anything non-rigorous about multiplication if it isn't insisted on that it's "really just" addition at work all along, so I don't at all see the analogy with the pencil-definition in calculus.

My point is that there is no need to teach kids anything other than multiplication is repeated addition. Let kids work out the intuition of multiplication, then break the news to them years later. If not, I see two alternatives:

1)Give the kids an axiomatic definition of multiplication, then explain how it works on the integers, and go from there.

I think this is a terrible idea. 6 year olds will not grasp that.

2)Don't tell the kids what multiplication is, just teach them how to do it. Make them memorize multiplication tables, and them teach them how to do "long multiplication".

Assuming that this would actually work in the first place, it is another terrible idea because it teaches the kids absolutely nothing and instead forces them to memorize how to multiply rather than getting an intutive grasp of multiplication.Either way, one can argue that there are downsides, I just think there are less downsides to teaching kids that multiplication is repeated addition.
 
  • #15
Repeated addition is merely a way to multiply integers, and this is what children are taught. That we can calculate 3 x 2 by adding: 3 + 3. They don't need to be told that multiplication always is repeated addition and that "it's just done differently, like so-and-so."

"This:
Code:
15 x 23
--------
     45
    30
--------
 =  345

is really just addition."


What is the point of this? What is the supposed intuition it provides? The addition part of multiplication is reserved to the addition of single-digit integer, and multiplication in general should be thought of as separate from the general aspect of addition. The important part is drawing the connections between multiplication and addition. Like distribution, or that it is possible to count the number of stones ordered like a rectangle by multiplying the number of stones on the sides. How is it better to say "this is really just the same thing"?

What something is is not equivalent to how it formally can be defined, and formal definitions is the last thing children should be taught.

When they calculate the area of rectangles or multiply rational numbers in decimal form, remembering the repeated-addition feature of multiplication is nothing more than a hinder for good intuition. What is added in 2 m * 3 m? What is added in 0.23 * 0.35 ?
 
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  • #16
Jarle said:
Repeated addition is merely a way to multiply integers, and this is what children are taught. They don't need to be told that multiplication always is repeated addition and that "it's just done differently, like so-and-so". What is the point of this? When we they to calculate the areas of rectangles or rational numbers in decimal form, remembering the repeated-addition feature of multiplication is nothing more than a hinder for good intuition. What is added in 2 m * 3 m? What is added in 0.23 * 0.35 ?

I agree that multiplication is not repeated addition. But I think it's the best to teach first-graders that it is repeated addition, because it will be much easier to grasp and to work with. I challenge everybody here to go to 6-year olds and explain multiplication without referring to repeated addition, in my opinion it's not possible...

Also, if you look back historically, then multiplication really did began as a shorthand to repeated addition. It is only much later that they multiplied by other numbers. So there's no lying if you tell children that multiplication of integers is repeated addition.

Of course, once you end up doing fractions, then the entire repeated addition story collapses, and we should tell the children that. But not before they're used to multiplying...
 
  • #17
micromass said:
I agree that multiplication is not repeated addition. But I think it's the best to teach first-graders that it is repeated addition, because it will be much easier to grasp and to work with. I challenge everybody here to go to 6-year olds and explain multiplication without referring to repeated addition, in my opinion it's not possible...

The route must obviously go through addition in basic multiplication, but that is not the point. This only part of learning how to multiply, not how to multiply in general. After learning basic multiplication at heart one can start learning multiplication in general. And then draw connections.

This is in my opinion intuitively better, because all along the line even up to university mathematics multiplication and addition are the two main algebraic operations with important relations to each other. Why confuse the two with each other for years?
 
  • #18
I think some of you guys are confusing "non-intuitive" or "awkward" with "logically absurd".

There is nothing logically absurd about adding something together "i" times; gongyae is right, it's merely non-intuitive. Same for 0.23 x 0.35; the equivalent addition is not absurd, just awkward.

The point is theoretically important because if, in any branch of mathematics (say, arithmetic), it transpires that multiplication can theoretically always be reduced to an operation of addition, that means multiplication is logically redundant (vide Occam's Razor).

In everyday life, of course, some such operation as this:

36
12
----
72
360
----
= 432
===

- is far more convenient, in the practical way, than adding 36 twelve times. But the calculation is merely a conjuring trick; the application of certain mental short-cuts which are known to give a correct result. It does not suffice to prove that there really is some mystical entity called "multiplication", of the same logical status as "addition", though different in kind.

How do you know that six twelves are seventy-two, without calculating it? Because you learned your multiplication tables in primary school. Could you think of a way to PROVE to a child that 6 x 12 = 72 without resorting to addition?

I suggest that addition is the more logically fundamental operation (at least in arithmetic) because

a) A multiplication operation is always theoretically reducible to an addition operation;

b) An addition operation is never reducible to a multiplication operation, except by employing formal rules which are themselves ultimately reducible to rules of addition.
 
  • #19
"Of course, once you end up doing fractions, then the entire repeated addition story collapses... "

I really don't think so, Micromass. Try it on piece of notepaper. If you try multiplying (say) one and three-quarters by five eighths, you will find that every step can be reduced to an addition (although, towards the end, you would need to use subtraction to reduce the denominator to its lowest number).
 
  • #20
Alan1000 said:
- is far more convenient, in the practical way, than adding 36 twelve times. But the calculation is merely a conjuring trick; the application of certain mental short-cuts which are known to give a correct result. It does not suffice to prove that there really is some mystical entity called "multiplication", of the same logical status as "addition", though different in kind.

What are you talking about? "Mystical entity called multiplication"? No one are talking about any mystical entities here. Not the "same logical status" as addition? Define logical status.
 
  • #21
Alan1000 said:
a) A multiplication operation is always theoretically reducible to an addition operation;

That's a view that will ultimately fail in modern mathematics. Historically, you are correct. But mathematics is far beyond that phase.

I challenge you to reduce i.i=-1 to an addition operation.
Or how do you reduce the scalar multiplication of vectors to an addition operation?
Or how do you reduce the multiplication of group rings to an addition operation?

Your point-of-view only works for the integers or things that ressemble the integers. But ultimately it breaks down.

Not to say that it isn't pedagogical! If you want to calculate the volume of a cube, then thinking of repeated addition really confuses most students. And if you want to explain 0.23*0.4 to students, then I really suggest not talking about repeated addition...
 
  • #22
Miromass, most of your objections were covered earlier in this thread.

>Or how do you reduce the scalar multiplication of vectors to an addition operation?

U dot V = UxUx+UyUy+UzUz = Ux added Ux times, followed by Uy added Uy times, followed by Uz added Uz times.

But I would say this is not really a multiplication. After all, anything times 1 is itself, but there is no meaning established for U dot 1. This is a different operation entirely. If the dot / inner product operator were truly multiplication then there should be some way to compute U^(3/2) where U is dotted with itself for a total of 1.5 times. But there's no such animal.
 
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  • #23
Goongyae said:
Miromass, most of your objections were covered earlier in this thread.

>Or how do you reduce the scalar multiplication of vectors to an addition operation?

U dot V = UxUx+UyUy+UzUz = Ux added Ux times, followed by Uy added Uy times, followed by Uz added Uz times.

That's the inproduct. I'm talking about scalar multiplication http://en.wikipedia.org/wiki/Scalar_multiplication ). That is, take a vector v, and multiply it by 0.5, which yields a vector which is only half as long as v. I'll be surprised if you can reduce this to an addition!

I say if your "multiplication" can't be reduced to an addition, then it's not a multiplication at all, regardless of your use of the word.

That's a very narrow point-of-view, and modern mathematics is far beyond this. For mathematicians, multiplication is just a defined operation (which is preferably distributive over addition). There are many instances where this kind of multiplication is handy. And please, tell me how multiplication by i, multiplication in Boolean algebras and group rings, multiplication of vectors, the vectorial product, etc. can be reduced to an addition. I believe that all these instances are worthy to be called multiplication, certainly the multiplication by i...
 
  • #24
Honestly, it doesn't really matter whether or not multiplication can be reduced to repeated addition in suitably weird scenarios. What matters is how we should teach it to kids. I think Devlin's idea was that we should instead teach them about it in terms of scaling, thus introducing them to geometry and continuity while at the same time getting the happy coincidence that it works like repeated addition on numbers.

I'm not sure if this would work -- we'd have to ask the education experts. It would be a cool idea, though. The real problem is that multiplication can mean a lot of different things that happen to work the same way. In the group/matrix setting, it's a way of thinking about composition of functions. If we choose to think of rings as coming from abelian group endomorphisms, then the same applies there, but if we don't, then it's this vague axiomatic thing that's just characterized by associativity and distributivity.

As an argument against teaching multiplication as repeated addition, think about exponentiation. I was taught this as repeated multiplication for the longest time, and I unsurprisingly grew okay with it. For fractional exponents, I would just think "the square root is the power of 1/2 because you need to multiply two of them to get the original number back," and so on. Irrational exponents get recovered in the limit. When I started learning complex numbers, I got completely lost -- I could manipulate the expressions fine, but in order to truly understand how complex exponents worked, I had to divest myself of the idea that this somehow came from multiplication. It's entirely possible that the same disconnect between multiplication and addition underlies others' problems in learning math.
 
  • #25
Jarle said:
What are you talking about? "Mystical entity called multiplication"? No one are talking about any mystical entities here. Not the "same logical status" as addition? Define logical status.

I guess I was unconsciously echoing Russell's comment about the cardinal numbers in the Introduction to Mathematical Philosophy. I agree it was a rash choice of words. But it was partly prompted by the fact that while quite a few people in this thread are asserting - and I emphasise the word 'asserting', as opposed to offering a definition or a reasoned argument - that multiplication is a genuine, distinct operation in its own right, noboby has actually said anything to defend or justify the position.

'The same logical status' here refers to the fact that multiplication and addition are both operations carried out upon a set of numbers in order to arrive at a total.
 
  • #26
micromass said:
That's a view that will ultimately fail in modern mathematics. Historically, you are correct. But mathematics is far beyond that phase.

I challenge you to reduce i.i=-1 to an addition operation.
Or how do you reduce the scalar multiplication of vectors to an addition operation?
Or how do you reduce the multiplication of group rings to an addition operation?

Your point-of-view only works for the integers or things that ressemble the integers. But ultimately it breaks down.

Not to say that it isn't pedagogical! If you want to calculate the volume of a cube, then thinking of repeated addition really confuses most students. And if you want to explain 0.23*0.4 to students, then I really suggest not talking about repeated addition...

I grant everything you say, Micromass. I set out to argue the case that the concept of 'multiplication' in ordinary arithmetic is not merely logically dispensible, it really does not exist, and I accept that I did not make my premises clear enough. And the occasional references to what we would or would not say to students have perhaps not been helpful, since these are practical considerations and have no bearing on the purely logical aspects of the argument.

Will someone go some way towards meeting my arguments by offering a rigorous definition of 'multiplication'? I have tried, and I cannot come up with one. But I don't have the mathematical training that some of you people have, so I expect you to do better.

Consider the arithmetic problem I posed in my earlier post, this time as a problem in mental arithmetic. If you asked an ordinary person "what is 12 x 36?", very likely they wouldn't be able to answer instantly; they would need a few moments to work it through in their heads. The process might go something like this:

"Right, 12 x 36... well, 10 x 36 = 360 (because 12 contains 10, and to multiply by 10, you just add 0 - learned that by rote in primary school); so it's the same as 360 + 2 x 36. Two threes are six (rote learning again), but these threes are really thirties, so add a zero (multiplication by 10 again), which gives 60, so we now have 360 + 60 + 2 x 6. Well, 2 x 6 =12 (rote learning again); so 360 + 60 + 12 = 432".

Obviously the articulation will vary between any two people, but notice that multiplication here reduces to three elements:

(1) values extracted from data sets (the 'times tables') which were learned by rote in primary school;

(2) the operation of addition; and

(3) the application of a formal rule which is not itself a method of calculation, but which is known to deliver correct results: 'to multiply a number by 10, just add a zero'. (Another well-known example: 'to multiply a two-digit number by 11...').

Though this is another purely practical example which does not constitute a logical proof, I adduce it to illustrate the point that in mathematics, very often what we think we are doing, and what we really are doing, are two very different things.
 
  • #27
Alan1000 said:
I grant everything you say, Micromass. I set out to argue the case that the concept of 'multiplication' in ordinary arithmetic is not merely logically dispensible, it really does not exist, and I accept that I did not make my premises clear enough. And the occasional references to what we would or would not say to students have perhaps not been helpful, since these are practical considerations and have no bearing on the purely logical aspects of the argument.

Will someone go some way towards meeting my arguments by offering a rigorous definition of 'multiplication'? I have tried, and I cannot come up with one. But I don't have the mathematical training that some of you people have, so I expect you to do better.

Consider the arithmetic problem I posed in my earlier post, this time as a problem in mental arithmetic. If you asked an ordinary person "what is 12 x 36?", very likely they wouldn't be able to answer instantly; they would need a few moments to work it through in their heads. The process might go something like this:

"Right, 12 x 36... well, 10 x 36 = 360 (because 12 contains 10, and to multiply by 10, you just add 0 - learned that by rote in primary school); so it's the same as 360 + 2 x 36. Two threes are six (rote learning again), but these threes are really thirties, so add a zero (multiplication by 10 again), which gives 60, so we now have 360 + 60 + 2 x 6. Well, 2 x 6 =12 (rote learning again); so 360 + 60 + 12 = 432".

Obviously the articulation will vary between any two people, but notice that multiplication here reduces to three elements:

(1) values extracted from data sets (the 'times tables') which were learned by rote in primary school;

(2) the operation of addition; and

(3) the application of a formal rule which is not itself a method of calculation, but which is known to deliver correct results: 'to multiply a number by 10, just add a zero'. (Another well-known example: 'to multiply a two-digit number by 11...').

Though this is another purely practical example which does not constitute a logical proof, I adduce it to illustrate the point that in mathematics, very often what we think we are doing, and what we really are doing, are two very different things.

I'm not sure what you mean when you say that "multiplication...does not exist". Now, we could get into some silly (in my opinion) discussion about whether Math exists outside the human mind or not, but let's not because in either case, multiplication does, in fact, exist.

At its heart, multiplication is a function. Say you have a set, F, of mathematical objects (it could be matrices, reals, integers, integers mod n, etc) then define the function M:(FXF) -> F by rule f(a,b)=c.

Now, we are not given an explicit formulation for the multiplication rule, but we are given some axioms that it must satisfy. Now, all of this is how multiplication is defined in rings and fields, and reals are a Field. So, ignoring the rest of the algebra stuff, just think of multiplication as a function from RxR to R. In this sense, it most certainly exists.

In a way that can be made mathematicly precise, the reals are an extension of the integers. Thus, the multiplcation that exists in R is the same multiplication that exists in Z, so it most certainly exists in ordinary arithmetic. The only "problem" is that in Z multiplication is THE SAME AS repeated addition in Z. Note that I said THE SAME AS, not "defined as". So, kids are taught that multiplication is just some extension to addition, not a logically distinct operation.

My argument is that this is a very sensible thing to do. Others disagree with me, which is fine; it wasn't until I took my first Abstract Algebra course that I really picked up on this.
but this is irrelevant to your post.
 
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  • #28
"At its heart, multiplication is a function. Say you have a set, F, of mathematical objects (it could be matrices, reals, integers, integers mod n, etc) then define the function M:(FXF) -> F by rule f(a,b)=c.

Now, we are not given an explicit formulation for the multiplication rule..."

Thanks for your reply, Robert.

I take your point about the use of functions, but when you say 'we are not given an explicit formulation for the multiplication rule', that's very much the part that bothers me! The part where knowledge and understanding give out, something seems to work, we know not why, we know not how, but it seems to give the results we want, so we continue to do it. That is blind faith and, if one step in a chain of deductive reasoning consists of blind faith rather than well-tested mathematical logic, surely that risks compromising the whole chain?

As to the relationship between our mathematics and the universe we observe, I like some of Richard Feynman's comments: 'everything we think we know is just an approximation'; 'my students don't understand it, because I don't understand it. Nobody does'.

Whenever we try to distil the mathematics of the cosmos, we invariably get it wrong; although our approximations are improving. Ptolemy's geocentric theory was a beautiful piece of mathematics, an intellectual tour de force in classical geometry, but nobody takes him seriously today. Nor can Newton's theory of gravity be seriously regarded as "the" way in which the universe works; there are too many important patterns it can neither predict nor explain. Relativity theory does somewhat better, if you can forget that it predicts the universe should consist of an isotropic soup of particles...

Anyway, thank you everybody for your comments; it's been very stimulating! I will probably bow out of this thread now, because I don't think I have anything more to contribute.
 
  • #29
I see that people such as micromass and others are getting at the point that multiplication by definition is applied to more than just real numbers, and in thinking about it I think that he (they) is (are) right.

For higher structures like matrices, quaternions, high level groups the definition kind of breaks down and means different things and in this context I agree that multiplication is not as trivial or easy to visualize as a general operation.

Something for you geometers out there if you're listening? In terms of things dealing with i I was thinking maybe you could use the Guddermanian to say go from tan to hanh, sin to sinh and cos to cosh and use appropriate inverses to work with quantities involving i without resorting to complex arithmetic. Any thoughts?
 
  • #30
Alan1000 said:
I take your point about the use of functions, but when you say 'we are not given an explicit formulation for the multiplication rule', that's very much the part that bothers me! The part where knowledge and understanding give out, something seems to work, we know not why, we know not how, but it seems to give the results we want, so we continue to do it. That is blind faith and, if one step in a chain of deductive reasoning consists of blind faith rather than well-tested mathematical logic, surely that risks compromising the whole chain?

No, you're missing the point. I am saying that multiplication is the name commonly given to an operation on a set such that this operation satisfies several axioms. The exact definition changes depending on what set you are using. I'm not saying that it is some "thing" out in the mathematical ethos that, for reasons unknown, "works".

So, multiplication does, in fact exist.
 
  • #31
Sorry if this has been said, already but I think we need to consider units.

If multiplication was repeated addition then:
If a and b had the units of 'm'.

a * b = a + kc

Would not tally dimensionally as 'kc' would have the same units as a, which is meters. Whereas (a*b) clearly has the unit of meter squared.
Therefore, multiplication cannot (IMO) be repeated addition.
 
  • #32
micromass said:
I challenge you to reduce i.i=-1 to an addition operation.
Easy: pi/2+pi/2=pi.
 
  • #33
As a contribution to the side digression about teaching multiplication.

"Go forth and multiply"

I think most youngsters have a pretty good idea what multiplication is without a formal definition and manage pretty well on the idea that

"To multiply is to make more"

Then they come multiply decimals, fractions and other animals that were not in the Ark and they falter.

The argument in Hurkyl,s post#4 looks impressive but isn't it rather circular?
Circular in that the properties of multiplication are implicitly assumed in the satement 1b = b?
 
  • #34
Harking back to the original challenge: i.i=-1 is a definition, not a deduction; or to be more pedantically exact, i=sqrt(-1) is the definition which underpins it, and this means that the whole proposition is true "by definition", not by deduction. It is not at all the outcome of a multiplicative procedure, it just happens to have that form. If mathematical propositions were banknotes, we would say that i.i=-1 is a very good forgery!

To recast it as an addition, I would simply say, sum i with zero i times, in accordance with the axioms of arithmetic.
 
  • #35
Alan1000 said:
Harking back to the original challenge: i.i=-1 is a definition, not a deduction; or to be more pedantically exact, i=sqrt(-1) is the definition which underpins it, and this means that the whole proposition is true "by definition", not by deduction. It is not at all the outcome of a multiplicative procedure, it just happens to have that form. If mathematical propositions were banknotes, we would say that i.i=-1 is a very good forgery!

To recast it as an addition, I would simply say, sum i with zero i times, in accordance with the axioms of arithmetic.

That's exactly my point. It is true by definition, so it has nothing to do with addition. The same is true with e.e, it is just a definition which has nothing to do with addition. Or (1/2).(1/2) is just by definition equal to 1/4. The point is that multiplication is just an addition, which just happens to be equal to addition in some cases!
And if a teacher explains multiplication by repeated addition, then students will be very confused when they learn i.i=-1. If we however, define multiplication as just an operation that comes naturally, and which happens to coincide with addition sometimes, then i.i=-1 is not all that difficult.

When I was still very young, I thought exponentiation was just repeated multiplication. But because of that, I had real troubles with 2^0=1 or 2^(-1)=1/2. I couldn't understand why this should be true. But now I realize that it is just a definition to make things work nice. We could have other definitions, but then things would be far uglier... I think every student should be told this (not immediately though, but after a little while).
 
<h2>1. What is multiplication?</h2><p>Multiplication is a mathematical operation that involves combining two or more numbers to get a total or product. It is represented by the symbol "x" or "*".</p><h2>2. How is multiplication different from addition?</h2><p>Multiplication is different from addition because it involves repeated addition of the same number. For example, 3 x 4 means adding 3 four times (3 + 3 + 3 + 3) which gives us a product of 12. In contrast, addition involves combining two or more numbers to get a sum.</p><h2>3. What are the basic properties of multiplication?</h2><p>The basic properties of multiplication are commutativity, associativity, and distributivity. Commutativity means that the order of the numbers being multiplied does not affect the product (e.g. 2 x 3 = 3 x 2). Associativity means that the grouping of numbers being multiplied does not affect the product (e.g. 2 x 3 x 4 = 2 x (3 x 4)). Distributivity means that multiplication can be distributed over addition (e.g. 2 x (3 + 4) = (2 x 3) + (2 x 4)).</p><h2>4. How is multiplication used in everyday life?</h2><p>Multiplication is used in everyday life for various tasks such as calculating the total cost of multiple items, determining the area of a rectangle or square, and converting units of measurement. It is also used in more complex calculations in fields such as science, engineering, and finance.</p><h2>5. What are some strategies for learning multiplication?</h2><p>Some strategies for learning multiplication include using visual aids such as arrays or number lines, memorizing multiplication tables, practicing with flashcards, and using real-life examples to understand the concept. It is also helpful to break down larger multiplication problems into smaller, more manageable ones.</p>

1. What is multiplication?

Multiplication is a mathematical operation that involves combining two or more numbers to get a total or product. It is represented by the symbol "x" or "*".

2. How is multiplication different from addition?

Multiplication is different from addition because it involves repeated addition of the same number. For example, 3 x 4 means adding 3 four times (3 + 3 + 3 + 3) which gives us a product of 12. In contrast, addition involves combining two or more numbers to get a sum.

3. What are the basic properties of multiplication?

The basic properties of multiplication are commutativity, associativity, and distributivity. Commutativity means that the order of the numbers being multiplied does not affect the product (e.g. 2 x 3 = 3 x 2). Associativity means that the grouping of numbers being multiplied does not affect the product (e.g. 2 x 3 x 4 = 2 x (3 x 4)). Distributivity means that multiplication can be distributed over addition (e.g. 2 x (3 + 4) = (2 x 3) + (2 x 4)).

4. How is multiplication used in everyday life?

Multiplication is used in everyday life for various tasks such as calculating the total cost of multiple items, determining the area of a rectangle or square, and converting units of measurement. It is also used in more complex calculations in fields such as science, engineering, and finance.

5. What are some strategies for learning multiplication?

Some strategies for learning multiplication include using visual aids such as arrays or number lines, memorizing multiplication tables, practicing with flashcards, and using real-life examples to understand the concept. It is also helpful to break down larger multiplication problems into smaller, more manageable ones.

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