Is Multiplication Really Just Repeated Addition?

[GuitarStrings]

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.

 

[Landau]

I challenge you to reduce i.i=-1 to an addition operation.

Easy: pi/2+pi/2=pi.

 

[Studiot]

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?

 

[Alan1000]

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.

 

[micromass]

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).

 

[disregardthat]

Easy: pi/2+pi/2=pi.

That's like saying 2*3 is log(2)+log(3).

 

[Benjamin113]

I consider Multiplication to be grouping. it's true that it IS an extension of addition, but I don't like thinking of it that way.

Consider the basic formula for Work : W = Fd
what many physics students fail to understand is the concept behind the formula...that A force F is being applied to a distance of d meters (F for each d)

or, on a more basic level, consider 12 * 3
while some could interpret this as 12 + 12 + 12, it is also three 12's (that is, to put it less vaguely, that you could read the problem as "there is a group of 3, and each one is worth 12")

as for division, some fail to realize that division is a whole that is being "grouped separately"
consider 12 / 3
this is saying that a whole (12) is being broken into 3 parts, and each part is worth 4.


I feel that if this method were taught in elementary schools, kids would have an easier time conceptualizing what they are looking at and struggling to figure out.

 

[Landau]

That's like saying 2*3 is log(2)+log(3).

How can a true statement be like a false statement?

Besides, I don't think my answer was so bad. It recognises the fundamental relation between multiplication of complex numbers and scaling rotation, i.e. the algebra representation

\mathbb{c}\to M_2(\mathbb{R})
z=x+yi\mapsto \left(\begin{array}{cc}x & -y\\ y & x \end{array}\right).

So if we stick to the unit circle, there is not scaling and only rotation. And the composition ('product') of two rotations (in 2d) amounts to adding the angles.

 

[disregardthat]

How can a true statement be like a false statement?

Besides, I don't think my answer was so bad. It recognises the fundamental relation between multiplication of complex numbers and scaling rotation, i.e. the algebra representation

\mathbb{c}\to M_2(\mathbb{R})
z=x+yi\mapsto \left(\begin{array}{cc}x & -y\\ y & x \end{array}\right).

So if we stick to the unit circle, there is not scaling and only rotation. And the composition ('product') of two rotations (in 2d) amounts to adding the angles.

Sorry for the late reply. Yes, there is a fundamental relation, but it is not a reduction per se. The point is that adding exponents is a wholly different operation than multiplying, though each will yield the same result. I can as easily say that addition of real numbers is really just a special case of multiplication, since 2^a*2^b = 2^(a+b). Isn't it curious to conclude that I by this have reduced addition to multiplication of powers of 2? What an operation is is the calculatory process of it. So if we are using different rules, we are doing a different operation.

 

[brydustin]

And I'm sure that this was very helpful to the first year engineering student...

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​

 

[Hurkyl]

And I'm sure that this was very helpful to the first year engineering student...

It's hard to say, since the first year engineering student never responded.

Any comprehensive response is going to have to include an explanation of why, whatever multiplication "is", one can still do a lot by thinking in terms of repeated addition. If you have a better way of communicating that, then by all means share.

 

[apeiron]

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.

No one seems to have tackled this bit of the question, which seems more telling.

Multiplication does seem just like repeated addition. It shares the same freedom of construction. You can set off and get somewhere either with a series of steps, or one big step that is the equivalent. Neither operation has to deal with the destination until it arrives at it.

But with division, you have to start off "somewhere" and find the regularity within. You are at the larger destination and want to recover the smaller steps that could have got there. You can no longer construct the answer freely. Without prior information (knowledge of the times tables which could be used inversely) there is no choice but to grope for a result, hazard a guess and see if it works out as a construction-based answer.

So you have three simple operations based on freely constructive methods, and a fourth that is different in a fundamental way it seems.

Division does appear to depend on a further usually unstated assumption about a global symmetry of the number line. As can be seen from the story on normed division algebras.

I would be interested in how this issue is usually handled in the philosophy of maths (so not the definitional story, but the motivational one).

 

[Hurkyl]

If you think of division only in terms of how it relates to multiplication, then naturally division will seem like it's working backwards.

 

[apeiron]

If you think of division only in terms of how it relates to multiplication, then naturally division will seem like it's working backwards.

But I wasn't. So if you care to offer a more constructive reply...

 

[Hurkyl]

But I wasn't. So if you care to offer a more constructive reply...

Then could you explain what you mean by starting at the destination and working backwards to figure out how to get there?

 

[apeiron]

Then could you explain what you mean by starting at the destination and working backwards to figure out how to get there?

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?

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.

If I gave you an additive, subtractive or multiplicative question, you could say hang on and I'll use this operation to crank out the answer. The size of each step, and the total number of steps, is specified. So no problems.

But with division, even if the number of steps has been specified in the question, the size of them isn't. It is precisely what you have to discover somehow.

 

[Studiot]

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?

You will find the answer in Euclid, dear soul.

It is a very simple and elementary construction that used to be taught to 11 year olds.

 

[Hurkyl]

If I gave you an additive, subtractive or multiplicative question, you could say hang on and I'll use this operation to crank out the answer. The size of each step, and the total number of steps, is specified. So no problems.

But with division, even if the number of steps has been specified in the question, the size of them isn't. It is precisely what you have to discover somehow.

I'm thoroughly confused with this. The way real numbers are normally specified, there is a straightforward division algorithm. Just follow the steps until you're done (or have enough precision, as the case may be).

 

[Hurkyl]

I'm thoroughly confused with this. The way real numbers are normally specified, there is a straightforward division algorithm. Just follow the steps until you're done (or have enough precision, as the case may be).

I just want to add that there are even ways of representing numbers that are especially convenient for division. e.g. representing any positive real number x by the decimal expansion of log(x). (I'm not being frivolous with this -- I have really seen this used) Also division is rather simple in the the prime factorization representation of rational numbers.

 

[apeiron]

You will find the answer in Euclid, dear soul.
It is a very simple and elementary construction that used to be taught to 11 year olds.

I'm thoroughly confused with this. The way real numbers are normally specified, there is a straightforward division algorithm. Just follow the steps until you're done (or have enough precision, as the case may be).

: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.

 

[apeiron]

I just want to add that there are even ways of representing numbers that are especially convenient for division. e.g. representing any positive real number x by the decimal expansion of log(x). (I'm not being frivolous with this -- I have really seen this used) Also division is rather simple in the the prime factorization representation of rational numbers.

Again, the question was not about clever ways around a problem, but about the problem.

How can division be considered a species of addition? (When multiplication does seem to be)

More attention to the OP please and less condescension to my requests for an answer.

 

[Hurkyl]

: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.

Please try to be far more specific than you have been. I have to guess at the fine details of what you mean.


Long division is an exact operation on decimal numerals. Every computable operation on real numbers will have to deal with precision issues of some sort. Even addition.

When applied to decimals that represent rational numbers (because at some point a sequence of digits repeats forever), a slight modification allows long division to terminate after a finite number of steps.


Incidentally, when applied to rational numbers represented as a quotient of integers, the division algorithm and multiplication algorithms are pretty much identical.

 

[apeiron]

Please try to be far more specific than you have been. I have to guess at the fine details of what you mean.

Long division is an exact operation on decimal numerals. Every computable operation on real numbers will have to deal with precision issues of some sort. Even addition.

When applied to decimals that represent rational numbers (because at some point a sequence of digits repeats forever), a slight modification allows long division to terminate after a finite number of steps.

Incidentally, when applied to rational numbers represented as a quotient of integers, the division algorithm and multiplication algorithms are pretty much identical.

OK, forget I mentioned real numbers at one point as irrational numbers are another example of how the simplistic notion of construction or addition breaks down in practice.
Limits have to be introduced as a further action.

In fact forget the whole question because you clearly are not interested in actually addressing it, just talking around it forever.

 

[Hurkyl]

In fact forget the whole question because you clearly are not interested in actually addressing it, just talking around it forever.

Eh? There were two and a half pages addressing the original question. You brought up a new claim -- that division is somehow working backwards from a destination to a construction, but yet you are not thinking in terms of division being an inverse of multiplication, and that this is somehow a fundamental difference between division and other arithmetic operations, rather than just being one of many ways to view division.

If "addressing" your point means unquestioningly buying into your assertion, then yes, I am uninterested in "addressing" it.

 

[Studiot]

In fact forget the whole question because you clearly are not interested in actually addressing it, just talking around it forever.

I gave you an answer that is absolutely precise, but you chose to do exactly what you are accusing others of - you ignored it.

 

[apeiron]

I gave you an answer that is absolutely precise, but you chose to do exactly what you are accusing others of - you ignored it.

Eh? You said...

You will find the answer in Euclid, dear soul.

It is a very simple and elementary construction that used to be taught to 11 year olds.

I don't know about precise, but that's the feyest attempt at an insult I've seen in a long time.

 

[apeiron]

Eh? There were two and a half pages addressing the original question. You brought up a new claim -- that division is somehow working backwards from a destination to a construction, but yet you are not thinking in terms of division being an inverse of multiplication, and that this is somehow a fundamental difference between division and other arithmetic operations, rather than just being one of many ways to view division.

If "addressing" your point means unquestioningly buying into your assertion, then yes, I am uninterested in "addressing" it.

The OP said...

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.

So that was what I was throwing out as a question. Your answer is that division is simply inverse multiplication. But that does not deal with the OP comment that division is not repeated subtraction.

Again, if you have nothing useful to say on the matter, just leave it to someone else.

 

[Studiot]

I don't know about precise, but that's the feyest attempt at an insult I've seen in a long time.

No insults were intended so I'm sorry if you feel insulted.

However the fact remains that up to the late 1960s boys in their first year in an English grammar school would be taught the construction, from Euclid, that I was referring to.

In those days such a construction was used by engineers and draughtsmen and a version appeared on many boxwood scales of that time.

You should also remember this is the pure maths section of the forum. In pure maths we are allowed the luxury, as was Euclid, of perfect constructions. Remember also there are very specific mathmatical rules governing perfect constructions.
But I assume you already know all this?

There is a further twist to your question. You have not specified what x is, but it cannot be any random real number, it can only be an integer. This makes the construction basic.

 

[apeiron]

However the fact remains that up to the late 1960s boys in their first year in an English grammar school would be taught the construction, from Euclid, that I was referring to.

In those days such a construction was used by engineers and draughtsmen and a version appeared on many boxwood scales of that time.

I seem to have lost my boxwood scales and skipped 1960s grammars, so you might actually have to state what it is you are referring to here. Does it have a name? Can you provide a link? Or would that be too shockingly precise?

 

[Studiot]

Construction to divide a line into n equal parts.

Draw the line (in your case equal to the random real number to be divided or mark any line at a random point if you like to create a random real number)

Draw an auxiliary line at a convenient angle (30^{o} is generally convenient) and crossing the original line at one end.

With compasses set to any convenient length mark off n steps along the auxiliary line, commencing at the intersection with the original line.

Join the mark representing the last step to the other end or marked point of the original line.
Through each mark along the auxiliary line draw a line parallel to this third line to intersect the original line.

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