Is Multiplication Really Just Repeated Addition?

[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.

 

[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.