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So, what is multiplication?

by JyN
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Benjamin113
#37
Mar7-11, 11:19 AM
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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
#38
Mar7-11, 11:36 AM
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Quote Quote by Jarle View Post
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

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

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
#39
Mar27-11, 04:16 AM
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Quote Quote by Landau View Post
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

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

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
#40
Mar28-11, 10:20 AM
P: 209
And I'm sure that this was very helpful to the first year engineering student....

Quote Quote by Hurkyl View Post
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
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Mar28-11, 04:01 PM
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Quote Quote by brydustin View Post
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
#42
Mar28-11, 05:30 PM
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Quote Quote by JyN View Post
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
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Mar28-11, 05:56 PM
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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
#44
Mar28-11, 05:59 PM
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Quote Quote by Hurkyl View Post
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
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Mar28-11, 06:05 PM
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Quote Quote by apeiron View Post
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
#46
Mar28-11, 06:24 PM
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Quote Quote by Hurkyl View Post
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
#47
Mar28-11, 06:32 PM
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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
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Mar28-11, 06:34 PM
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Quote Quote by apeiron View Post
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
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Mar28-11, 06:51 PM
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Quote Quote by Hurkyl View Post
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
#50
Mar28-11, 07:01 PM
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Quote Quote by Studiot View Post
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.
Quote Quote by Hurkyl View Post
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
#51
Mar28-11, 07:18 PM
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Quote Quote by Hurkyl View Post
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
#52
Mar28-11, 07:19 PM
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Quote Quote by apeiron View Post
: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
#53
Mar28-11, 07:36 PM
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Quote Quote by Hurkyl View Post
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
#54
Mar28-11, 07:54 PM
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Quote Quote by apeiron View Post
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.


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