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So, what is multiplication?by JyN
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#37
Mar711, 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. 


#38
Mar711, 11:36 AM

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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. 


#39
Mar2711, 04:16 AM

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#40
Mar2811, 10:20 AM

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



#41
Mar2811, 04:01 PM

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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. 


#42
Mar2811, 05:30 PM

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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 constructionbased 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). 


#43
Mar2811, 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.



#44
Mar2811, 05:59 PM

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#45
Mar2811, 06:05 PM

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#46
Mar2811, 06:24 PM

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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. 


#47
Mar2811, 06:32 PM

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


#48
Mar2811, 06:34 PM

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#49
Mar2811, 06:51 PM

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#50
Mar2811, 07:01 PM

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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 cutoff on the number of decimal places as a further pragmatic choice. 


#51
Mar2811, 07:18 PM

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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. 


#52
Mar2811, 07:19 PM

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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. 


#53
Mar2811, 07:36 PM

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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. 


#54
Mar2811, 07:54 PM

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


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