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I'm doing a bit of research for a tech presentation that touches on the subject of mathematical duality. (To be clear, my presentation is not on mathematics or duality, but mentions duality in passing.)

My question for the math folks is, is it correct to say multiplication is the mathematical dual of division? Or am I completely misunderstanding mathematical duality?

Wikipedia says,

the divides and multiple-of relations on the integers

is considered an example of an "order reversing duality".

So in layman's terms, could a person characterize multiplication/division as an example of mathematical duality?

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they are just words and "inverse" is better, more accurate. –  Patrick I-Z Jan 3 '11 at 20:31
(1) This is not an appropriate forum for this question. Try math stackexchange instead. (2) I really don't care to be addressed as "math geek", no matter how light-heartedly it's meant. This is a professional forum. If you were consulting a medical doctor online on a professional matter, would you address him or her as "medicine geek"? No? I didn't think so. –  Todd Trimble Jan 3 '11 at 20:32
No offense meant. I'm a software geek, so I thought I'd be among similar kin. :-) –  Judah Himango Jan 3 '11 at 20:42
I know you didn't mean to offend, but the point is that we don't know one another, and the presumption is a off-putting. No condescension intended, but please remember this is a professional forum. –  Todd Trimble Jan 3 '11 at 20:57
For what it's worth, when folks come into the professional forum of StackOverflow, we're addressed as geeks and it's a compliment. I had a mental picture in my head, perhaps planted there by XKCD, that math folks were similar kin, so I thought nothing of it. I now see this community is very much unlike what I had pictured, so I apologize for the apparent insult. –  Judah Himango Jan 3 '11 at 21:22
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closed as off topic by Todd Trimble, Noah Snyder, Mariano Suárez-Alvarez, José Figueroa-O'Farrill, Deane Yang Jan 3 '11 at 20:42

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2 Answers

up vote 5 down vote accepted

You misunderstand duality. A better example is: planes and lines through the origin (in 3-dimensional space). To each plane you can assign the line perpendicular to it, and vice versa. This is more just a pairing of two kinds of objects, it has interesting properties. For example, a collection of lines are contained in one plane if and only if the corresponding perpendicular planes contain one line. Or, the angle between two lines is equal to the angle between the corresponding perpendicular planes. This is where things start to become interesting: certain statements about lines through the origin (in 3-dimensional space) can be reformulated as statements about planes through the origin (in 3-dimensional space). In mathematics this is very useful. It happens, for example, that switching back and forth between the two "worlds" in a duality gradually enhances our understanding in both worlds.

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Thanks for the helpful (and non-condescending) response. –  Judah Himango Jan 3 '11 at 20:43
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I think that non-technically, duality is pretty unrestricted (if the dual of A is B, then the dual of B is A). But if you use it technically, it is important to say exactly what you mean by it.

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sorry this wasn't supposed to be condescending –  maxdev Jan 3 '11 at 20:45
This wasn't condescending. Some of the comments to my question, however, were. –  Judah Himango Jan 3 '11 at 20:51
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