Product of Negatives (2010)
susam.in
Product of Negatives (2010)
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Re: Product of Negatives (2010)
#2An alternative, "common sense proof" would be that you're undoing the taking away of things, meaning you have more than you started (i.e. a positive result).
Re: Product of Negatives (2010)
#3An important thing about numbers in general is that whenever somebody says “complex/negative numbers don’t actually exist”, they are somewhat right, in a sense. What exists is magnitude and phase
Does that mean we should abandon them? Absolutely not. Encoding phase (or in a much more common subset, parity) is so absolutely useful it’s no wonder we bake 90° intervals (-, i) into our notations: they can be intuitively dealt with. It’s still somewhat easy to skip over the property, however; as a student at least I seem to need to backtrack over signs at least once an hour when working with anything rigorous enough. I wonder if 2-tuple notation, eg (+, 23) or (-i, x²), would be more intuitive by making parity/phase explicit rather than implicit.
Complex numbers are a little more nuanced, but no less useful. I imagine you could develop an alternative notation to make things more intuitive, but thankfully it’s generally taken for given nowadays that they’re intrinsic to how we’ve explored nature.
Re: Product of Negatives (2010)
#4There is a discussion in this post's comments section⁽¹⁾ that this works for fields and rings too.
I know there are precise definitions for fields and rings but can someone here give me some good examples of fields and rings? Being a non-mathematician, I find it easy to manipulate examples than manipulate definitions.
Are the set of integers a field? I guess not because the multiplicative inverse of 2 is not present in this set.
Is the set of integers a ring? I think, yes.
For prime p, is Z_p = {0, 1, ..., p - 1} a field? I think, yes.
Are there any non-numeric rings where product of negatives is positive?
Re: Product of Negatives (2010)
#5An alternative, "common sense proof" would be that you're undoing the taking away of things, meaning you have more than you started (i.e. a positive result).
Here is a nice one I read sometime back.
Create a video of your friend walking 3 metres. Now play the video 4 times. Your friend walks 12 metres in the video. Play the video in reverse 4 times. Your friend walks 12 metres backwards in the video.
Create another video of your friend walking backwards 3 metres. Now play the video 4 times. Your friend walks 12 metres backwards in the video. Play the video in reverse 4 times. Your friend walks 12 metres forwards in the video.
Re: Product of Negatives (2010)
#6An important thing about numbers in general is that whenever somebody says “complex/negative numbers don’t actually exist”, they are somewhat right, in a sense. What exists is magnitude and phase Does that mean we should abandon them? Absolutely not. Encoding phase (or in a much more common subset, parity) is so absolutely useful it’s no wonder we bake 90° intervals (-, i) into our notations: they can be intuitively…
Complex numbers, the way they're used in most cases, is a tuple notation. They're a handy way of keeping your chocolate separate from your peanut butter, so to speak, as that little "times i" makes it difficult to accidentally get things mixed up. And that's the way I always explained it to my students: there are imaginary numbers in the original sense of fake roots that will go away if you ignore them long enough, and there are imaginary numbers in the sense that it makes some kinds of calculations easier to keep straight. I've never been convinced that they are the same thing. One is an annoying but temporary consequence of arithmetic, while the other is just a convention, when all's said and done.
Re: Product of Negatives (2010)
#7I strongly dislike these kinds of articles/posts due to one reason:
if you're going to prove such a fundamental thing, can you please provide the axioms that we start from? I.e. "we know" that a - a = 0, multiplication is distributive, and a x - b = - a x b. These seem arbitrary properties and "equally" fundamental to -a x -b = ab. Either start from peano and prove everything along the way, or tell the reader your assumptions. Don't just divine things along the way.
EDIT: Assumptions are in the third paragraph of the post. I highly doubt they were there when I wrote the comment. Either way, my concern has been resolved.
Re: Product of Negatives (2010)
#8This site had more intuitive explanations for things like this, including imaginary numbers, calculus, etc
https://betterexplained.com/articles/rethinking-arithmetic-a...
Re: Product of Negatives (2010)
#9There is a discussion in this post's comments section⁽¹⁾ that this works for fields and rings too. I know there are precise definitions for fields and rings but can someone here give me some good examples of fields and rings? Being a non-mathematician, I find it easy to manipulate examples than manipulate definitions. Are the set of integers a field? I guess not because the multiplicative inverse of 2 is not present…
The typical example of a field is the collection of rational numbers. These are still numeric, so they might not seem too exotic. Similarly, the typical example of a ring is the collection of single-variable polynomials with ringed (integer) coefficients. In both of these examples, the product of negatives is positive.
A more interesting example: If R is a ring, then R-valued square matrices of fixed size also give a ring, using addition and multiplication of matrices. Matrices aren't just positive, negative, or zero; they can have a mix of positive and negative entries. In these "matrix rings", the product of negatives isn't exactly positive, although I bet that somebody can make this more rigorous. (Come to think of it, this applies to the rings of polynomials, too.)
Re: Product of Negatives (2010)
#10summary:
-1*-1 =
-1*-1 + -1*1 + 1 =
-1*(-1 + 1) + 1 =
-1*0 + 1 =
1