We've banned the submitter, the site, and dozens of other accounts, including susam, for using a ring of accounts to manipulate HN. Such abuse is not tolerated.<p>All: if you notice fishy things (as a user did in this case), please let us know at hn@ycombinator.com. We catch a lot of abuse between software and moderation, but unfortunately not all. Vigilant users make a huge difference, and protecting the integrity of HN is a community effort.<p>(Please don't post insinuations about abuse in the threads, though, since most suspicions don't end up leading to real evidence. Send them to hn@ycombinator.com. This is in the site guidelines: <a href="https://news.ycombinator.com/newsguidelines.html" rel="nofollow">https://news.ycombinator.com/newsguidelines.html</a>)
An 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<p>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.<p>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.
I've always thought the best way to explain this was by analogy with the '90s TV show "The Crystal Maze" [0].<p>Contestents are put in a dome filled with gold and silver tickets being blown around by fans. For every gold ticket they collect, they get a point. For every silver ticket, they lose a point. If they collect enough points, they win a prize.<p>Sorting through the team's collection of tickets and throwing away a silver ticket (minus a -1) is just as good as adding another gold ticket (+1).<p>Not sure the kids these days are down with the crystal maze though. More loss to them - Richard O'Brien was a national treasure.<p>[0] <a href="https://en.wikipedia.org/wiki/The_Crystal_Maze" rel="nofollow">https://en.wikipedia.org/wiki/The_Crystal_Maze</a>
An 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).
I've seen a better explaination in this Mathologer video. In a bizarre twist it is now private (?!). Maybe it will work for you. But I suspect it was a takedown notice because he used a short clip from a movie famous among teachers.
<a href="https://www.youtube.com/watch?v=ij-EK-MZv2Q" rel="nofollow">https://www.youtube.com/watch?v=ij-EK-MZv2Q</a><p>The first number represents the amount of something. If it's negative, you have a debt.
The second number represents either a gain (if it's positive) or a loss (if it's negative).<p>From that point you can explain it to yourself using plain english. So, -4 * (-3) can be understood as "Lose a debt of 4, three times". If you have -4 * 3, you could be said to "gain a debt of 4 three times". 4 * -3 means (Lose 4 three times).<p>In the video Mathologer criticized exactly the kind of proofs like in this video. Just saying it's intuitive doesn't make it so. Fundamental things shouldn't be proven using a number of laws. They should be understood on the intuitive level and a proof is just to double check.
I strongly dislike these kinds of articles/posts due to one reason:<p>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.<p>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.
This is a decent intuitive explanation.<p>If you want an absolutely rigorous proof, you can view this Metamath proof: <a href="http://us.metamath.org/mpeuni/mulge0.html" rel="nofollow">http://us.metamath.org/mpeuni/mulge0.html</a> ; this has more far more steps, but is totally rigorous. It particular, its only axioms are those of classical logic and ZFC set theory (not even numbers are presumed, the system first proves "numbers exist and have these properties").
This site had more intuitive explanations for things like this, including imaginary numbers, calculus, etc<p><a href="https://betterexplained.com/articles/rethinking-arithmetic-a-visual-guide/" rel="nofollow">https://betterexplained.com/articles/rethinking-arithmetic-a...</a>
There is a discussion in this post's comments section⁽¹⁾ that this works for fields and rings too.<p>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.<p>Are the set of integers a field? I guess not because the multiplicative inverse of 2 is not present in this set.<p>Is the set of integers a ring? I think, yes.<p>For prime p, is Z_p = {0, 1, ..., p - 1} a field? I think, yes.<p>Are there any non-numeric rings where product of negatives is positive?<p>⁽¹⁾ <a href="https://susam.in/blog/product-of-negatives/comments/" rel="nofollow">https://susam.in/blog/product-of-negatives/comments/</a>
This is not a proof of why the product of negative numbers is positive. The reason why the product of negative numbers is positive is that we define multiplication to be that way.<p>Also, this post conflates the unary negation operator with negative numbers. The two are not the same. In so far as this post constitutes a proof (which IMO it does not), it is a proof about the behavior of the negation operator.<p>A good question to ask is why we made this specific choice of definition. Why should multiplication be defined such that -2*-3 = 6? This is a question that the post does shed some light on. If we'd chosen some other definition of multiplication, a lot of the "intuitive" properties of multiplication that hold over the natural numbers (such as the distributivity of multiplication over addition and subtraction) would no longer be true over the integers.