by polyphonicist on 3/1/20, 11:10 AM with 75 comments
by dang on 3/1/20, 9:06 PM
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.
(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: https://news.ycombinator.com/newsguidelines.html)
by yunruse on 3/1/20, 12:07 PM
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.
by soVeryTired on 3/1/20, 12:55 PM
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.
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).
Not sure the kids these days are down with the crystal maze though. More loss to them - Richard O'Brien was a national treasure.
by saagarjha on 3/1/20, 11:36 AM
by b0rsuk on 3/1/20, 12:56 PM
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).
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).
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.
by kidintech on 3/1/20, 12:26 PM
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.
by dwheeler on 3/1/20, 6:38 PM
If you want an absolutely rigorous proof, you can view this Metamath proof: http://us.metamath.org/mpeuni/mulge0.html ; 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").
by tromp on 3/1/20, 12:31 PM
-1*-1 =
-1*-1 + -1*1 + 1 =
-1*(-1 + 1) + 1 =
-1*0 + 1 =
1by edtechdev on 3/1/20, 12:29 PM
https://betterexplained.com/articles/rethinking-arithmetic-a...
by cousin_it on 3/1/20, 6:25 PM
by throwfermat on 3/1/20, 12:12 PM
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?
by wsxcde on 3/1/20, 1:56 PM
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.
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.