summary: -1*-1 = -1*-1 + -1*1 + 1 = -1*(-1 + 1) + 1 = -1*0 + 1 = 1
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Product of Negatives (2010)
11–20 of 79 posts
Re: Product of Negatives (2010)
#12There 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…
There's a subtle point to keep in mind when generalizing to rings/fields. The concept of 'positive' and 'negative' are defined in terms of an order relation,
e.g., 'positive' means >0 and 'negative' means For example, the integers mod n is a ring, so (-a) * (-b) = a * b holds, but it doesn't make sense to call a number mod n positive or negative, since -a mod n effectively means n - a mod n.
(posted an earlier version of this comment on susam.in.)
Re: Product of Negatives (2010)
#13summary: -1*-1 = -1*-1 + -1*1 + 1 = -1*(-1 + 1) + 1 = -1*0 + 1 = 1
The explanation that appeals to me:
If number N is an arrow on the number line from 0 to N, then multiplying N by -1 flips the arrow with the result -N. Multiplying by -1 again would be another flip, taking you back to N. So a flip followed by a flip is same as no change (i.e. multiplicative identity 1).
Re: Product of Negatives (2010)
#14I 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 as…
Like mentioned in another comment on this thread, the assumptions are well known field axioms. They form a good starting point.
And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.
Re: Product of Negatives (2010)
#15summary: -1*-1 = -1*-1 + -1*1 + 1 = -1*(-1 + 1) + 1 = -1*0 + 1 = 1
This would be clearer if the -1 was in parentheses (I can't do it because I'm on mobile). I think you are missing a plus sign on line two.
Re: Product of Negatives (2010)
#16I 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 as…
> or tell the reader your assumptions
It is right there in the first section of the article.
"In this discussion, we assume that we already know some basic properties of arithmetic operations such as the distributive property of multiplication over subtraction, existence of the additive inverse of real numbers, etc."
Re: Product of Negatives (2010)
#17I 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 as…
Like mentioned in another comment on this thread, the assumptions are well known field axioms. They form a good starting point. And why start with Peano axioms? They seem like a bad starting point because it would take pages upon pages of proof and it won't easily extend to other algebraic structures like rings and fields.
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Re: Product of Negatives (2010)
#18I've always thought the best way to explain this was by analogy with the '90s TV show "The Crystal Maze" [0].
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.
Re: Product of Negatives (2010)
#19I'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.
https://www.youtube.com/watch?v=ij-EK-MZv2Q
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.
Re: Product of Negatives (2010)
#20I 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 as…
> or tell the reader your assumptions It is right there in the first section of the article. "In this discussion, we assume that we already know some basic properties of arithmetic operations such as the distributive property of multiplication over subtraction, existence of the additive inverse of real numbers, etc."
Okay, I either didn't see it or that was added after my comment. Either way, that is exactly what I wanted.