/sci/ - Science & Math

[Anonymous No.9895863]

Brainlet here, I just started doing algebra after not having touched since rote learning it high school, and I had this thought:

Where does the axiom "the product of two negatives is a positive" come from?

What is it based off of?

Where do we see it in nature?

What is the correct perception of this axiom to have to truly conceptualize it?

The more I think about, the more I realize I have no idea why we do this. Why can't I grip this abstract idea, it almost seems arbitrary like there's no answer -- like we use this axiom simply because everything else is untrue.

Please help me understand this

[Anonymous No.9895867]

what is the opposite of the opposite of cold

[Anonymous No.9895888]

lets say you're on the racetrack. forwards is the positive direction and backwards is negative direction.
i lost the motivation to type out the rest but you can figure it out from here

[Anonymous No.9895890]

>>9895863
>https://www.youtube.com/watch?v=2k_jS1zVLWw
The post

[Anonymous No.9895892]

>>9895867
So you're saying that the axiom comes from the use of double negatives in languages, and that double negatives don't come from the axiom.

I.e. the use of double negatives in languages(in this case English) influenced our use of this axiom in the language of mathematics?

Spoken Language doesn't seem to have have any rhyme or reason to it. Why doesn't the rule apply to the phrase "two wrongs don't make a right?" What if we replaced "cold" with something that doesn't have a clear "opposite," like a triangle or infinity, what are the "opposites" of those?

What about languages where double negation is a negation?

You have basically told me nothing. I don't blame you though, because that's how I was taught to see it by my teachers.

[Anonymous No.9895893]

>>9895888

kek

[Anonymous No.9895902]

>>9895892
no. the double negative that's unique to english is something like "I ain't got no apples" which means "I dont have apples." (double negative means the same as a single negative).

The opposite of an opposite is the same as the original. This is the same in all languages.

[Anonymous No.9895909]

>>9895863
I guess it's easy enough to come up with an intuitive reason for why it should be true, but it would be interesting to know if there's some kind of proof for it.

It seems like we should be able to prove it by the fact that -1*x = -x.
That tells us that multiplying a negative by a negative, let's say a*b, gives us -1*-1*|a|*|b|.
Then all we have to do is prove that -1*-1 = 1. We know this is --1 from our fact above.
I don't know how to prove that --1=1, though it certainly seems obvious. (And no, I don't need to see a geometric "proof" or a real-world analogy)

[Anonymous No.9895914]

>>9895863
1 -1=0
multiply both sides by -1
-1(1 -1) = (-1)0
distribute
(-1)1 + (-1)(-1) = 0
since by (-1)1 = -1, we have
-1 + (-1)(-1) = 0
add 1 to both sides
(-1)(-1) = 1

Now generalize. Let a and b be positive.
(-a)(-b) = (-1)(-1)ab = ab
since ab is the product of two positive numbers, it is positive.

[Anonymous No.9895921]

>>9895902
>"I ain't got no apples"
You're incorrect. In proper spoken english "I ain't got no apples" means you DO have apples. What you're reffering to is Ebonics, a dialect of English. In African-american vernacular sure a double negative "means the same as a single negative" but not in the English.

Also, you'be just proven my point by using a sub-language of English, post-hoc, to try and prove your point--which was that using Human spoken language as a way to conceputalize the axiom is flawed because they're not all the same, steeped in our intuition and math isn't, math is abstract and not always based in reality.

[Anonymous No.9895922]

>>9895863
it's derived from the field axioms

[Anonymous No.9895925]

>>9895914
looks like a /thread to me

[Anonymous No.9895928]

>>9895914
I should add this isn't really the REASON that the product of two negatives is positive. That, I think, really comes from human intuition and practical usefulness. When it comes to such simple theorems, the proofs are post hoc justifications for what people have known without proof for hundreds of years. I mean, if someone came up with an axiomatic formulation of arithmetic that couldn't prove this result, or worse, proved it false, it just wouldn't be used, at least not practically. But this kind of proof is a good demonstration of the consistency of mathematics.

[Anonymous No.9895938]

>>9895914
You're basically telling me that the egg was because of the chicken. We're in a loop here

So you're using the distributive property as proof. I'm aware of this property. What if the reason that we chose "the product of two negatives to be a positive" was so that the distributive property didn't break?

What if you entered a world where (-1)(-1)=-1 like in Human spoken Language analogies you guys keep giving me. Then -1(1 -1) = -2 if we distribute but if we removed the property and simply did "order of operations" without distributing we would get 0, again

[Anonymous No.9895955]

>>9895938
So if you redefine the multiplication operator, you can create an inconsistent system that also doesn't fit with the real world. That's easy to do, but completely useless.
Your concern was already addressed in this post as well >>9895928

[Fuck everything t. OP No.9895957]

>>9895938
actually the "chicken or the egg" analogy might have been bad here since that can be answqered with evolution and natural selection. Or maybe it was perfect and answers my question.

Fuck this thread. I'm tired of thinking about this. I'll just continue to believe it's fact because of intuition and shit, just like everyone else.

This has been an excercise in procrastination. Instead of studying I got stuck thinking about this for hours. This wasn't even a noble distraction, I might as well have edged to trap porn for that amount of time, in the end I would have had the same unsatafied and shameful feeling. Thanks for nothing I'm out of here and Ill probably never question anything ever again.

[Anonymous No.9895988]

>>9895863
Multiplication in the complex plane is rotation around the origin so when you multiply -1 with itself you rotate 180' and another 180' ending up back where you started.

[Anonymous No.9896266]

>>9895888

>lets say you're on the racetrack. forwards is the positive direction and backwards is negative direction.
and then the garden direction is the path to the flower counsil. let's buy that road, it's neither forwards or backward. now we're counting, 1 2 3 4 5 ... 99 ... infinity -infinity -infinity+1 ...-2 -1 STOP. love and beg for love, we're here. almighty flower counsil, we are begging for the answer. FC says: "just have a rototom that you can distract yourself with".

[Anonymous No.9896273]

>>9896266
That was my second guess

[Anonymous No.9896292]

>>9895863
you're not wrong anon, it's a tough one to not misunderstand

[Anonymous No.9896293]

>>9895892
If you lose something negative (such as weight) that's a good thing. Two negatives make a positive

[Anonymous No.9896326]

>>9895914
I must be retarded: what are you doing exactly with that distribution? (I'm working up from babby algebra like OP)

[Anonymous No.9896871]

>>9896326

1. If x > 0, then 0 = x + (-x) > 0 + (-x) = (-x), so:
(x > 0) implies (-x < 0).

2. Also a + b = 0 implies a = -b since
a = a + 0 = a + (b + (-b)) = (a + b) + (-b) = 0 + (-b) = (-b).

3. Note that a(-b) + ab = a(b + (-b)) = 0 and apply 2. to get a(-b) = -(ab).

4. Note that (-a)(-b) + -(ab) = (-a)(-b) + a(-b) = (a + (-a))(-b) = 0 and apply 2. again to get ab = (-a)(-b).

The only thing I didn't show was a*0 = 0, but you can show that using 2. Simply choose a > 0, b > 0 to get what you asked.

[Anonymous No.9896895]

>>9895888
based

[Anonymous No.9897188]

>>9895914
>>9896871
Thanks, anon. That was far simpler than I made it out to be.

[Anonymous No.9897622]

>>9895892

Yes, there are languages where double negatives don't exist, but if you think about it logically, it's non intuitive.

example : opposite of the opposite of cold

cold = cold

opposite of cold = hot

opposite of the opposite of cold = opposite of hot = cold

get it?