Why is a negative number multiplied by a negative number positive?

>Why is a negative number multiplied by a negative number positive?
If you flip something twice you end up back where you started

Being completely real: because we defined it that way.

No joke, look up any set theoretical definition of the integral ring and you will see that the way that they define the operations makes what you say a fact. Basically we invent it.

But looking at it from a "muh math feels" perspective, that is how it should be. Lets think about how we should define multiplication with negative numbers. First, a negative times a positive.
Here is a quick example: 3*(-5)

Using our natural intuition we could say that this should be defined as:
(-5) + (-5) + -(5) = -15

Now, if you agree with this definition then good. To define multiplication we will apply this last property we decided to give negative numbers and see how far we can get before getting into trouble.

(-3)*(-5) = ((-1)*3)*(-5) = (-1)*(3*(-5)) = (-1)*(-5 + -5 + -5) = (-1)*(-5) + (-1)*(-5) + (-1)*(-5)

And here we reach "trouble". We are now multiplying two negative numbers as was our original problem, but now we have a more "fundamental" multiplication.

The question that arises here is
Should (-1)*a = -a?

And here -a is obviously what we define as a number, call it x, such that a + x = 0

Here we can argue that this is the fundamental reason for why we even want negative numbers. And thus we then conclude that our answer is 15.

To put this in summary with a little more math jargon, this is the case because if you want a theory of integers such that it is consistent when associativity, distributivity, and the "fundamental property of -1" then you will reach with a theory that says that negative times negative equals positive.

Nothing stops you from creating a different theory.

For example, you could try to find operations that do align with associativity and distributivity (thus making up a ring) but that completely disregards the "fundamental property of -1".

But maybe such an operation doesn't even exist. I am not an algebra person.

The point is: we pick and choose the things we like.

But that brings up the question of why multiplying something by a negative is equivalent to flipping.

I'm trying to find the video that I think explains this and I think it may be the one about group theory, but I feel like there might be a simpler one.

Because of stupid standardized testing, people are taught what numbers do without any reason why they do it.

>Because of stupid standardized testing, people are taught what numbers do without any reason why they do it.

What branch of mathematics i learn the reasons of "why they do it"? I have so many questions like these!

>groups
Lrn2group fgt pls

Are you OK with -5 * 1 being -5?

1 * x = x
This is the identity of multiplication, so adding negative numbers has to preserve this.

Now if you look at the number line as a 1 dimensional plane, you'll see that the the positive side flips to the other direction when multiplied by a negative, and negative * positive has to be negative because of the above formula. In order to preserve the integrity of this plane, a negative * negative must also flip to the other side; it must be positive.

>But that brings up the question of why multiplying something by a negative is equivalent to flipping.
Because the definition of -1 is that 1+(-1) = 0. If you go from 0 to 1 and then want to go back what do you do? Go the same length in the opposite direction.

More algebraically
1+(-1) = 0
(-1) + (-(-1)) = 0
So -(-1) = 1

Since -1/1=1/-1
(-1)(-1)=1*1
-*-=+
QED

What about imaginary exponents?

Why does 2^2i = 0.183456975 + 0.98302774 i?

and this also begs the question why two negatives makes a positive

It's a consequence of the field axioms.

What happens when you flip something backwards twice?

2*5 = 10 because 2+2+2+2+2 = 10
and 5+5 = 10.

we can have 5 groups of 2 or 2 groups of 5.

2*-5 can = -10 because -5+-5 = -10, but we can't create -5 groups of 2.

-2*-5 makes no sense to me because i don't see how we're creating -2 groups of -5 and getting 10 or vice versa.

Do you understand why
A* (B + C) = AB + BC
?

Because in order for this to remain consistent with the addition of negative numbers, a negative times a negative must equal a positive.

>A* (B + C) = AB + BC
FUCK, I meant
A* (B + C) = AB + AC

think of a number line, -1, 0, 1...
negative backward, positive forward
it's taking a backward value, then reversing(negating) it, however many steps(times)

'maths' are shorthand for visual descriptions

also, the earth is flat
youtu.be/m0dDw-8Nhow

>visually show me
neck yourself brainlet

When you have 5 groups, you have 5 groups more, hence plus. When you have -5 groups, you have 5 groups less, hence minus.

>but it gets confusing when you apply the commutative property and try to form -2 groups of 5, you can't have -2 groups of a number.
>what's going on?
What's going on here is that you have zero groups of 5, minus two groups of 5. For a total of (-2 * 5) = ((0 - 2) * 5) = (0 * 5) - (2 * 5) = 0 - 10 = -10.

When you have -2 * -5, you have zero groups of -5, minus 2 groups of -5. So (-2 * -5) = ((0 - 2) * -5) = (0 * -5) - (2 * -5) = 0 - (-10) = 0 + 10 = 10.

...Now you have to imagine that :
i2 = -1

How fucked are you ?