Can you mathematically prove that -1 x -1 = +1?

Question

It's really hard to explain, but, using math can you prove it? This guy apparently thinks it's wrong: http://www.timecube.com but he's insane anyway, so who cares.

The question asked is, "If positive one times positive one equals positve one, then why does negative one plus negative one equal positve one?"

The math he tries to prove it with is this:

-9 + -9 + -9 + -9 + -9 + -9 + -9 + -9 + -9 = -81

Do you see the flaw he made? there are a positive number of negative nine there, and it was multiplied by a negative number, so you must subtract -9 a bunch of times:

-9 - -9 - -9 - -9 - -9 - -9 - -9 - -9 - -9 = +63

But that's still wrong, and do you know why?

Because you always start with 0

here's the positive again:

+9 x +9 = +81
0 + +9 + +9 + +9 + +9 + +9 + +9 + +9 + +9 + +9 = +81

so,

what's the math then that proves this?

sorry the question should have read, "...then why does negative one TIMES negative one equal positive one?"

You're all giving me answers that assume the negatives cancel out each other. You've got to prove they do that without actually doing that. I already gave you a way to do it... just look at the way I started with 0 for the positives. It's really pretty simple. Oh, and I do have a life. I come here for inspiration when I'm working on my grphic designs.

Answer 1

-1*-1 = 1 this can be simply proved as beow
we know 1 -1 = 0
multiply by -1 on both sides
-1(1-1) = 0
-1 *1 -1(-1) = 0
-1 - 1(-1) = 0 as 1 is multiplicative identity -1 *1 = -1
add 1 on both sides
-1(-1) = 1
I have not done any tricks above (simple maths)
Additionally going by the statement in http://www.timecube.com
-1 is opposite.
so -1(-1) is opposite of opposite.
is opposite of opposite not same

Answer 2

(x+x²)/(x+1) = x(1+x)/(x+1) = x(x+1)/(x+1) = x

Answer 3

because of the x you can say that x can be equals to -2 and so multiply -2 by -1 its equals positive 2 plus minus 1 equals +1 ot looks something like this
-1x-1=1
-1(-2)-1=1
2-1=+1

Answer 4

1 * 1 = 1

-1 * 1 = -1

-1 * - 1 = -( -1) = 1

Answer 5

If you read -1*n as 'subtract 1 from 0 n times' then it's pretty clear that -1*(-1) => 0 - (-1) = +1

But, as you said, if he's nutz anyway, who gives a raggedy rodents rectum?


Doug

Answer 6

let's look at this more generally, with elements a and b.

we know that 0b=0 (since (0+0)b=0b gives us 0b + 0b = 0b and subtracting 0b from both sides give 0b=0).

So look at (-a)b:

ab + (-a)b = (a-a)b=0b=0

Add -ab to both sides to get (-a)b=-ab

Now look at (-a)(-b) = -a(-b) from the first part.

-a(-b)-ab=-a(-b+b)=-a0=0

Add ab to both sides to get (-a)(-b)=ab.

Now observe that I assumed nothing about a and b, except that they satisfy the distributive property (which is true in any ring). Take a=b=1 to get your desired result (I'm not bothering to say "in a ring with 1" because I don't think at this level we need to make the distinction).

Answer 7

Let a and b be any two real numbers. Consider the number x defined by

x = ab + (-a)(b) + (-a)(-b).


We can write

x = ab + (-a)[ (b) + (-b) ] (factor out -a)
= ab + (-a)(0)
= ab + 0
= ab.

Also,

x = [ a + (-a) ]b + (-a)(-b) (factor out b)
= 0 * b + (-a)(-b)
= 0 + (-a)(-b)
= (-a)(-b).

So we have

   x = ab

and

   x = (-a)(-b)

Hence, by the transitivity of equality, we have

   ab = (-a)(-b).

Answer 8

Oy that's a hard question, it's just a rule. I have never seen the deviation of it before. I don't even think our math teachers know that answer.

Answer 9

there's a way to prove it but i've forgotten it.even there's a way to prove : 0*x=0 or 1*x=x etc

Answer 10

omg, shoot ur self now, geta life! go take that extra time and get a girlfriend, jack off or someting