Prove(-1)(-1) = 1?

Question

That is, prove negative times negative is positive.

Tom's is a bona fide proof. He even finished it up with QED. Nice logical structure, thanks.

Answer 1

Firstly, we need to state the additive inverse law:

For any number a, there is a number -a such that -a + a = 0 or a-a = 0.

Secondly, we need to prove that a negative times a positive is negative:

a(b-b)=0
ab + (a)(-b) =0
Therefore (a)(-b) must be -ab since it is the
additive inverse of ab.

Thirdly, we can now show that the product of
two negatives is a positive as follows:

-a(b-b)=0
(-a)(b) + (-a)(-b) = 0
We already proved in step two that
(-a)(b) = -ab. Now since (-a)(-b) is the
additive inverse of -ab, it follows that
(-a)(-b) must be ab.
Q.E.D.

Locomexican89: You are assuming that the square of any number is positive, but what about the square of i ? i^2 = -1. Hmmm. In proofs you cannot make any assumptions.

Answer 2

use the inverse of the multiplicative identity

a x 1 = a by multiplicative identity

-1 x 1 = -1 by substitution of unit -1 for a

-1 x -1 = 1 by the definition of negative multiplication

That's all I got.

Answer 3

a * 1 = a math rule that anything multiplied by 1 is the same number
thus,
-1 * 1 = -1
therefore, by deduction,
1 = -1 / -1
which must mean that negative divided by negative gives you a positive number

that didn't get me anywhere...
let's try this
-1 * 1 = -1
now, divide both sides by -1
(-1) (1) / (-1) = -1 / -1
on the left
(-1) (1) / (-1)
if a = -1, then
a * 1 / a = a^1 * 1 * a^-1
thus you get a^0 * 1, which is 1
on the right side, it was previously proved that -1/-1 is 1

Answer 4

The product of two negative numbers is the product of their absolute value.

(- 1)(- 1) = |- 1| x |- 1| = 1

Examples of absolute value

| -1| = 1

|- 2| = 2

|- 3| = 3

| - 4| = 4

When you multiply two negative numbers their produce will be positive

Answer 5

Here it is mathematically
-x-y = (-1)x(-1)y= (-1)^2xy
so the product is positive
with ur example
(-1)(-1) = (-1)(1)(-1)(1)= (-1)^2(1)^2=1

Answer 6

(-1)(-1) = 1
(-1)(-1)/(-1) = 1/(-1)
(-1)(1) = 1/(-1)
-1 = -1

Answer 7

Begin with

1 = 1

Multiply both sides by (-1)

(-1) x 1 = (-1) x 1

Rearrange,

(-1) x 1/(-1) = 1

(-1) x (-1) = 1

done.

Answer 8

I typed it in my calculator and it is 1

Answer 9

because two wrongs make a right =D

orrrrrrrrrrrrrrrrr think of it like this...
-1 is the inverse of 1.
the inverse of an inverse is itself.

Answer 10

think of it this way. If a couple is the same than they are a good couple( positive). if they are different then they are a bad couple(negative)