So after a discussion with TruthWorthy in another thread (where he attempted to show that a double positive made a negative), I looked into why a double negative makes a positive.
Essentially, we must first agree that a negative number is simply a positive number multiplied by -1. In other words, to get the negative of 2, we multiply by -1. -1 * 2 = -2.
If we accept this as correct, then we can write the multiplication of two negative numbers like so:
-2 * -3 = (-1)(2)(-1)(3)
-2 * -3 = (-1)(-1)(2)(3)
-2 * -3 = (-1)(-1) * 6
So, the question is, (-1)(-1) = ?
The convention (-1)(-1) = +1 has been adopted because anything else causes the distributive property of multiplication to break for negative numbers.
For example, let's assume (-1)(-1) = -1.
(-1)(1 + -1) = (-1)(1) + (-1)(-1)
(1 + -1) on the left equals 0, so the equation can be written:
(-1)(0) = (-1)(1) + (-1)(-1)
Since we agreed that any positive number multiplied by negative 1 is the negative form, and that (-1)(-1) = -1, we can further simplify this:
(-1)(0) = -1 + -1
So:
0 = -2???
If we place any other number (apart from +1) as the answer to the calculation (-1)(-1) = ?, the same thing happens. This is why two negatives make a positive in mathematics.
Reproduced for your enjoyment from http://mathforum.org/dr.math/faq/faq.negxneg.html
Essentially, we must first agree that a negative number is simply a positive number multiplied by -1. In other words, to get the negative of 2, we multiply by -1. -1 * 2 = -2.
If we accept this as correct, then we can write the multiplication of two negative numbers like so:
-2 * -3 = (-1)(2)(-1)(3)
-2 * -3 = (-1)(-1)(2)(3)
-2 * -3 = (-1)(-1) * 6
So, the question is, (-1)(-1) = ?
The convention (-1)(-1) = +1 has been adopted because anything else causes the distributive property of multiplication to break for negative numbers.
For example, let's assume (-1)(-1) = -1.
(-1)(1 + -1) = (-1)(1) + (-1)(-1)
(1 + -1) on the left equals 0, so the equation can be written:
(-1)(0) = (-1)(1) + (-1)(-1)
Since we agreed that any positive number multiplied by negative 1 is the negative form, and that (-1)(-1) = -1, we can further simplify this:
(-1)(0) = -1 + -1
So:
0 = -2???
If we place any other number (apart from +1) as the answer to the calculation (-1)(-1) = ?, the same thing happens. This is why two negatives make a positive in mathematics.
Reproduced for your enjoyment from http://mathforum.org/dr.math/faq/faq.negxneg.html