Prove 2 negative multiples = a positive number?

Question

Bueller, Bueller, anyone??!??!

Answer 1

At risk of repeating myself, or saying the same thing over again, or possibly being redundant or overly repetitious:

Let K be any ordered field, and let a and b be elements of K such that a<0 and b<0. Then ab>0.

Proof:

First note that by the distributive law, (-a)(-b) + (-a)b = (-a)(-b + b) = (-a)*0 = 0. Likewise, ab + (-a)b = (a+(-a))b = 0*b = 0. Since both quantities are equal to 0, (-a)(-b) + (-a)b = ab + (-a)b. Since these are equal, they will remain equal upon subtracting (-a)b from both sides, so (-a)(-b) = ab. Now, since K is an ordered field, the order relation is compatible with addition, so we may add (-a) to both sides of the inequality a<0 to obtain 0<(-a). Similarly, 0<(-b). Since both (-a) and (-b) are greater than zero, their product is also greater than zero (the fact that the product of two positive quantities is positive is an axiom of ordered fields). Therefore (-a)(-b) > 0. However, (-a)(-b) = ab, so ab>0, and the product of two negatives is positive. Q.E.D.

Answer 2

This is not a proof, but it shows how it works.

Use any multiplication factors and follow the pattern.
-6(5) = -30
-6(4) = -24
-6(3) =-18
-6(2)=-12
-6(1) = -6
-6(0)= 0
-6(-1) =6
-6(-2)=12
-6(-3) =18
If you follow the pattern, each time you are adding 6 to get the next answer. You could do this with any non zero multiples and start and end wherever you want to.

Answer 3

When you multiply a negative times a positive, you take the opposite of that number. For instance, -1 times 6 is the opposite of 6, or -6, and -3 times 5 is the opposite of 3 times 5, or -15. When you multiply a negative times a negative, you're taking the opposite of it. For instance, -1 times -6 is the opposite of 1 times -6, or 6, and -3 times 05 is the opposite of 3 times -5, or 15.