# Thread: - * - = +

1. ## - * - = +

Why minus mutliply by minus eqaual is plus?

2. ## Re: - * - = +

There are several different ways to answer that. The simplest is to observe that if a< b then, for negative c, ac> bc. That is, multiplying by a negative number reverses the inequality: 2< 3 but -2> -3 as you would see by marking 2, 3, -2, and -3 on a number line. On a number line, multiplying a and b by positive c just changes the distance between the points. 2< 8 and the distance between them is 8- 2= 6. Multiply by positive c, we have 2c< 8c and the distance between them is 8c- 2c= 6c. If c< 1 the distance between the points has become smaller, if c> 1 the distance has become larger. But if c is negative, say -4, then 8c= -32 and 2c= -8. On a number line we now have -32< -8. The order has been reversed. But if we do twice, multiplying by both negative c and negative d, then we have reversed twice- the order is back the same as before. 2< 8. Multiplying by c= -4, we have 2(-4)= -8> -32= 8(-4). If we now multiply by, say d= -3, we have -3(-32)= 96> -3(-8)= 24. Multiplying by -3(-4) is the same as multiplying by 12: 12(2)= 24< 12(8)= 96.

Algebraically, we have the "distributive law" a(b+ c)= ab+ ac. suppose c= -b. Then b+ c= b- b= 0 so we have a(b- b)= ab- ab= 0. In particular, taking a= -3, b= 4 so c= -b= -4, a(b- b)= -3(4- 4)= -3(4)+ (-3)(-4)= 0. You don't ask about "negative times positive" so I guess you have no problem with the fact that -3(4)= -12. -12+ (-3)(-4)= 0 so (-3)(-4) must be positive. Specifically, (-3)(-4)= 12.