Why does multiplying two negative numbers equal a positive number?
In: 1210
This is a bit beyond ELI5, but I guess I will go for it.
So let’s think about what a negative number is. A negative number is a number so that when you add it to its positive counterpart, it adds up to zero. Due to the structure we are dealing with (technically called a ring), we also force that there is **only one such number.** So the number that you have to add to number *a* to get zero is *-a.* This also implies, since we have to add *-a* to *a* to get zero *-(-a)=a.*
So let’s take it a bit more general. We are going to focus on multiplying two numbers. So let’s now consider the following addition
*(-ab + (-a)(-b))*
But using the distributive property, we can rewrite this.
*(-ab + (-a)(-b)) = ((-a)b + (-a)(-b)) = (-a)(b + (-b))*
But *-b* **means** the number that when we add it to *b* we get zero. So we end up getting
(-*ab + (-a)(-b)) = (-a)0 = 0*
So we have shown that (*-a)(-b)* is precisely that number that when we add it to *-ab* we get zero. But by definition, that number is *ab*, so
*(-a)(-b) = ab*
Q.E.D.
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