I.e., (-2)*1=-2 but (-2)*(-1)=2
In: Mathematics
Let’s just figure out what (-1)(-1) is, because then you can do any other. Now, the only thing that matters about -1, and in particular negative numbers in general, is that it makes the equation 1+(-1)=0 true. **Nothing else matters about -1 except that when you add it to 1, you get zero.** This seems like an important property, so we probably should use it.
So what is (-1)(-1)? I want to use the equation 1+(-1)=0 so why not start with it. I can then take this equation and multiply it through by -1 to get
* (-1) + (-1)(-1) = 0
And we’re done with all of the work, we only need some interpretation. What this equation says is that **(-1)(-1) is a number so that, when -1 is added to it we get zero**. Compare the two bolded sentences. The first says that -1 is the number so that 1+ it is zero. The second says that (-1)(-1) is the number so that (-1)+ it is zero. But we already know that (-1)+1=0. Therefore, (-1)(-1) has to be 1.
We can see, then, why -1 flips things back and forth. If we multiply the equation 1+(-1)=0 through by (-1), then since 1*(-1)=(-1) it turns 1 into -1, and since the equation remains true throughout, it must also be that (-1) turns into 1. Multiplication through by -1 effectively swaps the order of the expression 1+(-1) *because* it turns 1 into (-1).
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