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).
think of the number line. positive numbers to the East of Zero and Negative Numbers to the West of Zero.
when there is a negative sign, we change the direction. When there is a positive sign, we do not change the direction.
so, when I multiply two positives, no change in direction.
when I multiple two negatives, there’s twice change in direction, hence the result is positive.
this link (from where I summarised above) explains is really at ELI5 https://www.mathsisfun.com/multiplying-negatives.html
When you multiply by a negative number, that’s like turning around 180 degrees. If you multiply by two negative numbers, you turn around 180 degrees twice, which is 360 degrees. If you multiply by 4 or 6 or 8 negative numbers, you’re just turning around 360 degrees multiple times, but end up facing the same way.
Think about it like walking on the numbers line. A positive number means you go to the right, along the positive x-axis; and a netative number means you go to the left, along the negative direction. When you move -3 spits, you could say that you’re moving 3×(-1) spots. Similarly, movibg 3×(-4) can be seen as 3×(-1)×4. And because multiplication is commutative, it can be seen as (-1)×3×4, or -(3×4), because multiplication is associative. So when you multiply with a negative number, you can see it as turning around, and moving in the direction you’re now facing. This means that something like -3×(-4) can be seen as:
(-1)×3×(-1)×4=
(-1)×(-1)×(3×4)=
-(-(3×4))
By default, you’re looking in the positive direction. Now you need to move a total of 3×4 strps in some direction. Initially, it’s in the positive way (because of PEMDAS: parentheses first). Then you move outward, and get to a minus sign, do you turn around. After that you encounter another minus sign. So you turn around again. And now you’re facing back in the original direction.
Separate it out. -2(-2) = -1*-1*2*2. Multiplying -1 by -1 inverts the negative to a positive, leaving 1*2*2, or just 2*2.
Multiplying by a negative acts as a toggle switch. On-off-on-off. Every 2 switches you are back to the original side.
>How exactly is a rainbow made? How exactly does a sun set? How exactly does a posi-trac rear-end on a Plymouth work? It just does.
I’ll provide a real answer:
Multiplication is basically a series of repetitive addition. 5×5=5+5+5+5+5 right? This is easily applied when only one number is negative, but when both numbers are negative, the equation becomes illogical. The question shifts from “How does this work?”, to “How do we *want* this to work?”
There is no practical situation that I can think of where you would multiply a negative by a negative. Basically, it’s to create structure in math. Purely for the sake of argument.
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