I'm having trouble understanding why negative numbers turn positive when they are multiplied or divided. I understood that negative multiplied positive is negative because I was given the following example:
I am in a submarine and going down -2m every second so in 3 seconds I will be at -6m.
I was trying to find an example in which I could undertand why negative X negative = positive.
Could someone give me an example?
I've asked this question almost a year ago, but this question has started to bother me again. Now, I just automatically change negative numbers to positive when they are multiplied.
But the more I think about it, I think a little differently. When you multiply positive numbers by each other they get bigger. So, if you mulyiply negative numbers by one another shouldn't they get smaller?
Does multipliying and dividing have a deeper meaning that I'm not grasping? What's the meaning of negative numbers? I know that they exist on the number line.
Draw yourself a 10 x 10 grid.
Say you wanted to find out what 9 x 9 is...
It might make sense to start with 10 x 10, and then take away one whole row of 10 and one whole column of 10 (so you are left with the 9 x 9 grid)...
The problem with taking one row of 10 and one column of 10 is that you've taken away 1 too many (you've taken away the one in the corner twice). So you have to add it back on...
What we have shown is a pictorial representation of the distributive law...
9 x 9 = (10 - 1) x (10 - 1)
= (10 x 10) - (10 x 1) - (1 x 10) - 1 x (-1)
But the only way that this makes sense is if the -1 x -1 = + 1.
= (10 x 10) - (10 x 1) - (1 x 10) + 1.
Therefore, negative times negative is positive.
+ x +
- x +
+ x -
and - x -
from multiplication as it is defined to be (i.e. the area of a rectangle).
It also makes sense if you think of + as "keep the same" and - as "change".
So + + means "keep + the same", i.e. +
+ - means "keep - the same", i.e. -
- + means "change the +", i.e. -
- - means "change the -", i.e. +.