I have to teach my $29$ year old brother math. He remembers basic arithmetic from school, but he always hated math, which is why I want to start him from the beginning with intuitive explanations for everything.
I am thinking of starting off with the number line. Adding two positive numbers is easy enough to explain, because you can represent positive number as lengths; if you want to add $5$ to $7$, you take the compass and measure $5$. Then you put the compass down at $7$, and the result will be $12$. However, how would I explain addition with negative numbers? The simplest way I can think of is to represent numbers as vectors in $2D$. Positive numbers are vectors pointing right, and negative numbers are vectors pointing left. With this representation, when we add two numbers, we add their vectors tip-to-tail. And when we have the negative of a number, we keep the length but just reverse the direction of the arrow.
I am not very satisfied with the approach above, as I think it overcomplicates things a bit. I would very much prefer to be able to return to representing numbers as just lengths. Is there a better way to explain this?