How to understand operations with negative numbers in a simple, intuitive way? 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?
 A: For addition and subtraction, think of walking forward and backward along a straight line without turning around. You measure from your starting point (origin).
A: Negative numbers probably originated in accounting.  If your bank account balance is positive, you have money there; if it is negative, then you owe money.  If you owe one person $\$300$ and another $\$200$ then you owe a total of $\$(200+200)$.  Thus $-200-300 = -500$.
A: I've never tried teaching anybody maths (and I'd be absolutely hopeless at it!), but what I myself find intuitive, in a seemingly naive way, are the ideas of: (a) an invertible operation; and (b) forming a new operation by repeating (iterating) a given operation.
Examples of invertible operations might be: (i) moving something by a fixed distance in a fixed direction (inverted by moving it the same distance back in the opposite direction); (ii) transferring a handful of beads from one pile (or jar) into another pile or jar (inverted by transferring the same number of beads back into the first pile, or jar - note that you can do this even if you only have the idea of transferring a single bead, and the idea of repeating an operation a certain number of times, and you haven't explicitly introduced the idea of one-to-one correspondence).
Adding whole numbers (positive, negative, or zero), $m$ and $n$, is explained by saying that "do something $m + n$ times" means "do it $m$ times, and then do it another $n$ times".
Doing something $0$ times means doing nothing.
Doing something $-1$ times means inverting (undoing) it.
Doing something $1$ time(s) just means doing it, of course.
Doing something $m \times n$ times means doing the operation of doing the same thing $n$ times, $m$ times. (Or perhaps this should be the other way round?)
So, for example, doing something $(-1) \times (-1)$ times means undoing undoing it, which just means doing it; and this is why $(-1) \times (-1) = 1$.
Doing something $(-3) + 2$ times means undoing it $3$ times, then doing it twice, so you might as well have just undone it once, which is why $(-3) + 2 = (-1)$. E.g. making $3$ moves to the left (or up) followed by $2$ steps to the right (or down) has the same cumulative effect as a single move to the left (or up).
(And so on.)
I'll get me coat.
A: The approach with my children was to use temperatures to introduce negative integers (it works better with Celsius degrees, since negative temperatures in Fahrenheit are uncommon). For instance,
the temperature is +2°C and we loose 3 degrees, to arrive to -1°C: children can visualise the difference on a thermometer. 
I have to confess that this method has some drawbacks. First, it works mainly in winter. Secondly, global warming may reduce the opportunities to have practical exercises with negative degrees. Thirdly, electronic thermometers are a killer for this method...
