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?

• Having taught basic math like that for quite a few years, that is the pretty standard visual approach. If you do not need a visual, money is always a good way to talk about positive numbers (earning) and negative numbers (spending) too since we all use it! – fullyhip Mar 7 '16 at 0:14
• I would suggest starting with negative numbers in themselves (eg distance in the opposite direction), not negative numbers as the result of operations. – David Mar 7 '16 at 0:15
• @fullyhip Do you think I should introduce subtraction as an operation, or just represent subtraction as the addition of a negative number? – Ovi Mar 7 '16 at 0:21
• @David Please see my comment above – Ovi Mar 7 '16 at 0:22
• Good question! Personally I always tell my students to think of subtraction as addition of a negative until they are VERY comfortable with the idea. – fullyhip Mar 7 '16 at 0:24

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$-200-300 = -500$. For addition and subtraction, think of walking forward and backward along a straight line without turning around. You measure from your starting point (origin). 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).