Question

prove that a negative number divided by a positive number is negative

Answer 1

Proof by contradiction:

Suppose not. Then a negative number N divided by a positive number P would be positive or zero. (I've assumed incidentally that all numbers are positive, negative or zero, and that zero is neither positive nor negative.)

Assume N/P is positive. Call that positive number Q, so N/P=Q. Multiply both sides by P. Then P*(N/P) = P*Q. But P*(N/P)=N, so N=P*Q. A positive number times a positive number is positive, so you now have a negative number (N) equals a positive number (P*Q). That's not possible. Contradiction.

Now suppose N/P = 0 (the other possibility we originally identified). Multiply both sides by P. Then N=0*P=0, which isn't a negative number. Contradiction.

So the original supposition must have been wrong.

(I don't like this "proof" however, since I've assumed among other things that a positive number times a positive number is positive - which though intuitively obvious is possibly as contentious as the original proposition. Does anyone have a better idea?)