Q: Why is it that when you multiply a positive number with a negative number you get a negative number?

Mathematician: Consider what would happen if when we multiplied a positive number by a negative number we got a positive number. For example, suppose that

3*-5 = 15

rather than the usual

3*5 = 15.

Now, if all the other standard rules of math still applied, then we could write

-15 = -(3*5) = 3*-5 = 15

and hence we would find that -15 and 15 are equal to each other. Depending on how you think about it, this is either a contradiction (if we fundamentally believe that -15 is not the same as 15) or we have destroyed all negative numbers entirely (if we simply interpret this formula to mean that there is no difference between negative and positive numbers). In either case, clearly something has gone screwy.

Perhaps an even simpler way to think about this is to observe that by using the rules of addition and the definition of multiplication, we have for example that

3*(-2) = -2 + -2 + -2 = -6.

More intuitively though, why should we expect that negative numbers multiplying positive ones should lead to negative numbers? Well, consider a financial example. If we denote positive amounts of dollars with positive numbers (i.e. money owned), it makes sense that negative numbers would correspond to money owed (i.e. debt). Then, for example, if one bank has $100 and it merges with a bank that has -$80 then the new, combined bank has a total of $20 in net assets (100-80 = 20). So where does multiplication come into play? Well, let’s now imagine that three banks merge, each of which has -$20. In that case, we should expect the total amount of net assets that the new merged bank has is just three times that of the individual banks, given by the mathematical expression 3*(-20) = -60. Of course, this only works if multiplying the positive number 3 with the negative number -20 leads to a negative number (in this case -60).