Why is negative times negative positive?

While I was philosophizing, I forgot how this could happen XD! So I had to rethink how it works.

So multiplication can be thought of as repeated addition. This even true for non-integer real numbers, just that sometimes you are adding it only by a non-integer proportion sometimes.

Addition is commutative, so you can explain adding A B times also as adding B A times.

With negative numbers, when we take B from A and if B is greater than A, we can say that the result is a negative number with the value of taking A from B, so that the borrowed amount is represented as the difference between the larger and smaller number.

Since we have established the origin of negative numbers through subtraction, we can add a positive number to a negative number to bring it closer to the positive realm. Adding a postive number reduces the borrow from the negative number, until it reaches zero, and then the familiar addition of applies.

Since we have established subtraction as removal earlier, we can use the concept of multiplication as repeated addition. When multiplying a positive number A with a negative number with value of B, we can think of it as borrowing the number B A times. This would result in a negative number A times as large as B was. By the commutative property, we can also think of it in the other way as borrowing A B times, since the negative operator will shift the process of multiplication from repeated addition to repeated subtraction.

Now, if we consider two negative numbers with the value of A and B, multiplying then would have the same effect as unborrowing the borrow. A being a negative number, when multiplied by B, another negative number, will have to be unborrowed B times, which in other words mean turning of the repeated subtraction into repeated addition because of the negative operator. The same happens the other way round.

I hope this was explanation was satisying to read!

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