Product of negative numbers

Question

Why is a negative number multiplied by a negative number a positive number?

I'm trying to know what does multiplying by a negative number mean.

If you think of multiplication as a "groups of" ($3 \times 5$ meaning $3$ groups of $5$ then it's confusing what a "negative number of groupsof $X$" could mean.

A lot of mathematics can be and was done without reference to "negative" numbers—whatever that unfathomable silliness might mean! The first several chapters of Stillwell's Mathematics and its History cover rational points on the circle, irrational numbers, distance, conic sections, rational curves, Pell's equation, chords and tangents, prime numbers, Chinese remainder theorem all without reference to negative numbers. You could do Platonic solids or spiric sections as well without ever venturing in that direction.

Answer 1

There are many ways to show the result of multiplying negative numbers. Here is an example that preserves order.$$3(-3)=-9$$ $$2(-3)=-6$$ $$1(-3)=-3$$ $$0(-3)=0$$ $$-1(-3)=3$$ $$-2(-3)=6$$ Think you can see where this is going.

Answer 2

Logically you can think of it as 'removing an absence'. For instance, say someone dug $4$ holes, removing 3 cubic meters of dirt from each. The change in amount of dirt is $4 \times -3 = -12$. Now say you want to undo this work... you want to remove the holes. This can be represented as taking away $4$ holes, each of which is missing $3$ cubic meters of dirt, so you get a change in the amount of dirt being $-4 \times -3 = 12$. This is equivalent to adding $4$ mounds of $3$ cubic meters of dirt; removing an absence is the same as adding a presence.