Limited to a numeric sense, multiplication is an operation which began life as repeated addition, and which had to then be extended in various ways as we extend

Limited to a numeric sense, multiplication is an operation which began life as repeated addition, and which had to then be extended in various ways as we extended our number systems. We know, for instance, that integral multiplication x * y provides the area of a rectangle with sides of length x, y. This applies without modification to the rationals and reals as well, so we know from this that multiplication remains consistent without having to rely on the concept of repeated addition. Similar extensions happened with negative arguments. At a certain point, pure intuition vanishes, and instead you are left with the task of simply creating a set of rules that define a consistent mathematics. Often times, the approach you take is limited by this very requirement; our mathematics is only consistent if a negative multiplied by a negative results in a positive, for instance. This topic hints at the gap between applied mathematics rooted in concrete reality, as it were, and theoretical mathematics, which of course deals purely with abstract structures.