Thread: Why do we want the distributive property?

From reading Edmund Landau's Foundations of Analysis, I see that he starts with the natural numbers and proves all their properties, such as the distributive property and later moves on to integers. He defines negative times negative as positive before he proves the distributive property for integers. However, why do we want the distributive property?
Is it for the same reason that we define how to multiply fractions, not out of necessity but that it allows for us to just use the same rules as we did for natural numbers and therefore can use the same techniques to solve algebra problems regardless of the number set?

I think that you can get motivation for our arithmetic rules from geometry, interpreting multiplication as finding the area of rectangles. Suppose we want to compute . On one hand, this is , finding the area of a rectangle. On the other hand, we can break it up into a and rectangle. We should expect the two values to be the same, because we simply took a figure and cut it into two parts.

Thanks, it's good to remember that. But I should have asked why do we want the distributive property to work with all integers and not just natural numbers?