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?