What is the distributive property of multiplication over addition?

Answer 1

The distributive law of multiplication over addition says that if x, y and z are any numbers, then

x(y+z) = xy + xz

. It's true not only for integers, but also for real numbers or even complex numbers.

Here's a proof / graphical explanation:

__________

|*****|***| ^

|*****|***| |

|*****|***| |x

|*****|***| |

|*****|***| v

---------------

<--y--><-z->

Consider the ASCII rectangle above. It has height x and width y+z, therefore its area is x(y+z). Alternatively, it can be viewed as two smaller rectangles joined together. The one on the left has height x and width y, so its area is xy. Similarly, the one on the right has area xz. So the total area is xy + xz. Therefore x(y+z) = xy + xz.