This said, how can you visualize multiplication by -1? (By definition, (-1)*(5) = -5.) This is a reflection on the number line of a number in 0: 5 times -1 equals -5. Say this aloud: "Multiplication by -1 is a reflection in zero on the number line." It'sounds just right, in part because it is true if the number being multiplied is positive. To keep it true for other (negative) numbers we need to define multiplication by -1 of any number (positive or negative) as a reflection in zero. Thus multiplying -5 by -1 we are getting its reflection, which is 5.

What needs to be proved is that these definitions comply with other laws of arithmetic, like associative, commutative, distributive. But this is a different matter.