The additive inverse of a number is the number that when added with the first, results in zero.

For example, the additive inverse of 5 is
negative 5 (-5), because 5 + (-5) = 0

The additive inverse of 3 is -3, because
3 + (-3) = 0

Another name for additive inverse is "opposite".  So the "opposite" of 4 is -4.

You can see from the above number line that both 3 and (-3) are three units from zero, therefore they are equal distances from zero, and    3 + (-3) = 0

Why do you need to know additive inverses?  Where do you use it in mathematics? 

One place you use them is when solving equations; e.g.  to solve  x + 3 = 9

       x + 3 = 9          (you need x by itself; you need to "move" the 3)
x + 3 + -3 = 9 + -3   (add the opposite of 3 to each side)
       x + 0 = 6          (a number plus its opposite equals 0)
             x = 6