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Two complex numbers are added by adding their real parts and their imaginary parts separately. So
(a+bi) + (c+di) = (a+c) + (b+d)i

For example, if z1 = 2+6i and z2 = 3+i then z1+z2 = 5+7i

Ordinary numbers can also be treated as complex numbers, for example 5 is 5+0i

If we are thinking of the number z=a+bi as a point in the complex plane, then what happens to the point when we add another number to it? The diagram below shows some examples.

If we had a picture in the plane, and we added the same number to every point in the picture, then the whole picture would move.

Now it is your turn to try it out. Enter a number is the boxes and press Go to move Penny. If Penny disappears out of the picture you can get her back by pressing Reset.

Go Reset Penny is at 0

When we move every point according to the same formula, we call it a mapping of the plane. In this case we are adding the same number to every point. If the number to be added is the complex number k then we can write this as
z |-> z + k

We might also want to give a name to our mapping, so that we can refer to it later. Suppose we want to call it T, then we would write
T : z |-> z + k
and read this as "T maps z onto z+k"