On the Argand plane the complex number z = a + ib is simply the coordinate point (a, b). Thus z* = a - ib is the point (a, -b), or the point (a, b) reflected over the "x" (real) axis.Originally Posted bykingkaisai2

You can think of the complex numbers as vectors in the Argand plane. So by multiplying z by a real number k you are simply stretching the vector by a factor of k.

Now it's simply a matter of showing that the operations of stretching and reflecting the vector over the real axis are commutative, which should be obvious.

-Dan