When dealing with complex numbers they can be presented as vectors, at least that is stated in my textbook. And the addition operation defined for complex numbers:

$$z_1 + z_2 = x_1 + x_2 + i(y_1 + y_2)$$

fully corresponds with the rules for vector addition.

geometric representation of complex number addition

But why the multiplication operation does not have a geometric (vector) representation? I wonder, because my textbook states that $\sqrt{i} = e^{\frac{i\pi}{4}}, e^{\frac{i5\pi}{4}}$, that is:

complex roots of imaginary one

But if I try to multiply $W_1 \times W_2$ I will get a zero vector. And even if $W_1$ and $W_2$ were not collinear I would have get a vector which lies in another plane.

So, could anyone give me an answer about whether I'm correct at all and if I am whether any explanation of this situation exists, or may be I just have to live with it?

Thank you.

  • 2
    $\begingroup$ If you can get hold of Needham's book Visual Complex Analysis, this is very well explained in section 1.I (see especially 1.I.5 on pp. 8–10). $\endgroup$ Commented Feb 4, 2015 at 11:12

2 Answers 2


In order to understand multiplication of complex numbers, it is better to represent complex numbers in "polar" form, namely $$ z=r(\cos(\theta)+i\sin(\theta)) $$ where $r$ is a positive real number representing the length of the vector defined by $z$ in the complex plane and $\theta$ the angle it forms with the real axis.

When you multiply $z$ with a similarly written $z^\prime$ and perform all the algebra you eventually get $$ zz^\prime=rr^\prime(\cos(\theta+\theta^\prime)+i\sin(\theta+\theta^\prime)) $$ The geometrical interpretation of multiplication is now very visible. E.g. if $|z^\prime|=1$ (i.e. $r^\prime=1$) multiplication by $z^\prime$ is just rotation by some angle.


Complex multiplication has no connection to the cross product of vectors.

You can understand complex multiplication as an addition in the logarithmic domain (the log of a complex number is the log of its module plus $i$ times the argument): you multiply the modules together and you add the arguments.

In geometric terms, one of the numbers applies a similarity transform to the other (scaling and rotation). Unfortunately, this interpretation is asymmetric. It is constructible with a ruler and a compass.

enter image description here

Take the green ($z_1$) and black ($1$) vectors and rotate them to align on the blue ($z_2$) one; this adds the arguments. Then by an homothety, you multiply the lengths and get the red ($z_1z_2$) vector.

  • $\begingroup$ okay, thank you, so I've understood that the statement that a complex number can be represented by a vector is incorrect. Please correct me, if I'm wrong $\endgroup$
    – d.k
    Commented Feb 4, 2015 at 10:38
  • 2
    $\begingroup$ Complex numbers are commonly represented as 2D vectors, but the usual dot and cross products do not apply. (Actually, they are the real and imaginary parts of $z_1z_2^*$.) $\endgroup$
    – user65203
    Commented Feb 4, 2015 at 11:05

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