Let $z = a+bi$, $w = c+di$ and $s = z+w$.

It is very easy and straightforward to calculate $s$, $\arg(s)$ and $|s|$, but what I want to know, is how $\arg(s)$ and $|s|$ are related to $z$ and $w$.

What is $\arg(s)$ in terms of $\arg(z)$ and $\arg(w)$?
And likewise, what is $|s|$ in terms of $|z|$ and $|w|$?

Funny enough, complex addition seems to be more difficult than multiplication, when we think about $\arg(z)$ and $|z|$. Even complex multiplication is easier, a simple formula for that using only $|x|$, $Arg(x)$ and other basic operations, can be written like this: $$zw = |z||w|e^{(Arg(z)+Arg(w))i}$$ And for complex exponentiation it's a bit more complicated and complex (literally): $$z^w = \frac{|z|^{Re(w)}}{e^{Arg(z)Im(w)}}e^{(Arg(z)Re(w)+ln(|z|)Im(w))i}$$

  • Not much can be said about that, at best all you can say is that $|s| \leq |z| + |w|$ – Naweed Seldon Nov 8 at 19:34
  • Looks like a duplicate of math.stackexchange.com/questions/2412297/… – Naweed Seldon Nov 8 at 19:36
  • 1
    You can't say much about $\arg(s)$ in terms of $\arg(z)$ and $\arg(w)$, and you can't say much about $|s|$ in terms of $|z|$ and $|w|$. However, in terms of $\arg(z)$ and $\arg(w)$ and $|z|$ and $|w|$, you can say everything you want to, using basic trigonometry. – Arthur Nov 8 at 19:38
  • Yes that post is similar, but im not just asking about that, im asking how to express arg(s) and |s| in terms of |z|, arg(z), |w| and arg(w). Calculating arg(z+w) is just atan((Im(z)+Im(w))/(Re(z)+Re(w)) + or minus pi depending on wether Re(z+w) and Im(z+w) is positive or not. @NaweedSeldon – Nils Phillip Talgö Nov 8 at 19:50
  • Maybe expressing arg(s) is easier by using a combination of arg(z) and |z|? When we think about it, we both know the angles of z and w, and we also know their lenghts, doesnt that form a triangle? link – Nils Phillip Talgö Nov 8 at 19:58
up vote 0 down vote accepted

There is no shortcut. Convert to Cartesian, add and back to polar.

This yields

$$|z+w|=\sqrt{(|z|\cos(\text{Arg }z)+|w|\cos(\text{Arg }w))^2+(|z|\sin(\text{Arg }z)+|w|\sin(\text{Arg }w))^2}=\sqrt{|z|^2+2|z||w|\cos(\text{Arg }z-\text{Arg }w)+|w|^2}$$

and similar for the argument.

  • If you compare the formula for multiplication and exponentiation to this, addition actually seems more complicated. Who would have thought that in some way, $2+3$ is actually harder than $2*3$ and $2^3$. – Nils Phillip Talgö Nov 9 at 9:49
  • @NilsPhillipTalgö: have a look at $(2+3i)^{1-2i}$... – Yves Daoust Nov 9 at 9:50
  • Actually, the polar form is very close to the logarithm. $\log z=\log|z|+i\text{Arg }z$. This explains why the product in polar form is easy, $\log zw=\log z+\log w$. And also why addition doesn't work: $\log(z+w)=???$. – Yves Daoust Nov 9 at 9:53

Your Answer


By clicking "Post Your Answer", you acknowledge that you have read our updated terms of service, privacy policy and cookie policy, and that your continued use of the website is subject to these policies.

Not the answer you're looking for? Browse other questions tagged or ask your own question.