# What is the angle and the length of the sum of two complex numbers?

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 G. Seldon Nov 8 '18 at 19:34
• Looks like a duplicate of math.stackexchange.com/questions/2412297/… – Naweed G. Seldon Nov 8 '18 at 19:36
• 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 '18 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 '18 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 '18 at 19:58

$$|z+w|=\sqrt{(|z|\cos(\text{Arg }z)+|w|\cos(\text{Arg }w))^2+(|z|\sin(\text{Arg }z)+|w|\sin(\text{Arg }w))^2}=\sqrt{|z|^2+2|z||w|\cos(\text{Arg }z-\text{Arg }w)+|w|^2}$$
• 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 '18 at 9:49
• @NilsPhillipTalgö: have a look at $(2+3i)^{1-2i}$... – Yves Daoust Nov 9 '18 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 '18 at 9:53