# Questions tagged [complex-numbers]

Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

12,645 questions
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### For $z \in \mathbb{C}$, the only point where the derivative for $f(z)=|z|^{2}$ exist is the origin.

Let $z=x+iy$, as $f(z)=z\bar{z}$ we have $f(z)=x^2+y^2$. Then we can define: $$f(x,y)=u(x,y)+i v(x,y).$$ Where $u(x,y)=x^2$ and $v(x,y)=y^2$. To see $f$ has derivative in the origin I can see the ...
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### How is $\sum_{k=0}^{5} {}^5C_k\sin(kx)\cos(5-k)x=16\sin(5x)$?

The original question was if $\sum_{k=0}^{5} {}^5C_k\sin(kx)\cos(5-k)x=N\sin(5x)$ then what is the value of N? I plugged $\frac{\pi}{2}$ in place of x and N came out to be 16. But is there any ...
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### Prove: If $z\in \mathbb{C}$ and $|z|=1$, then $|z+1|^2+|z-1|^2=4$.

If $z\in \mathbb{C}$ and $|z|=1$, then $|z+1|^2+|z-1|^2=4$. My attempt: $|z|=1 \Leftrightarrow z=1^2=1$ Hence $|1+1|^2+|1-1|^2=4$ Is that enough?
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### sums of series involving complex numbers

I dont know how to use latex any good rescources to quickly learn from scratch, hence why I have photographed the question. The question is as follows question question final part of working The ...
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### Square root of a negative number

Correct me if I am wrong, $\sqrt{-4}=2i$. But how do you explain it to a student? We know $\sqrt{-1}=i$, but one cannot say $\sqrt{-4}=\sqrt{-1}\sqrt{4}=2i$ as the laws of indices can only be ...
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### Find a basis for a set involving complex conjugates

How can I find a basis for this vector space over ℝ: A={(u, v, v̄, -u) : u,v ∈ 𝐂} (so u and v are complex numbers of the form a + bi and v̄ is the complex conjugate of v). It's just the complex ...