# Questions tagged [complex-numbers]

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

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### Are Imaginary Numbers as $Real$ as Real Numbers?

In my Abstract Algebra book, A First Course in Abstract Algebra,'' by Fraleigh, the author seems to suggest that imaginary numbers are as $real$ as the real numbers, by asserting, for example, that ...
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### Consider $az^2+bz+c=0$ where $a,b,c$ are all complex numbers

Consider $az^2+bz+c=0$ where $a,b,c$ are all complex numbers. What is the condition (ie the relation between $a,b,c$) for which the given quadratic has both real roots? I took the conjugate of the ...
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### What is $5$ taken to the power iota? [closed]

Could someone please solve this. I tried to put it in Euler's form but that wasn't any use because $$5 = 5e^{i\cdot0}=5.$$
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### Region $S$ represented by $a+b\omega +c{ \omega }^{ 2 }$

Let $\omega =-\dfrac { 1 }{ 2 } +\dfrac { \sqrt { 3 } }{ 2 } i$ and $S$ denote the set of all the complex numbers in the Argand plane of form $a+b\omega +c{ \omega }^{ 2 }$ , where $a,b\in [0,1]$ ...
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### Let $z=x+iy$, where $x,y\in \mathbb{Z}$. Find the area of the octagon whose vertices are roots of $(z\bar z)|z^2-\bar z^2|=1200$

$$|z^2||z-\bar z||z+\bar z|=1200$$ $$(x^2+y^2)(|2x|)(|2y|)=1200$$ $$(x^2+y^2)(|xy|)=300$$ I don’t think there is any realistic way to obtain $x$ and $y$ other than hit and trial. I am asking this ...
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### complex numbers , college mathematics [closed]

Find all the values of $log(1-i)$ and $\log(3-2i)$ Find all values for the following expressions: $5^i$ and $\log(1+i)^{πi}$
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### Domain of $\operatorname{Arg}(1/z)$: $\operatorname{Re}(z) \neq 0$

The book I'm using is Complex Variables and Applications, 9th Ed by Brown and Churchill. I'm confused about exercise 14.1.b (pg 43): For each of the functions below, describe the domain of definition ...
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### Finding the rank of a matrix $A$

Find the rank of the $n \times n$ matrix $A = [i + j]_{i,j \le n}$ (over $C$). C here should be the complex space; although i am having trouble interpreting what A exactly is, I do not understand the ...
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### Can $\operatorname{Re}(a+bi)^n$ be overlapped with $a,b\in\mathbb{Q}$ fixed? (Just changed the $\mathbb Z$ in my earlier question to $\mathbb Q$)

This is a generalization of my earlier question, Can $\operatorname{Re}(a+bi)^{n}$ be overlapped with $a,b\in\mathbb{Z}$ fixed?, just changing $\mathbb Z$ into $\mathbb Q$. Is there any non-trivial ...
### Let $s,t,r$ be non zero complex numbers and $L$ is is set of solutions of $z=x+iy$ of the equation $sz + t\bar z+r=0$
Prove that $L$ is a singleton set if $|s|\not =|t|$ And, prove that $z$ is a straight line if $L$ is not singleton Solving the equation, I got $z=\frac{\bar s r -\bar r t}{|t|^2-|s|^2}$ I personally ...
I have $(1+i)^{13}$ and I need to find the principal argument. I did this: $(1+i)^{13}$ = $(2)^{13/2}(cos(\frac{13pi}{4}) + isin(\frac{13pi}{4}))$ using De Moivre's Theorem, but I dont know where to ...