# Tagged Questions

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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### $z^n = a_n + b_ni$ Show that $b_{n+2} - 2b_{n+1} + 5b_n = 0$ (complex numbers)

$$z = 1+2i \ (complex \ number) \\ z^n = a_n + b_ni \ \ \ (a_n, b_n \in \mathbb{Z}, n \in \mathbb{N}^*)$$ Prove that $b_{n+2} - 2b_{n+1} + 5b_n = 0$ How can I solve this? Thank you! EDIT: Or ...
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### Solve $z^2+i\bar{z} = 0$

Need to solve: $$z^2+i\bar{z} = 0$$ I have tried to use the same method for the other exercise here: Solve $z^2+iz=0$ but I do not know how to manage the $\bar{z}$ Any help?
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### Real roots of $z^2+\alpha z + \beta=0$

Question:- If equation $z^2+\alpha z + \beta=0$ has a real root, prove that $$(\alpha\bar{\beta}-\beta\bar{\alpha})(\bar{\alpha}-\alpha)=(\beta-\bar{\beta})^2$$ I tried goofing around with the ...
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### Hi! Just wondering if any one can help me out with this roots question? [closed]

(i). Factorise $z^2 - 5z + 6$ and hence, solve the equation $z^2 - 5z + 6 = 0$ (ii). Show that $z^2 - 5z + 6$ is a factor of $z^3 + (-4 + i)z^2 + (1 - 5i)z + 6(1 + i)$. (iii). Find the three roots ...
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### About matrix diagonalization in C from the characteristic polynomial.

Ok the excercise is: You have one characteristic polynomial, it's: $\lambda^4 + \lambda^2$ Find two matrixes with this polynomial, one of them diagolalizable in C and the other one not. so the ...
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### solutions of $\bar z = |z-2\Im(z)|^2$.

I need to find all the solutions of $\bar z = |z-2\Im(z)|^2$. I know that $z=x+iy$ and $\bar z=x-iy$ and then $2\Im(z)=2y$. But can someone show the algebra for what I do next?
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### Adjoin complex numbers to an arbitrary field? [closed]

This is probably nonsense but I'm throwing it out there. I don't think I can even explain the question very well: Has anyone seen bizarre things such as adjoining, say $i$ or $\pi$, to say a finite ...
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### Imaginary Golden Ratio

While playing with the results of defining a new operation, I came across a number of interesting properties with little literature surrounding it; the link to my original post is here: Finding ...
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### n-th roots of unity summing to $0$

Let $\zeta = e^{2\pi i/n}$ be an $n$-th root of unity, and let $S = \{\zeta^m|m=0,1,\ldots,n-1\}$ be the corresponding sets of all $n$-th roots of unity. Let $k \leq z$. Let $C \subseteq S$ such ...
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### Localization of roots of complex quadratic equations

Let $a,b,c\in\mathbb C\setminus\{0\}$ be complex numbers such that $$b^2-4ac \neq 0.$$ We consider the equation $$ax^2+bx+c=0.$$ I am interested in general statements about the roots of this equation ...
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### Complex multiplication definition

$$\left(a,b\right)\left(c,d\right)=\left(ac-bd,ad+bc\right)$$ I'm in a book dealing with quaternions and it says the above is the definition of multiplication for complex numbers. Can someone show ...
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### Explaining difference between natural numbers, integers, rationals, reals, complex numbers, Gaussian integers [closed]

As so far as usage in elementary number theory goes, what is the difference between the natural numbers, the integers, the rational numbers, the complex numbers, and the Gaussian integers?
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### Inequality with complex root and positive imaginary part

Let $z$ be a complex number with $\mathrm{Im}(z)>0$, and we consider $$w:=\frac{-z+\sqrt{z^2-4}}{2}.$$ It is written that "we take the square root so that $\mathrm{Im}(w)>0".$ I want to prove ...
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### How can I solve $y''=\frac{a}{y^2}$ where a is a (positive) constant?

Actually, I found out a way to solve that, but I can't get rid of complex numbers. And it does not make sense when it comes to complex numbers as the original question that involves this differential ...
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### How is $\cos(x)={e^{jx}+e^{-jx}\over 2}$? [closed]

How to prove the following equation? What is proof for $$\cos(x)=\dfrac{e^{jx}+e^{-jx}}{ 2} \qquad \qquad j=\sqrt{-1}$$?
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### How to find relations between the roots of a fourth degree polynomial which has only complex roots?

Let f(x) is a fourth degree polynomial such that $f(x) = x^4+x^3+x^2+x+1$. Let p be a root of f(x). Then which one of the following cannot be root of f(x): $p^2$ $p^3$ $p^4$ $p^5$ I know f(x) has ...
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### How to turn the reflection about $y=x$ into a rotation.

If we reflect $(x,y)$ about $y=x$ then we get $(y,x)$. And because $x^2+y^2=y^2+x^2$ this can also be represented by a rotation. Using this we get: $$(x,y)•(y,x)=2xy=(x^2+y^2)\cos (\theta)$$ Hence ...
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### Real Numbers Raised to Imaginary Powers? [closed]

What is a real number to the power of an imaginary or complex number? e.g. 3i. I have searched through sites about imaginary numbers, but none seem to say anything about imaginary indices. Examples ...
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### locus of complex number 2

Que: If $arg(\frac{z-z_1}{z-z_2})=\pi$ then what is the locus of $z?$ Doubt In my textbook it is written that it represents the straight line joining $A(Z_1)$ and $B(Z_2)$ but excluding the ...
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### If $f$ has pole at $0$ then show that $e^f$ can't have pole at $0$.

i am trying to show that if $f$ has a pole at $0$ then $e^f$ can't have removable singularity at $0$ ? I tried to show that but i have a problem . I assume that $e^f$ has removable singularity ...
### Can we solve for $c$ in the equation $\sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\right)=0$?
Let $N\geq 1$ and $0\leq k\leq N-1$ be fixed numbers, and $c>0$ be unknown. Suppose we have \begin{eqnarray} \sum\limits_{i=0}^{N-1} \exp\left(-jc\sin\left(\frac{2\pi i}{N}-\frac{\pi k}{N}\right)\...
### Circle and line construction of a compex number $z\in\mathbb C$
Let $C\subseteq\mathbb C$ be the field of constructible complex numbers; that is, it includes only the elements $z\in\mathbb C$ which can be constructed with circles and lines. The field \$E\subseteq \...