Questions tagged [complex-numbers]

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

A complex number is a number in the form $z=a + bi$, where $a$ and $b$ are real numbers and $i$ is the imaginary unit, or alternatively, $z=r\cdot e^{i\theta}$, with $r$ called the magnitude and $\theta$ called the argument.

The complex conjugate, $\overline z$, is $a-bi$ or $r\cdot e^{-i\theta}$.

Read more about complex numbers and their properties here.

19229 questions
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Complex number proof

Let f(x), g(x) $\in \mathbb C[x].$ Prove that if f(x) | g(x) and g(x) | f(x), then there exists a nonzero $c \in \mathbb C$ such that $f(x) = c * g(x)$ (You may use the fact that for any p(x), q(x) $\in \mathbb C [x],$ deg(p(x)q(x)) = deg(p(x)) +…
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Why $A\sin(2\pi ft) =\frac{A}{2j}(e^{j2\pi ft}-e^{-j2\pi ft})$ but not $\frac{A}{2}(e^{j2\pi ft}-e^{-j2\pi ft})$?

$$v_{\mathrm{in}}(t)=A\sin(2\pi ft) =\frac{A}{2j}\left(e^{j2\pi ft}-e^{-j2\pi ft}\right) \\ |H(f)|=|H(-f)|;\angle H(f) = -\angle H(f) \\ v_{\mathrm{out}}=H(f)v_{\mathrm{in}}=\frac{A}{2j}H(f)e^{j2\pi ft} - \frac{A}{2j}H(-f)e^{-j2\pi ft}$$ Here I…
aukxn
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Factorising a complex polynomial over $\mathbb{C}$

I'm given $f(z)=z^6-1$ to factorise over $\mathbb{C}$. My working is as follows up to the point I don't understand: $f(-1)=0$ and $f(1)=0$ So $(z+1)$ and $(z-1)$ are factors $(z+1)(z-1)=z^2-1$ $(z^2-1)(z^4+pz^3+qz^2+rz+s)=z^6-1$ …
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General formula for a square in complex numbers

I need to find a general formulae for a square, with its interior included, in terms of complex numbers. Note that your general square should have (general centre, side-length and orientation.) I do not know how to deal with the orientation i.e.…
Student
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showing that $(z_1^2z_2)^2$ is a real number

Given $z_1=a+bi,z_2=c+di,\frac{b}{a}=\frac{d}{c}=\frac{1}{\sqrt3}$, $a,b,c,d$ are real numbers; $z_1,z_2$ are complex numbers. Need to prove that $(z_1^2z_2)^2$ is a real number. So i figured that…
debi
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Prove there is no complex z such that $|z|=|z + i\sqrt5| = 1$

This is a question in introduction to pure mathematics. I am pretty sure I am close to the answer but I can't quite decide why this proves that there is no complex numbers: $$|z| = |z + i√5| = 1$$ $$\sqrt{cos²Θ + (isinΘ + i\sqrt5)²} =…
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How do I reduce (2i+2)/(1-i) with step-by-step please?

I need a step by step answer on how to do this. What I've been doing is converting the top to $2e^{i(\pi/4)}$ and the bottom to $\sqrt2e^{i(-\pi/4)}$. I know the answer is $2e^{i(\pi/2)}$ and the angle makes sense but obviously I'm doing something…
Bourezg
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viewing complex numbers as a linear transformation

I am studying an intro to complex analysis and geometry book in order to become more adept with complex numbers and hopefully eventually the basics of complex analysis. I love the explanations but I am having a lot of trouble with the exercises. The…
mark leeds
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solving complicated complex numbers

Find values for $a$ and $b$ so that $z=a+bi$ satisfies $\displaystyle \frac{z+i}{z+2}=i$. Below are my workings: so far i simplify $\displaystyle \frac{z+i}{z+2}=i$ to $z=zi+i$ which $a=i$, $b=z$
carry
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Real part of a quotient

Is there some fast way to know the real part of a quotient? $$\Re\left(\frac{z_1}{z_2}\right)$$ $z_i\in \mathbb{C}$
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Finding all roots in a complex system/equation

I dont understand where to begin, or how to approach this question. it asks: find all the roots of: $$(1 + \sqrt{3}i) ^{1/2}$$ should I put it into polar form first? $$z = re^{ix}$$ what throws me off on this question is that it is raised to the…
JLL
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Unit circle traversed once in the punctured plane not homotopic to the same circle traversed twice.

On page 185 of Saff, Fundamentals of Complex Analysis, the author states that in the punctured plane ($\mathbb{C}- \{0\}$), the unit circle traversed once in the positive direction is NOT continuously deformable to the unit circle traversed twice in…
BlueBuck
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Principal (and secondary) square roots of a complex number

This is a follow-up of the post here: using phasors to handle complex numbers I have decided to create a new post as now I am considering a deeper issue. Say if we want to compute $\sqrt{-5}$. If I want to find its principal square root then I can…
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New numbers, same rules

I have been studying complex numbers. But I don't understand why we are following almost all the rules of real number system, since this is a new system. Why so?
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Expressing polar complex numbers in cartesian form

I need to express $z = 4e^{-i\pi/3}$ in the form of $x + yi$ and represent it on the Argand diagram. I think that $4 = \sqrt{x^{2} + y^{2}}$ and that $\tan (\pi/3) = y/x$ but I haven't been able to do anything useful with this information... Is this…
jm22b
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