Mathematics Stack Exchange
Mathematics Stack Exchange

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 conjugate to the power proof.

How can I proof that: $$(z^n)^* = (z^*)^n$$ Where: z is a complex number, n is a positive whole number * is the complex conjugate
user100545
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Writing $((i-\sqrt3 )/(i-1))^{24}$ in algebrical form

$$ \left(\frac{i-\sqrt3 }{i-1}\right)^{24} $$ is to be written in algebrical form. I'd like to forget about the exponent for the moment, and focus on the base. If I'm right it can be rewritten…
Fra H
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What property of complex numbers allow us to represent it in the plane?

Why can trigonometry as a geometrically defined concept be used to algebraic operations between complex numbers? What connects the two things together and how ?
trashbin4
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Minimum Value of a complex expression

Find Minimum value of $\dfrac {\mathrm{Im}~ z^5} {(\mathrm{Im}~ z)^5}$ for $z = x+iy$. Here I started with $z = r \exp(i \theta)$ solve the expression finally i got the answer in terms of $\tan$ which i find little difficult to get along with
Milan Amrut Joshi
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Convergence of the power of a complex number

Given any complex number $z$, I am interested in investigating the convergence properties of $z$ to some power $n$. We have three cases: i) If $|z| <1$, then we can write $z^n=\mathrm{e}^{nx}\mathrm{e}^{iny}$. Letting $n \to \infty$ we get that…
Numbersandsoon
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Value of expression $\frac{1}{1+z^{8k}+z^{16k}}$

If $z$ is a complex number such that $z^{23}=1.$ Then $\displaystyle \sum^{22}_{k=1}\frac{1}{1+z^{8k}+z^{16k}}$ What I try : Using $z^{23}=1\Longrightarrow z^{24}=z\Longrightarrow z^{24k}=z^k$ So $\displaystyle…
jacky
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If $w=\operatorname{cis}\frac{2\pi}n$, then how can i prove by induction that $1+2w+3w^2+⋯+nw^{n−1}$ is equal to $\frac n{w-1}$?

I tried to solve it in the simple way by adding on both sides $(n+1)w^n$ but I was unsuccessful
Alexandre Barros
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Equilateral triangle in complex plane

Prove that the points $a_1,a_2,a_3$ are vertices of an equilateral triangle if and only if $a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_3a_1$. I rewrite the equation as $2a_1^2+2a_2^2+2a_3^2-2a_1a_2-2a_2a_3-2a_3a_1=0$, which is…
Paul S.
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Maximum modulus principle.

Find the maximum value of $|f(z)|$ on the closed complex disk of radius 2, where $f(z)$$=$$z^4\over{z^2+10}$. Usually I approach these problems by calculating the modulus squared and simplifying, but here it seems it will slow things down a bit.
Johnny Apple
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Complex solutions to $1^x = 2$

I saw a YouTube video where the presenter claimed that there were solutions to the equation $1^x = 2.$ To summarize their argument: Clearly, $1^x = 1$ for all real $x.$ However, there may be complex solutions. $$ \begin{align} &1 = e^{2k\pi i}…
user317176
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Simplify $(1-\sqrt{11}i)\sqrt{5-\sqrt{11}i}$

Simplify $(1-\sqrt{11}i)\sqrt{5-\sqrt{11}i}$ My solution: By using half angle formula of the tangent function for the second number in the product we…
Bob Dobbs
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For distinct non-zero complex $z_1$, $z_2$, $z_3$ satisfying $|z-1|=1$ and $z_2^2=z_1z_3$, which of the following are true?

$z_1, z_2, z_3$ are three non zero distinct points satisfying $|z-1|=1 \space \& \space z_2^2=z_1 z_3$ then $\qquad$ (A) $\displaystyle\frac{z_3-z_2}{z_2+z_3-2}$ is purely imaginary $\qquad$ (B)…
Samar Imam Zaidi
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$a,b,c$ be three complex numbers satisfying $|a|=|a-b|=3\sqrt3,|a+b+c|=21$,$a^2+b^2+c^2-ab-bc-ca=0$. Find $|b|^2+|c|^2$

Let $a,b,c$ be three complex numbers satisfying $$|a|=|a-b|=3\sqrt3\\|a+b+c|=21\\a^2+b^2+c^2-ab-bc-ca=0$$ Find $|b|^2+|c|^2$. I started by squaring the first relation to get $|b|^2=a\bar{b}+b\bar{a}.$ Then $$|a+b+c|^2=|a|^2+|b|^2+|c|^2+a\bar b…
Maths
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Difficulty dealing with lots of complex numbers

I've been trying to solve this question for a while, tried different approaches, but I can't get to the answer. The question is: Define $\, \, \,e^{i\beta \Phi}=\rho \left ( \dfrac{\tau_0}{\tau_1} \right )^2$ with $\Phi = \Phi_1 + i\Phi_2$, and…
LeviT
  • 35
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Prove the complex conjugate

If $w$ and $z$ are two complex number, prove that $$\left(\frac{w}{z}\right)^* = \frac{w^*}{z^*}$$ My attempt:$$\left(\frac{w}{z}\right)^* = \left(w \cdot \frac{1}{z}\right)^*$$ $$\left(w \cdot \frac{1}{z}\right)^* = w^* \cdot…
James Thomson
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