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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Does $a+ib=0$ imply $a=b=0$?

If we have a complex number number $a + ib = 0$, then we can write as $a + bi = 0 + 0\cdot i$. Equating corresponding terms, we get $a = b = 0$. What is this? Where am I going wrong ?
shraman shraman
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For what values of $n$ is $(e^{i(\frac{2\pi}{3})})^n$ real?

So this is my thought process: $$ \begin{align} (e^{i(\frac{2\pi}{3})})^n &= (e^{i(\frac{2\pi n}{3})}) \\ &= \cos{(\frac{2\pi n}{3})} + i \sin{(\frac{2\pi n}{3})} \\ \end{align} $$ This is real when $$ \begin{align} \sin{(\frac{2\pi n}{3})} &= 0…
Skeleton Bow
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Proof of Euler's theorem in complex numbers

I am an engineer and not a mathematician, but I have used the formula stated below many number of times.$$e^{i\theta} = \cos{\theta} + i\sin{\theta}$$ Just out of curiosity, what is the proof to this theorem? I know that adding the Taylor series of…
ubuntu_noob
  • 537
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Solve the equation $z^{3}=18+26i$, where $z=x+yi$ and $x,y$ are integers.

$(x+yi)^{3}=(x^{3}\text{−}3xy^{2})+(3x^{2}y\text{−}y^{3})i=18+26i$ which leads to $$\begin{cases} x^{3}\text{−}3xy^{2} & =18\\ 3x^{2}y\text{−}y^{3} & =26 \end{cases}$$ The problem here is that I don't know to precede further. I've been told to…
mohamez
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If $\sum\limits_{j=1}^n | w_j|^2 \leq 1$ implies $ \left| \sum\limits_{j=1}^n z_j w_j \right| \leq 1$, then $\sum\limits_{j=1}^n | z_j|^2 \leq 1$

I want to show following. Fix $n$ a positive integer. Suppose that $z_1, \cdots z_n$ are complex numbers satisfying \begin{align} \left| \sum_{j=1}^n z_j w_j \right| \leq 1 \end{align} for all $w_1, \cdots, w_n\in \mathbb{C}$ such that…
phy_math
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$(z-1)^n+ (z+ 1)^n=0$?

Let $n$ be a positive integer, and let $z∈\mathbb{C}$ satisfy $(z-1)^n+ (z+ 1)^n=0$. a, I have to show that $z = (1+w)/(1-w)$, where $w^n = -1$ b, Show that $w \bar w=1$ c, Deduce that $z$ lies on the imaginary axis. I got the a part by rearranging…
szantamano
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How to break $\frac{1}{z^2}$ into real and imaginary parts?

$$ \frac{1}{(x+iy)^2}=\frac{1}{x^2+i2xy-y^2}=\frac{x^2}{(x^2+y^2)}-\frac{2ixy}{(x^2+y^2)}-\frac{y^2}{(x^2+y^2)}$$ So I thought I could just say: $$ Re(\frac{1}{z^2})=\frac{x^2}{(x^2+y^2)}-\frac{y^2}{(x^2+y^2)}$$ and $$…
User3910
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What is the square root of $3 + 2\sqrt{10}i$?

I need to compute the square root of $3 + 2\sqrt{10}i$. I know how to solve it, but for some reason I'm not getting the correct answer. I attempted to solve it like this: $$ \sqrt{3 + 2\sqrt{10}i} = x + iy \quad \longrightarrow \quad 3 +…
jimpix
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If $\frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then $|z_{1}+z_{2}+z_{3}|$

If $z_{1},z_{2},z_{3}$ are three complex number such that $|z_{1}| = |z_{2}| = |z_{3}| = 1$ and $\displaystyle \frac{z^2_{1}}{z_{2}z_{3}}+\frac{z^2_{2}}{z_{3}z_{1}}+\frac{z^2_{3}}{z_{1}z_{2}} = -1.$Then possible values of…
juantheron
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Express real number in exponential form

Express $ z=-32 $ in exponential form. My reasoning: $ z=-32 $ is the same as $ z=-32+0i $ Exponential form should look like $ z=Re^{\theta i} $ $ R =\sqrt{(-32)^2 + 0^2} = 32 $ $ \theta = \tan^{-1}(\frac{0}{-32}) = 0 $ So answer becomes: $ z =…
Edward Ruchevits
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When is $(2cis\frac{2\pi}{3})^n$ real?

When is $(2cis\frac{2\pi}{3})^n$ real? Using de Moivre's theorem: $$(2cis\frac{2\pi}{3})^n = 2^ncos(\frac{2\pi}{3}n) + i2^nsin(\frac{2\pi}{3}n)$$ $$\therefore sin(\frac{2\pi}{3}n) = 0 = sin(0), sin(\pi), sin(2\pi)...$$ $$\therefore \frac{2\pi}{3}n…
StopReadingThisUsername
  • 1,553
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Find $w$ such that $w^8=15-15i$

Find the complex number, lying in the second quadrant, and having the smallest possible real part, which satisfies the equation $$w^8=15-15i$$
ryantata
  • 53
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How can I calculate $(1+i)^{5404}$?

I saw a pattern while evaluating some other powers of similar complex number so I tried to calculate the above question by expanding it, please tell me if it is correct...? $(1+i)^{2} = 2i$ $(1+i)^{4}$ = $(2i)^2$ = $-4$ $(1+i)^{8} = (-4)^2 =…
shivani
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Taking Mod on both sides, mathematically correct?

When given a equation containing complex numbers such as $$ \frac{a+ib}{c+id} = x + iy$$ and required to prove $$ \frac{a^2 +b^2}{c^2+d^2} = x^2 + y^2$$ Is taking the mod of both sides a legal mathematical step? I ask so because my textbook first…
noobAtMaths
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Linear algebra : Solving $i \cdot\bar{z} = 2 +2i$

$i\cdot\bar{z} = 2+2i$ I know that $\bar{z} = a-bi$ so then i get $i(a-bi)=2+2i$ Then $ai+b=2+2i$ (because $i^2=-1$) When 2 complex numbers are equal you usually can equal their parts Ex: $2+2i=a+bi$ so $a=2$ and $b=2$ But in this case I aint got…
maude
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