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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Imaginary to the power of fractional number

I am sorry if it's stupid, but I don't know how to calculate imaginary number "$i$" power to fractional number. I know: $$\begin{aligned} i^2 &= -1\\ i^3 &= \;\;\;i \\ i^4 &= -1 \end{aligned}$$ and so on. I know that multiplying a complex number by…
pajczur
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condition for two complex number $a$ and $b$ to be $|a+b| = |a| + |b|$

If $a$ and $b$ are two complex numbers, and $a \neq 0$, then how to show that the condition required for $|a+b| = |a| + |b|$ is $b/a$ is real and non-negative. I did the following and I got stuck $ \hspace{12 mm}|a+b|^2 = (|a| + |b|)^2…
hasExams
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Find the complex roots of the equation and do a check

Find the complex roots of the equation and do a check: $$ x^{2} + (2+i)x-1+7i=0 $$ What I've done: $$D=b^2-4ac=(2+i)^2-4(-1+7i)=4+4i+4-28i=8-24i$$ $$\sqrt{8-24i}=\pm(\sqrt{(640-8)/2}+i(640+8)/2) = \sqrt{316}+18i$$ Here the wrong calculation. And…
xukerey
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Prove: $(\frac{1+i\sqrt{7}}{2})^4+(\frac{1-i\sqrt{7}}{2})^4=1$

$$\left(\frac{1+i\sqrt{7}}{2}\right)^4+\left(\frac{1-i\sqrt{7}}{2}\right)^4=1$$ I tried moving the left exponent to the RHS to then make difference of squares exp. $(x^2)^2$. Didn't get the same on both sides though. Any help?
RiktasMath
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What is the range of $r$ here?

I am new to complex numbers. Let $C_1=\arg(z)=\frac\pi6$ and $C_2=\vert z-i2\sqrt3\vert=r$ be two curves which intersect at two points. Then I want to find the value (may be range) of $r$. How to find that? I assumed $z=a+ib$ then $r=\sqrt{a^2 +…
Fghj
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Find all complex numbers $z$ such that ($z^6 - i) \in \mathbb R$

Find all complex numbers $z$ such that ($z^6 - i) \in \mathbb R$ My solution: Let's set $x^6 = (z - i)^6$. Then $$x^6 = |x| e^{6\theta i} \\ x^6 \in \mathbb R \iff 6\theta = k\pi \land k\in \mathbb Z$$ $$\theta = \frac{k\pi}{6}$$ Therefore $z -…
Aemilius
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$P(z)=(z^n-1)$ has roots $1, a_1,a_2,...,a_{n-1} $. Prove that $(1-a_1)(1-a_2)...(1-a_{n-1})=n$

$P(z)=(z^n-1)$ has roots $1, a_1,a_2,...,a_{n-1} $. Prove that $(1-a_1)(1-a_2)...(1-a_{n-1})=n$ No idea how to prove this. Would really appreciate any help, hints... Thanks in advance :)
kjhg
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Conjugate to $\cos z$ is not equal to $\cos z$ conjugate $\overline{\cos(z)}\neq\cos(\overline{z})$

I have a dilemma. I have a task where I'm supposed to show that $|\cos z|^2+|\sin z|^2=1$ if and only if $z\in \mathbb{R} $. In my argument I have that $$\cos z\cdot\overline{\cos z} + \sin z\cdot\overline{\sin z}=\cos z\cdot \cos\overline{z} + \sin…
Mathiaspilot123
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Why does $\mathrm{cis}\left(2\pi x\right)$ not equal $1$?

$$ \begin{align} \mathrm{cis}\left(\theta\right) &= \cos\left(\theta\right)+i\sin\left(\theta\right) \\ &= e^{i\theta} \end{align}\\ $$$$ \text{Let } \theta = 2\pi x \\ $$$$ \begin{align} \mathrm{cis}\left(2\pi…
Ontonator
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In what quadrant is the complex number $z?$

Problem: For a number $z\in\mathbb{C},$ the following conditions apply: $z+\bar{z}>0.$ $iz+\bar{iz}<0.$ Determine in which quadrant $z$ lies. Attempt: I simply substituted $z=a+bi, \quad a,b>0$ and computed both…
Parseval
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Equality to be proved using Demoivre's theorem

If $\sin (A)+\sin (B)+\sin (C)=\cos (A)+\cos (B)+\cos (C)=0$ then prove that $\sin (2A)+\sin (2B)+\sin (2C)=0$ using Demoivre's theorem. $$\text {Attempt} $$ From above information we know that $e^{iA}+e^{iB}+e^{iC}=0$ squaring we get an…
Archis Welankar
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Argument of $-1+\sqrt3i$

Find the Argument of $-1+\sqrt3 i$. My book has given the answer is $\frac{2\pi}3$. But I got $\frac{4\pi}3$. Since $$\theta =\tan^{-1}(-\sqrt3)=\frac{-\pi}3$$ As it lies in the second quadrant therefore $$\theta…
dr.rise
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How to get the argument of a complex number?

Given an complex number $z=a+bi$, we could find the polar form. For example $z=1+\sqrt{3}i$ has radius $2$ and $\arg z=\frac{\pi}{3}$. My question is: Do I have to memorize all the possible sin and cosine values for every possible argument…
Applied mathematician
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Generalization of conjugate $a-b$ for $a+b$

I wonder whether there is a pattern that goes on and on: $$(a+b)\,(a-b) = a^2 - b^2$$ $$(a+b)\,(a+(-1/2 + i \sqrt{3}/2)b)\,(a+(-1/2 - i \sqrt{3}/2)b) = a^3 + b^3$$ $$(a+b)\,(a+i b)\,(a-b)\,(a-i b) = a^4 - b^4$$ The general product would be as…
user4414
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How $m^4+4 = 0 \Rightarrow m = 1 \pm i,-1\pm i$?

This is given in my module as a part of a problem's solution: $$m^4 + 4 = 0 $$ $$\Rightarrow m = 1 \pm i,-1\pm i$$ I am not getting how this conversion is taking place,could somebody explain?
Quixotic
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