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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Deducing the square root of $2i$ from the square root of $i$

I'm studying a set of lecture notes on complex numbers, which derive the fact that $\sqrt{i} = \pm \frac{1 + i}{\sqrt{2}}$. I'm fine with this result, but they then comment that, knowing this fact, we can deduce that $\sqrt{2i} = \pm (1 + i)$. I'm…
Cardinality
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Why there are exactly $n$ distinct $n$-th roots?

I was solving the following exercise: Given a complex number $z \neq 0$, write $z = re^{i\theta}$ where $\theta = \arg(z)$. Let $z_1 = Re^{i\alpha}$, where $R = r^{1/n}$ and $\alpha = \theta/n$, and let $\epsilon = e^{2\pi i/n}$, where $n$ is a…
Gold
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Let $\omega=\frac{\sqrt{3}i-1}{2}$. Prove $\frac{\omega a-\omega}{1+a+\omega}\in\Bbb{R}$ for all $a\in\Bbb C$ with $|a|=1$, $a\not=\omega^2$.

$\newcommand{\i}{\mathrm i}$Let $\omega=\frac{\sqrt3\i-1}2$, and $a\in\Bbb C$ with $|a|=1$, $a\not=\omega^2$. Prove that $z=\dfrac{\omega a-\omega}{1+a+\omega}\in\Bbb R$. There's an obvious bash: let $a=\cos x+\i\sin x$. So \[\frac{\omega…
youthdoo
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Let $z\in\mathbb{C}, z\ne0$, if $\left|z-\frac{1}{z}\right|=2$, then what is the maximum value of $|z| ?$(Solve preferably without using geometry)

Let $z$ be a non-zero complex number such that $\left|z-\frac{1}{z}\right|=2$. What is the maximum value of $|z| ?$ (Solve preferably without using geometry.) My attempt: I tried to simplify things down by writing $\left|z-\frac{1}{z}\right|=2$ as…
Stephen Smith
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Complex number inequality on a disk

Let $z_0 \in D(0,1) \subseteq \mathbb{C}$ and $\epsilon_0>0$ such that $\bar D(z_0, \epsilon_0) \subset D(0,1)$. Show that $\forall z \in \bar D(z_0, \epsilon_0), \forall t \in [0,1]: |1-zt|\ge 1-\epsilon_0-|z_0|>0$ I have been struggling with…
TOP STIN ELLADA
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Universal property of complex numbers

I know that $\mathbb R$ is - up to a unique isomorphism of ordered fields - the unique complete ordered field. Is there a similar characterization of the complex numbers? I guess that $\mathbb C$ can not simply be defined as a field with particular…
Filippo
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Complex Numbers, Argand Diagrams

Is it possible to tell if |3-4i| is greater than |1+2i| based entirely from viewing their plotted positions on an Argand diagram?
Roy Sheehan
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Complex number $\ln(i)$ into rectangular form?

I am going through the Mathematics of the DFT book by Julius O. Smith III. One of the questions ask the following: How would you convert the complex number $\ln(i)$ into the form $x + yi$, where $x$ is the real part and $y$ is the imaginary…
edderic
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Can the rule $\sqrt{xy}=\sqrt{x}\sqrt{y}$ over complex numbers be salvaged through multivaluation?

Wikipedia mentions the following arithmetic fallacy: $$x<0 \land y <0 \land \sqrt{x \times y}= \sqrt{x} \times \sqrt{y}, $$ since this would lead to $-1=i^2=1$. So, the above rule is ommitted from the rules of arithmetic of complex numbers. Now, if…
Zuhair
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Expression should evaluate to 1 but I get 1 only for certain cases

The expression is $(a - j \mu_1 b) \div (z - \sqrt{z^2 - a^2 - \mu_1^2 b^2})$ $z = a$, the above expression reduces to $(a - j \mu_1 b) \div (a - \sqrt{- \mu_1^2 b^2})$ One way to refactor the above expression is using $j^2 = -1$, the above…
user3873617
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$|z_1+z_2+…+z_n|\ge \frac{1}{\sqrt{1+t^2}}(|z_1|+|z_2|+…|z_n|)$

Let $z_1,z_2,…,z_n\in \mathbb C$, such that $|y_i|\le tx_i$ for some positive number $t$ ($z_i=x_i+iy_i$). I want to verify the inequality $|z_1+z_2+…+z_n|\ge \frac{1}{\sqrt{1+t^2}}(|z_1|+|z_2|+…|z_n|)$. I found that…
user1007173
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How to find all values of $z$ such that $z^3=−8$

The question is Solve the equation $z^3=-8$ My attempt I attempt to write it out in polar co-ordinates since $z = r(\sin(x) + i\sin(x)) \\ z^3 = r^3(\cos(3x) + i\sin(3x))$ then $r^3\sin(3x) = -8$ and $r^3\cos(3x) = 0$ but from here I'm not really…
hngyb hjynu mgbtv f
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Help plotting an Argand Diagram

I am stuck on a past exam question. I don't have a clue what it's on about and would appreciate any help. In this question, $w$ denotes the complex number $cos{\frac{2}{5}\pi} + isin{\frac{2}{5}\pi}$ i) Express $w^2$, $w^3$ and $w^*$ in polar form,…
Lerp
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Is $\sqrt{-a -ib} = i\sqrt{a + ib}$ true?

I am grade 12th student. I am learning complex numbers through recorded lectures of my mentor. He taught us that $\sqrt{ab} = \sqrt{a}\sqrt{b}$ if atleast one of them is positive. He told us that this operation can't be performed on imaginary…
user12137152
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De Moivre's Theorem: prove that $\sum_{j=0}^{50}(-1)^j\binom{101}{2j}\cos^{101-2j}(\frac {\pi}{101})\sin^{2j}(\frac {\pi}{101})=-1$

(#):Suppose $\theta$ is a real number, and $z=\cos(\theta)+i\sin(\theta).$ Then for any $m \in \Bbb Z, z^m+\bar z^m=2\cos(m\theta)$ and $z^m-\bar z^m=2i\sin(m\theta).$ (b): Applying statement (#), prove that…
sunny
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