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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.

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Difference between $|z^n|$ and $|z|^n$ where $z$ is a complex number

I got a question asking to prove that $|z^n|$ and $|z|^n$ are the same thing. But what is the difference between keeping the exponent in or out of the modulus?
7up234
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Shading the inequality $|z|<|z-2i|$

$|z| < |z-2i|$ Help. I know that each of the absolute values represents a circle with a radius that is unknown. However, I do not understand how to interpret the inequality and shade the corresponding region on the diagram.
Shooting Stars
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Is the imaginary unit equivalent to its negative counterpart as far as algebra goes?

My math professor made what appears to me to be a sweeping remark that if we have a long and complicated equation involving the imaginary unit 'i', and we replace it everywhere with '-i', as factors of terms and even in the exponents,the equation…
Ad Nazir
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Basic complex number property: $z\in \mathbb{C}$, $|z|>0$ if and only if $z\neq 0$ and $|z|^2=z\bar{z}$

what I've done, let $|z|=\sqrt{x^2+y^2}>0$ then we can not have x,y equally zero hence $z\neq 0$ , now $z\bar{z}=x^2+y^2=|z|^2$. conversely, let $z\neq 0$ and $|z|^2=z\bar{z}$, then $x=0,$or $y=0$ but not the at the same time,hence $|z|>0$.
Remu X
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An inequality problem with complex numbers

Knowing that $p$ and $q$ are complex numbers, $|p| < 1$ and $|q|<1$ show that $|\frac{p - q}{1 - q\bar{p}}| < 1$. I've tried to write: $p=x + yi$ and $q=a+bi$ which led me to $x^2 + y^2 + a^2 + b^2 < 1 + (x^2 + y^2)(a^2 + b^2)$ after some algebra,…
Thiago
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Find the image of the parabola $x = \frac{9}{4} − \frac{y^2}{9}$ under the principal square root mapping $w = z^{1/2}$ with $z \in \mathbb{C}$.

Find the image of the parabola $x = \frac{9}{4} − \frac{y^2}{9}$ under the principal square root mapping $w = z^{1/2}$ with $z \in \mathbb{C}$. Definition of principal square root mapping: $z^{1/2} = \sqrt{|z|}e^{i\operatorname{Arg}(z)/2}$. Arg is…
user796511
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Why can the $\sqrt{-1}$ not be a real number, but the $\sqrt{i}$ can just be complex?

If I understand correctly, imaginary numbers were invented in order to expand the domain of the square root function into the negative numbers. Curiously though, no such expansion from the complex numbers is necessary to define the square root (or…
AEStudent51345
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A geometric question on complex numbers

Suppose $z_1,z_2,\cdots,z_n$ be $n$ complex numbers and $$ r=\max | z_i-z_j|,\;i,j=1,2,\dots,n\;i \neq j.$$Further let $$z=\frac{z_1+z_2+\cdots+ z_n}{n}.$$ Is it true that for all $k=1,2,..,n$,$$ |z-z_k| \leq r?$$ If so how can we prove it? Any…
AgnostMystic
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Let $z$ be a complex number, s.t. $z^2-z+1=0$. Find $z^5 + z$.

Let $z$ be a complex number, s.t. $z^2-z+1=0$. Find $z^5 + z$. As $z= z^2+1$, so $z\neq 0$, also $z^2\neq 0$, as $z\neq 1$. Anyway, we can divide $z^2-z+1=0$ by $z$ to get: $z+\frac1z-1=0$ We need to derive $z^5 + z$ in terms of $z+\frac1z$, so on…
jiten
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multiplying two multi-valued quantities

We know that $\sqrt {-1}=\{i,-i\}$, then $\sqrt {-1} \sqrt {-1}=\{i,-i\}\{i,-i\}=\{1,-1\}$ or $\sqrt {-1} \sqrt {-1}=\{i^2, (-i)^2\}=\{-1\}$?
Arvind
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Singularities of $f(z)=\dfrac{z}{\sin\pi z^{2}}$?

Consider the function $f(z) = \dfrac{z}{\sin\pi z^{2}}$. It has a simple pole at $z=0$. There are also 4 poles at $z=\pm\sqrt{n}$ and $z=\pm i\sqrt{n}$, where $n\in Z^{+}$. What is the order of the pole?
Bapi Basak
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Showing the value the expression takes using another approach

Let $1, \omega, \omega^{2}$ be the cube roots of unity. Then the product $$ \left(1-\omega+\omega^{2}\right)\left(1-\omega^{2}+\omega^{2^{2}}\right)\left(1-\omega^{2^{2}}+\omega^{2^{3}}\right) \cdots\left(1-\omega^{2^{9}}+\omega^{2^{10}}\right) $$…
ProblemDestroyer
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Determining a complex number

I started learn complex numbers and solve complex numbers. I tried to do this example but I don't really know and don't understand how to do it. I know that $z = a + bi$ $a= \text{Re}$ and $b=\text{Im}$ Determine the complex number $$ if $_1= 2 +…
VTack
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Modulus power rule

I am not sure whether the following equality holds for complex number: Let $x$ be complex number and $n$ be positive integer. Then $|x^n|=|x\cdot …\cdot x|=|x|\cdot …\cdot |x|=|x|^n$. I think the case in real is trivial and I have seen someone tried…
sym sym
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How to express the solution in exponential form and plot on the complex plane

So the equation is $$\frac{z^4}{i^{15}+i^{80}}=-1$$ I`ve tried to solve it by this way $$z^4+iz^4=-2$$ => $$(1+i)z^4=-2$$ then I multiplied both sides by $$(1+i) => z^4=-1+i$$ From this I have $$\sqrt{2}e^{i(3\pi/4)} = z^4.$$ Then I got…
Ulan Duishenaliev
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