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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Principal and General Argument product/division relations

In books we have seen that $\arg{zw} = \arg{z} + \arg{w}$ (z and w are complex numbers), is here the arg referred to the general argument not concerning the principal one only ? So for principal argument it would be $\def\Arg{\operatorname{Arg}}…
Paracetamol
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$\sqrt{x}+\sqrt{-x}=2$. The answer is $x=-2i$ or $x=\pm2i$

For the prolem(1) $$\sqrt{x}+\sqrt{-x}=2 \qquad (1)$$ I take $\sqrt{-x}$ as $\sqrt{x}\times i$. In the end, I get $$\sqrt{x}=1-i=\sqrt{2} e^{i( \frac{7}{4} \pi +2n\pi)}$$ $$x=-2i=2e^{i( \frac{7}{2} \pi + 4n\pi)}$$ But if I take $\sqrt{-x}$ as…
StRichLee
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How would you find the inverse of a complex function?

Let's say you had a function $f \colon \mathbb{C} \longrightarrow \mathbb{C}$ defined by $$ f(x+ \iota y) = (3+6x)+ \iota (2−3y). $$ How would you go about finding its inverse? This provoked my interest and curiosity, as it seems you can't simply…
The DeadCreator
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Solving $\overline {(z+1+5i)}= 3z+9−9i$

Solve for $z$, and give your answer in the form $a+bi$. $$\overline {(z+1+5i)}= 3z+9−9i$$ I have gotten to where I have: $$z=3\bar z+8+4i$$ But I am stuck here and could use some guidance.
LTD429
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Show that $\frac{3}{5} + \frac{4}{5}i$ number in multiplicative complex numbers field(apart from $0)$ has infinite order

Show that $\frac{3}{5} + \frac{4}{5}i$ number in multiplicative complex numbers field(apart from $0)$ has infinite order and prove that $\frac{1}{\pi}\arctan(\frac{4}{3})$ is irrational. by contradiction $\exists n$ s.t.…
unit 1991
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Help with a complex number equation

I have been trying to solve the following equation: $z^2z^*−3z|z|+2z=0$ (z* is the conjugate of z) I tried as following: $z^2z^*−3z|z|+2z=0$ $z|z|^2−3z|z|+2z=0$ $z(|z|^2−3|z|+2)=0$ And from here I dont really know what I got to do (?). Also I tried…
eojpyd
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how to calculate the cube root of this complex number

Can anyone help me solve this exercise? calculate the cube roots of $\frac{1}{(2-2i)}$ I started by rationalising by doing $\frac{1}{(2-2i)}$= $\frac{1}{(2-2i)}$ * $\frac{(2+2i)}{(2+2i)}$ ---> $\frac{(2(1+i)}{8}$ ----> $\frac{(1+i)}{4}$ then how can…
user972251
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Solving $z^3 =-1$

I need to find the solution of $z^{3} =-1$. Let$$z=[ r,\theta ]$$ then using de Moivre's theorm$$ z^{3} =\left[ r^{3} ,3\theta \right] $$ write the number $-1$ in modulus argument from, $$[ 1,( 2n-1) \pi ]$$ the we can write$$\left[ r^{3} ,3\theta…
RGen
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Modulus of complex number that satisfies $z+\frac{k}{z}=a+bi$

Given that $z$ is a complex number that satisfies $$z + \frac{k}{z} = a + bi$$ for $k,a,b \in \mathbb{R}$, are there any ways to find the modulus of $z$ without solving the equation for $z$?
P. BNIAC
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Show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$

The problem is : if $z$ lies on a circle with diameter having endpoints $z_1$ and $z_2$ then show that $|z-z_1|^2 + |z-z_2|^2 = |z_1 - z_2|^2$ where $z, z_1, z_2 \in \mathbb{C}$. The angle subtended by the diameter on any point on the circle is a…
Alraxite
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First degree complex equations

My question is: are first degree complex equations solved in the same way as "ordinary" first degree equations. For instance $(2+i)z=1+3i$. I would say that $z=\frac{1+3i}{2+i}$. Is this general for first degree complex equations? Thanks.
omem
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Why can't we use this technique to compare complex numbers?

I have heard that you can't compare complex numbers... Well it is true because it's not defined on the real number line, you can't compare imaginary quantities... But I just wanted to know that why can't we compare them using their modulus? For…
Param Varsha
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finding the argument of $1-\cos 2\theta -i\sin 2 \theta$

"Let z = $1-\cos 2\theta -i\sin 2 \theta$, $0\leq \theta \leq \pi$. Find the modulus and argument of z in terms of $\theta$ in their simplest forms." The modulus is pretty easy, I got $2\sin \theta$ which is correct. The problem is, when finding the…
Ray
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Drawing points on Argand plane

The points $5 + 5i$, $1− 3i$, $− 4 + 2i$ and $−2 + 6i$ in the Argand plane are: (a) Collinear (b) Concyclic (c) The vertices of a parallelogram (d) The vertices of a square So when I drew the diagram, I got an rectangle in the 1st and 2nd…
joey rohan
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Minimum of $|z^3+w^3|$ given that $|z+w|=10$ and $|z^2+w^2|=11$.

I'm kind of stuck at this. My progress so far: Suppose $w,z\in\mathbb{C}$ are such that $|w+z|=10$ and $|w^2+z^2|=11$. Let $a=z+w$ and $b=z^2+w^2$. One can observe that $\frac{a^2-b}{2}=zw$, so we…
Bryan Castro
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