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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.

19229 questions
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Is $(2+i)^n + (2-i)^n $ a real number ($\in \Bbb R$)?

The question: $$\forall n \in \Bbb N$$ is the number $$(2+i)^n+(2-i)^n$$ in the real numbers ($\Bbb R$)? My try for solution using Newton binom: $$(2+i)^n = \sum_{k=0}^{n}\binom{n}{k}2^{n-k}i^{k}$$ $$(2-i)^n =…
Alon
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Let $z$ be a complex number of maximum amplitude satisfying $\vert z-3\vert=Re(z)$

Let z be the complex number of maximum amplitude (argument) satisfying $$|z-3|=Re(z),$$ then I need the value of $|z-3|$ So I proceeded with substituting $z=x+iy$, and got the following:- $$\sqrt{(x-3)^2+y^2}=x$$, and squaring it I got the equation…
Techie5879
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Difficult exam question - finding complex numbers from a quadratic.

This appeared in a past exam paper for the complex calculus course I am taking. I am required to find all solutions for it and plot them in the complex plane. Below is my working. I stopped before calculating $z$ because it clearly won't get…
user244685
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Real part of a complex number

Compute $\operatorname{Re}\left(\frac{1}{z+1}\right)$ when when $|z| = 1$. The only way I could think to go about this is to simply go by definitions. If $z\in \Bbb C$, then $z\bar z$ = $|z|^2$. Now $$z=\frac{|z|^2}{\bar z}$$and$$z=\frac{1^2}{\bar…
In the blind
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complex number - modulus problem

If $|z_1+z_2|=|z_1|+|z_2|$ where $z_1 ; z_2$ are different non zero complex numbers, then (a) $Re(\frac{z_1}{z_2})=0$ (b) $Im(\frac{z_1}{z_2})=0$ (c) $z_1+z_2=0$ Please guide how to proceed...
Sachin
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Finding the conjugate of a complex number

We know that the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign.  So, given a certain complex number, it is possibile to find its conjugate by writing it…
Kinka-Byo
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Prove $3|z-1|^2 = |z+1|^2 \iff |z-2|^2 = 3$

I'm struggling with this prove question. I tried starting with let $z = x + iy$, and substituting the three in the first eqn with $|z-2|^2$, with the intention of eventually equating the LHS with the RHS, and I ended up with something most likely…
CuandoNotebook
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Double inequality containing the argument of a complex number and two angle values

Thanks for taking your time to read this question. I have come across an exercise which requires me to plot the area which satisfies these two conditions: $$ |z|^2 -5|z| +6<0$$ $$ \frac{π}3≤arg(z)≤π$$ Now, I know how to solve the first one but…
l0ner9
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Trouble figuring out a complex number equation

I'm struggling to prove this question, an help is greatly appreciated! If $a=cis(\pi/5)$, prove that: $$a^7=-a^2$$ and $$a^9=-a^4$$
spuddy
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Show that $z(t)=\frac{1+it}{1-it}$ describes a circumference on the complex plane.

I am asked to prove that $z(t)=\frac{1+it}{1-it}$ describes a circumference on the complex plane when $t$ takes every value in the extended real number line. That is, $\mathbb{R} \cup\{\pm\infty\}$. I don't have any idea how to proceed with this.…
user403851
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Complex conjugate of a variable raised to the power $n$

What would be the complex conjugate for these three. Assuming $i$ is always $${\sqrt{-1}}$$ $$i^{11}$$ $$(2-3i)^3$$ $$\frac{3-i}{2i+5}$$
user1730308
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why is $i$ not like $0$?

Consider the following proof: $$e^{i(\theta+2n\pi)}=e^{i(\theta)}$$ where $n$ is a non-zero integer $$ i(\theta+2n\pi)=i(\theta) $$ $$ \theta+2n\pi=\theta $$ $$ 2n\pi=0 $$ From the answers that I have read so far, it seems the error is introduced in…
o c
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If $S(n)=i^n + i^{-n}$ n is a positive integer, then the total number of distinct values of $S(n)$ are

I wrote it in the form $$S(n)=\frac{i^{2n}+1}{i^n}$$ $$=\frac{(-1)^n+1}{i^n}$$ If n is odd, then the value will be zero. If n is even then it will become $$\frac{2}{-1}$$ So I can find only 2 distinct values of $S(n)$ but the answer says there are…
Aditya
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The real part of $e^{e^{i\theta}}$?

I know that a solution for this question exists on SE, I have seen it, but it doesn’t satisfy my query. I couldn’t understand the solution. Here is an excerpt from one of the answers $$z = e^{e^{i \theta} }$$ $$\implies z=e^{\cos \theta +i\sin…
Aditya
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3 answers

If $\omega =\cos 40 + i\sin 40$ then $|\omega + 2\omega ^2 + 3\omega ^3....9\omega ^9|^{-1}$

If $\omega =\cos 40 + i\sin 40$ then $|\omega + 2\omega ^2 + 3\omega ^3....9\omega ^9|^{-1}$ All angles are in degrees I will be writing $\omega$ as w to make it easier to type From the value of $w$ it is clear that $w^9=1$ Solving the arithmetic…
Aditya
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