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
2
votes
2 answers

Complex number equations ( process)

Among the exercises I was solving these are the ones I don't understand and I don't know if the process or solutions are ok, so please correct whenever I went wrong. Thanks everyone. (I'm supposed to solve by converting to trigonometric form when…
user484696
2
votes
1 answer

Locus of a complex number

Let $f$ be a mapping of ℂ in ℂ and M be a point of affix $z= x+iy$. Find the set of points M, such that:$f(z)$ is a pure imaginary number and $f(z)=z^2 + z + 1$. I'm new to complex numbers but what I can imply from this question is that $z^2 + z +…
A.Tea
  • 27
2
votes
2 answers

How do I determine which complex value corresponds to which term

Given that $z$ is a non real cube root of 1. Find the exact values of $a = (1+2z+3z^2)$ and $b= (1+3z+2z^2)$. I ended up getting $a+b = =-3$ and $a*b=3$. Thus solving simultaneously I conceived $z=\frac{-3±i\sqrt3}{2}$. The problem is I am not…
allquiet1984
  • 149
2
votes
3 answers

How to solve the complex equation $ω^2=-11/4+15i$

The question is stated as following: "First, solve the equation: $ω^2=-11/4+15i$ and after, with the help of that, solve: $z^2-(3-2i)z+(4-18i)=0$" The problem for me lies in solving the system of equations for ω; $Re:a^2-b^2=-11/4$ and…
J. Dilla
  • 21
2
votes
2 answers

Solving a complex equation more efficiently

I want to solve: $(z+2-i)^6=27i$ My thought was expressing it as: $(z+2-i)^6=27 e^{i(\frac{\pi}{2}+ 2\pi k)}$ where $k \in \mathbb{Z}$ $(z+2-i)=\pm \sqrt{3} e^{i(\frac{\pi}{12}+ \frac{1}{3}\pi k)}$ But if I now try to write this in standard form It…
user459879
2
votes
2 answers

Graph complex inequality $|1/z|<1$, when $z \in \mathbb{C}$

Problem Graph complex inequality $|1/z|<1$, when $z \in \mathbb{C}$ Attempt to solve if i set $z=x+iy$ when $(x,y) \in \mathbb{R}, z \in \mathbb{C}$ $$ |\frac{1}{x+iy}|<1 $$ Trying to multiply denominator and nominator with complex conjugate. $$…
Tuki
  • 2,237
2
votes
3 answers

How to solve the equation $(z+1)^7 = z^7$ for all $z$?

Find all $z$ for the equation $$(z+1)^7 = z^7$$ The different solutions can be unsimplified and both rectangular or exponential. I have the lead that I subsitute the $z+1$ term with a root of unity. Then, I got lost when I looked at that $z^7$.…
ilikepi314
  • 181
2
votes
2 answers

compute $(\sqrt 3 + i)^{14} + ( \sqrt 3 - i)^{14}$

I want to compute $(\sqrt 3 + i)^{14} + ( \sqrt 3 - i)^{14}$. My attempts: I was thinking about De Moivre's theorem $$(\cos\theta + i \sin\theta)^n= \cos(n\theta) + i \sin(n\theta)$$ but I don't know how it can be applied here. Any…
jasmine
  • 14,457
2
votes
0 answers

Algebaric closure - Are polynomials with complex _powers_ also complex?

If I'm understanding correctly, the field of real numbers is not algebraically closed, because you can set up a polynomial with real coefficients that has roots that are not in the reals. (e.g. the roots of ${x^2 + 1 = 0}$ is not in the reals, since…
Gavin Smith
  • 21
2
votes
2 answers

Writing $(\sqrt{3}+3i)^{18}$ and $(i-1)^{-11}$ in the form $a+bi$

I have two current math problems I just can't solve. I'm to express the following in the form $a+bi$: $(\sqrt{3}+3i)^{18} \qquad\text{and}\qquad (i-1)^{-11}$ The answer for the first one is $12^9$; and for the second, it's $\frac{1}{64}(1-i)$. The…
j.doe
  • 19
2
votes
4 answers

Given that $z=\cos\theta+i\sin\theta$, show that $Re\left(\frac{z-1}{z+1}\right)=0, \quad z\ne-1$

Given that $z=\cos\theta+i\sin\theta$, show that $Re\left(\frac{z-1}{z+1}\right)=0, \quad z\ne-1$ For this question I had to show that the real part of $\frac{z-1}{z+1}=0$ To find that I first substituted $z$ with $\cos\theta+i\sin\theta$ to…
Pablo
  • 548
2
votes
1 answer

Prove that $\frac{1}{2}+ \sum_{k=1}^n \cos (k\theta) = \sin((n+ \frac{1}{2})\theta)/2\sin\frac{\theta}{2}$

Suppose $\sin \frac{\theta}{2} \neq 0$ . Prove that $$\frac{1}{2}+ \sum_{k=1}^n \cos (k\theta) = \frac{\sin[(n+ \frac{1}{2})\theta]}{2\sin\frac{\theta}{2}}$$ The question also give the hint, $$z=\cos\theta + i\sin\theta =…
Steve
  • 397
2
votes
2 answers

Compute $(1+\alpha^4)(1+\alpha^3)(1+\alpha^2)(1+\alpha)$ where $\alpha$ is the complex 5th root of unity with the smallest positive principal argument

I just started on the topic Complex Numbers and there is a question that I am stuck on. The question is: If $\alpha$ is a complex 5th root of unity with the smallest positive principal argument, determine the value of …
Steve
  • 397
2
votes
5 answers

Raising complex number to high power - Cartesian form

My question is about raising a complex number to a high power, I know how to do that with De Moivre law, but i need to get the result in cartesian form, like $z=x+iy$. and without trigonometric terms. The problem exactly is: Write the following…
D.Rotnemer
  • 459
2
votes
3 answers

Existance of solution for a simple complex equation?

Does any $u \in \mathbb{C}$ exist such that: $$\frac{u}{\sqrt{-u^2}}=1$$ If yes, give an example please. UPDATE: OK, I thought a little about that myself and I think it goes like this ($m,n\in\mathbb{N}$ and…
bollty
  • 21
1 2 3
…
99 100