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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Solve: $(\bar{z})^4+z^2=16i$

I was trying to solve this equation: $$(\bar{z})^4+z^2=16i$$ but do not know where to start, I tried to carry out the powers, but then I do not know to continue, in my book there is not enough information. where do I start?
malloc
  • 407
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6 answers

Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$

I have a problem to solve this exercise, I hope someone help me. Find all couple $(x,y)$ for satisfy $\frac{x+iy}{x-iy}=(x-iy)$
Salvattore
  • 551
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3 answers

Prove an equation in complex numbers

Prove an equation ($n \in N$, $z \in C$, $a \in R$): $$ z^n + \frac1{z^n} = 2 \cos\alpha n$$ if $$ z + \frac1{z} = 2 \cos\alpha$$ I tried math induction, but did not solve.
Vladislav Brylinskiy
  • 159
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Complex numbers question (finding the Im)

So I've been wrestling with this for about an hour and am still not sure how to solve it. The task is to find the Im part of the complex number (it has to be as short as possible, or else it'd be easy): $$\frac{1}{(z^2 + zi)}$$ for…
Seifer
  • 21
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5 answers

Locating possible complex solutions to an equation involving a square root

Note after reviewing answers. This question illustrates in a non-trivial way how the choice of how to compute the square root in the complex plane can make real difference to the answer you get. In this question finding all the solutions came down…
Mark Bennet
  • 100,194
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7 answers

Solve for $z$ in $z^5=32$

This was the last question on my Year 11 Complex Numbers/Matrices Exam Name all 5 possible values for $z$ in the equation $z^5=32$ I could only figure out $2$. How would I go about figuring this on paper?
VikeStep
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Let $a \in \mathbb{C}$, how many $n$'th roots of $a$ have nonnegative imaginary part?

Let $a \in \mathbb{C}$. It might as well be on the unit circle, so $a=e^{i \theta}$. I'm interesting in finding, for $n \ge 2$, how many $n$-th roots, $\omega_i$ (with $ 0 \le i \le n-1$), have $\Im (\omega _i) \ge0$. Intuitively, the answer should…
Spine Feast
  • 4,770
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Find $n$ where $\displaystyle \bigg|i+2i^2+3i^3+\cdots \cdots +ni^n\bigg|=18\sqrt{2}$.

Suppose that $n$ is a natural number such that $\displaystyle \bigg|i+2i^2+3i^3+\cdots \cdots +ni^n\bigg|=18\sqrt{2}$. Find the value of $n$. What I try : Let $\displaystyle S =i+2i^2+3i^3+\cdots +ni^n\cdots (1)$ Then $\displaystyle iS…
jacky
  • 5,194
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Given that $|z+w| = 1$ and $|z^2+w^2| = 14$, find the smallest possible value of $|w^3+z^3|$.

Given $w$ and $z$ two complex numbers such that $|z+w| = 1$ and $|z^2+w^2| = 14$. Find the smallest possible value of $|w^3+z^3|$, where |.| denotes the absolute value of a complex number, given by $|a+bi| = \sqrt{a^2 + b^2}$. My…
Chhaya gupta
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Complex numbers of modulus 1 Summing to 0

Problem: Let $z_1, z_2, ..., z_n$ be complex numbers with a modulus of $1$. Given the condition $S = \sum_{j=1}^{n} z_j = 0$. Please compute all possible values for $z_j$. My idea: Assume $$z_j = e^{i\theta_j},\quad \theta_j \in [0,2\pi),\quad…
sunset
  • 21
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Periodicity of complex logarithms multiplied by a real scalar

In a problem that I was given it asks to show that $\text{ln}(i^3) = 3\cdot\text{ln}(i)$ where this lower case "$\text{ln}$" is defined as $\text{ln}(z) = \text{log}_e|z| + i\cdot\text{arg}(z)$, since $\text{arg}(z) = \theta+2n\pi$ there is a…
Conor_Meise
  • 195
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4 answers

If $z=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$ show that $z^{3n}+z^{3m+1}+z^{3k+2}=0$ where $k,m,n\in\Bbb N$

If $z=-\frac{1}{2}-\frac{\sqrt{3}}{2}i$ show that $z^{3n}+z^{3m+1}+z^{3k+2}=0$ where $k,m,n$ are natural numbers. I'm not allowed to use Euler's formula. I reached this $$\text{cis}(-\pi n)+\text{cis}(-\pi m - \frac{\pi}{3})+\text{cis}(-\pi k -…
The Bulletbroof Joker
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I am having a doubt in complex numbers.

A question in my text book states as follows: Find all possible values of $\sqrt i + \sqrt{-i}$. Firsly I found the value of $\sqrt i$ which came out to be $\pm(1/\sqrt 2 + i(1/\sqrt 2))$. Then I wrote $\sqrt{-i}$ as $\sqrt{-1} \sqrt{i} =…
Madhav Mittal
  • 21
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2 answers

How to express $|a + b + 2ab|^{2}$ where $a\in\mathbb{C}$ and $b\in\mathbb{C}$ satisfy $|a| = |b| = 1$?

I have recently started a problem related to complex numbers. The topic is new to me, but I understand how do they work though. However there is one concept which is a little bit "intricate" to me, and this is the square of the magnitude of a…
fikooo
  • 117
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Are the two expressions $a \neq \pm i\sqrt{2}b$ and $b \neq \pm i \frac{1}{\sqrt{2}}a$ equivalent?

Are these two expressions equivalent: $a \neq \pm i\sqrt{2}b$ and $b \neq \pm i \frac{1}{\sqrt{2}}a$ ? I would assume the follwing shows that they are: $a \neq \pm i\sqrt{2}b \iff a \frac{1}{\sqrt{2}}\neq \pm ib \iff ia \frac{1}{\sqrt{2}} \neq \pm b…
Spink
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