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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What is the circumference of a complex unit circle?

What is the circumference of a unit circle in a complex coordinate system $ e^{i \phi} $ ?
Moonraker
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For three complex numbers we have $|z_1|=1$ ,$|z_2|=2$ ,$|z_3|=3$ and $|9z​_1z_2 + 4z_1z_3 + z_2z_3|=12$

For three complex numbers we have: $|z_1|=1$ ,$|z_2|=2$ ,$|z_3|=3$ and $|9z​_1z_2 + 4z_1z_3 + z_2z_3|=12$ Then find value of $|z_1 + z_2 + z_3|$ I took $z_1=1(\cos A+i\sin A),z_2=2(\cos B+i\sin B),z_3=3(\cos A+i\sin A)$ but it doesn't help much.…
Ananya
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$z=e^{2\pi i/5}$ solves $1+z+z^2+z^3+z^4=0$

What is the best way to verify that $$1+z+z^2+z^3+z^4=0$$ given $z=e^{2\pi i/5}$? I tried using Euler's formula before substituting this in, but the work got messy real fast.
user321401
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Finding the abs. value and argument of a complex number?

Given the following complex number: I'm asked to find the abs. value and the argument, I found both in rectangular form, is that considered incorrect? Or must I convert it to polar form then solve $|z|$ and $\theta$? I solved each individual…
Pupil
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Equation in complex numbers $z^5=\bar{z}$, equivalence issue

It's a classic problem in many textbooks: Solve in complex numbers the equation $$z^5=\bar{z} \tag{1}$$ The solution is a as follows: We apply the modulus, and obtain that $|z|=0$, so $z=0$, which satisfies and second case $|z|=1$,so…
amarius8312
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Exponential of a complex variable

Can someone please tell me if I am approaching this correctly? Given the following and asked to solve for the complex variable z: $$[e^z]^3-5e^z=0$$ My approach was purely algebraic and is why I have my doubts: $$[e^z]^3=5e^z$$…
NotSoSN
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get magnitude of addition of complex numbers in trigonometric form

My problem is that I have multiple complex number in trigonometric form and I want to add those and get the magnitude of the result. I am aware that the normal route would be to calculate the rectangular form, add the numbers and then I can easily…
Kilsen
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How to justify $a=(a,0)$ in Theorem $\mathbf{1.29}$ in Baby Rudin?

Rudin says in page fourtheen in theorem 1.29 : If $a$ and $b$ are real, then $(a,b)=a+bi$. Proof he gives: $$a+bi=(a,0)+(b,0)(0,1)\\=(a,0)+(0,b)=(a,b)$$ of course this is correct (if we accept $a=(a,0)$), however i'm interested in how to justify…
user153330
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Finding roots of complex number

The problem is specific as an example from hw. But it is more the concept/process I could use clarification on. Given a complex number $$\Big(\frac{-2}{1-i\sqrt3}\Big)^{\frac{1}{4}}$$ Find all possible roots. I know the method is to change into the…
NotSoSN
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What is the motivation behind the solution of this problem involving complex numbers?

The problem is Suppose for three distinct complex numbers $a, b, c$ such that $|a|=|b|=|c|>0$ all of the three numbers $a+bc, b+ac, c+ab$ are purely real. Prove that $abc=1$ By playing with the identities, I came up with a solution. Let…
Gerard
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Pulling a negative out of a square root

The following is invalid, since the operation is not defined when $a, b < 0$: $\sqrt{-1}\sqrt{-1} = \sqrt{(-1)(-1)} = \sqrt{(-1)^2} = \sqrt{1} = 1$. This is not correct, because $ii = -1$. This shows that $\sqrt{a}\sqrt{b} = \sqrt{ab}$ is invalid…
Wesley
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Plotting a complex argument arc

I am having trouble sketching a complex argument arc $$ \text{Sketch the following on an arcand diagram:}\\ \arg\left(\frac{w+1}{w}\right)=\frac{\pi}{6}$$ I've tried to devise a method on my own looking at questions and answers but it has failed…
εν οίδα ότι ουδέν οίδα
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How would one graph $x^2+y^2=-1$

I'm interested in graphing equations with no real solutions. I know in the real plane no such pair $(x,y)$ will satisfy the equation, and I can list out a few solution pairs. However this does me no good as graphing a complex number by itself is…
Ahmed S. Attaalla
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Showing $(-1)^{\frac{(n-1)(n-2)}{2}} = -\frac{1}{2} [ (1+i) i^n + (1-i)(-i)^n]$

I want to show following equation holds. \begin{align} (-1)^{\frac{(n-1)(n-2)}{2}} = -\frac{1}{2} [ (1+i) i^n + (1-i)(-i)^n] \end{align} Can you give some hint or any practical method to compute this?
phy_math
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How do I derive these roots

Let $z = \cos(\frac{\pi k}{5}) + i\sin(\frac{\pi k}{5})$ Consider the imaginary part of $z^5$, and deduce that $x^4 - 3x^2 + 1 = 0$ has solutions: $$2\cos(\frac{\pi}{5}), ~2\cos(\frac{2\pi}{5}), ~2\cos(\frac{3\pi}{5}),…
stariz77
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