Mathematics Stack Exchange
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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.

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Euler notation of complex numbers

If I use the Euler notation for a complex number, then a complex number $z$ can the following exponential form: $$e^z=e^{x+iy}=e^x e^{iy}=e^x(\cos y+i\sin y)$$ So $e^x$ and $y$ represent the module and the angle of polar form of $z$, respectively?
AnalyticHarmony
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Ordering in $\mathbb{C}$

Saff and Snider in their book define order axioms as below: If $\alpha\neq 0$ then either $\alpha>0$ or $\alpha<0$. If $\alpha>0$ and $\beta>0$ then $\alpha+\beta>0$. If $\alpha>0$ and $\beta>0$ then $\alpha\beta>0$. I have to prove the…
matrixx
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Plotting complex arcs on an Argand diagram

What would $arg(\frac{z-4i}{z+2i}) = \frac{\pi}{4}$ look like on an Argand diagram and why? I don't know how to check my answer because Desmos does not support complex numbers.
PhysicsMathsLove
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Range of $f(z) = |1-z|+|1+z^2|$, where $z$ is a complex number

If $z$ is a complex number such that $|z| = 1.$ then range of $f(z) = |1-z|+|1+z^2|$ Attempt: assuming $z=x+iy$ and $|z| = 1$ so $x^2+y^2$ so $f(x,y) = \sqrt{(1-x)^2+y^2}+\sqrt{(1+x^2-y^2)^2+4x^2y^2}$ could some help me, thanks
DXT
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Reflection of a point about a complex line

On a website It is written as The concept of reflection points of a straight line or reflection on a number line is important and a bit tricky too. It is entirely different from the point reflection. Two points say P and Q are said to be the…
Koolman
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Finding out the range of values of the argument of a complex number

Let $z,z^2,z^3,z^4$ be the four complex numbers.If these taken in order form a cyclic quadrilateral then the question is to find out the range of values of $\theta$ where $\theta$=$arg(z)$ and $\theta €(0,2\pi)$. Since it forms a cyclic…
Navin
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For given equation, calculate modulus of complex number $z$

If $z$ is a complex number such that Real part of $z\neq 2$ and $$z^2=4z+|z|^2+\frac{16}{|z|^3}$$ I assumed $z=x+iy$ and tried equation real and imaginary part on both sides. After equating imaginary part, I got $x=2$ or $y=0$ but when I equal real…
user383014
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Complex numbers- Number of values satisfying z :|z|=z+1+2i

Find the number of complex numbers satisfying $|z|=z+1+2i$ . My method: I know $|z|$ is real. So, the imaginary part of the RHS should be equal to $0$. So, $z$ should be of the form $x-2i$. Using that I am getting an imaginary value for x itself!…
Jamil Ahmed
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Find the value of $z_1^2 + z_2^2 + z_3^2$ given that $z_1, z_2, z_3 \in \mathbb{C}$, $z_1 + z_2 + z_3 = 0$ and $|z_1| = |z_2| = |z_3| = 1$.

So I'm given the task of finding the value of $$z_1^2 + z_2^2 + z_3^2$$ For $z_1, z_2, z_3 \in \mathbb{C}$ such that $z_1 + z_2 + z_3 = 0$ and $|z_1| = |z_2| = |z_3| = 1$. (Edit: of course $|z|$ represents the modulus of $z$) I have done some…
D. Brito
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Cube Roots are Complex?

I am in DE and finding singular points for power series. For a singular point at $x^3=-1$, I said $x=-1$ and called it a day. Turns out this is wrong (and can be shown via Euler's formula). When are you supposed to learn this stuff in math? Euler's…
Alex G
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How to relate the $|z^3+z^{-3}|$ with $|z+z^{-1}|$?

I got stuck on this seemingly simple question: If $z$ is a complex number satisfying $|z^3+z^{-3}| \le 2$, then the maximum possible value of $|z+z^{-1}|$ is: (A) $2$ (B) $2^{1/3}$ (C) $2\sqrt 2$ (D) $1$ Using the AM-GM inequality, I…
FreezingFire
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Draw in a complex plane

Draw z in a complex plane, for which: $Re\left( \frac{1}{z-1} + \frac{1}{z+1}\right)>0$ No idea. Tried it all, but it becomes too complicated to sketch it.
Spideyyyy
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a complex analysis problem

Let $\alpha ,\beta$ be two complex numbers with $\beta \neq 0$ , and $f(z)$ a polynomial function on $\mathbb{C} $ such that $f(z)=\alpha$ whenever $z^5 = \beta$. What can you say about the degree of the polynomial $f(z)$ ? How can I solve the…
hum
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Show that $|z| = 1$ if and only if $\bar{z} = \frac{1}{z}$.

Maybe a very stupid question but I am stuck. Show that $|z| = 1$ if and only if $\bar{z} = \frac{1}{z}$. Is it enough to simply multiply, i.e. $z\bar{z} = \frac{1\times z}{z} = 1$? Showhow I feel this is not correct. I know that if $z = \pm 1$ or $z…
user376984
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Find the values of $n$ such that $(1+\sqrt3i)^n$ is a real number

Find the values of $n$ such that $z^n=(1+\sqrt3i)^n$ is a real number. My reasoning: The power will be real iff $\sin\arg z=0$. Since $\sin 0,\sin\pm\pi,\sin\pm2\pi,\dots=0$, $3\mid n$. Is it…
BananaBuisness
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