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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Prove there exist a complex number that satisfies the given conditions

Prove that there exists a complex number $z=a+ib$ on the complex plane such that there are no two complex numbers $z_1=x_1+i y_1$ and $z_2=x_2+i y_2$ such that $\left|z_1-z\right|=\left|z_2-z\right|$ where $a, b \in \mathbb{R}$ and $x_1, x_2, y_1,…
轻型八神
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Will locus of $arg(Z - Z_0) = \theta$ include $Z_0$? ($Z_0$ is a constant complex number but $Z$ is a variable)

My mathematics teacher taught us that if we draw the locus of $|Z-Z_0| = \theta $ ($Z$ is a variable complex number and $Z_0$ is a constant complex number), it will be a ray originating from the point $P$ denoting $Z_0$ and make an angle of $\theta$…
I am a Human
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Is $i$ a rotation?

Question: Is $i$ (the imaginary number) a rotation? This might be a very a loose question to ask but I can't find a better place to ask. Now a friend of mine told me this when we were discussing complex numbers, but I just can't figure out how is…
轻型八神
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$f(z) = \frac{z^2}{1 − \cos z}$ at $z = 0$. What is the easiest way to determined that $z=0$ is a removable singularity?

$f$ is a complex valued function. $f(z) = \frac{z^2}{1 − \cos z}$ at $z = 0$. What is the easiest way to determined that $z=0$ is a removable singularity. Is this correct justifation? The numerator and denominator both have a zero of order two at…
jenny9
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Resolving cubic equation where domain is complex number

I unable to resolve the following equation (I don't have any idea how even start resolving this equation): $$8z^3-12z^2+6z-1-i=0$$
M.Naszko
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Convert $2^{11/2} e^{(3 \pi/4) i}$ where $ - \pi < \theta \le \pi$

Express complex number $(i^{10} + i^{-7})^{11}$ in Euler form, where $r > 0$ and $ - \pi < \theta \le \pi$, where $\theta$ is in the form of $k\pi$ I convert $(i^{10} + i^{-7})^{11}$ to $(-1 + i)^{11}$ In Euler form, $ (\sqrt{2} e^{(3 \pi/4)…
user307640
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Points that satisfy the equation $|z + 3| + |z + 1| = 4$

I am trying to find the points that satisfy the equation $$|z+3|+|z+1|=4$$ Substituting the value of $z$ and evaluating its modulus gives me $$\sqrt{(x+3)^2+y^2}+\sqrt{(x+1)^2+y^2}=4$$ What I tried to do is to square both sides giving me…
Abdulrahman
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Creating a complex number so that its norm equals to 1

I would like to create a complex number c so that its norm is equal to some number a (for the purpose of this question let's assume a = 1) if I already have either its real or imaginary part. I know that: $$\lVert \mathbf{c} \rVert =…
Slan
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How to simplify these complex expressions?

I have expressions: $F(s)=\sqrt{\left[\left(a+b\right)s+c\right]\mp\sqrt{\left[\left(a-b\right)s+c\right]^{2}+d}}$ where $s$ is a complex number; and $a,b,c,d$ are reals such that $a>0,\,b>0,\, a\neq b,\, c>0,\, d\geq0$. I am trying to figure…
Leszek
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Solving the equation $\overline z-z^2=i(\overline z+z^2)$ in $\mathbb{C}$

Let $\overline z$ denote the complex conjugate of a complex number z and let $i= \sqrt{-1}$. In the set of complex numbers, the number of distinct roots of the equation $\overline z-z^2=i(\overline z+z^2)$ is _____________. My approach is as…
Samar Imam Zaidi
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Why Does changing the placement of the I switch the fraction from positive to negative

So I am doing an exercise on khan academy and I answer this question and ask for the explanation and I don't understand this Step. Why does changing the placement of the i change the fraction from negative to positive?
Cuppa Milk
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Let $z_1, z_2, z_3$ be complex numbers with arguments $2\pi/3$ apart and each has modulus of at most $A>0$. Maximize $|z_1 + z_2 + z_3|$.

Let $a_1 = |z_1|$, $a_2 = |z_2|$, and $a_3 = |z_3|$. Without loss of generality, take $z_1 = a_1 $ $z_2 = a_2 e^{2\pi i/3} $ $z_3 = a_3 e^{-2\pi i/3}$ We want to maximize $|z_1 + z_2 + z_3|$ subject to $|a_1| \leq A$ $|a_2| \leq A$ $|a_3| \leq…
s114
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In what space can we have points with coordinates $(x_1 + y_1i,x_2 + y_2i)$?

Bear with me on this one please. I am learning about complex numbers. We have cartesian coordinates made up of real numbers for points in cartesian plane. Is there any plane in which we can have cartesian coordinates made up of complex number for…
KawaiKx
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Find maximum value of $\left|u-2i\right|^2+2\left|v+i\right|^2$ with conditions.

$u$ and $v$ are two complex numbers such that $\left|u^2+uv-2v^2-3i\left(u+2v+2\right)+8\right|=44$ and $\left|u+2v\right|=6$. Find maximum value of $T=\left|u-2i\right|^2+2\left|v+i\right|^2$. I have tried and get some interesting things, but…
user628755
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Maxima $|1+z|+|1-z+z^2|$ when $|z| = 1$

I was solving this problem 5 from book namely Complex Analysis A to Z , here we have to find range of $$ y=|1+z|+|1-z+z^2|$$ with $|z|=1$ . First of all i didn't get the step $|1-z+z^2|=\sqrt{|7-2t^2|}$ . Then i tried to solve this question by…
Rishi
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