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.

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How to find point of perpendicular bisector given equation of line and point?

It would be really awesome if you guys could give me a hand on this question! Question: Given that the equation of a line is $y=-\sqrt{3}x$. This line can be also expressed in $|z-1|=|z-k|$, where $k \in\Bbb C$. Find $k$ in cartesian form. What I've…
Munchies
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Find z s.t. $z+\frac{1}{z}$ is real. Is my solution good?

I must find all z s.t. $z+\frac{1}{z}$ is real. I know that $z = a + bi$ is real when the Imaginary part is 0. So, there we go: $$z+\frac{1}{z}=…
cocacola
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$1$, $z$, and $z^{2019}$ are vertices of an equilateral triangle

Find all complex numbers $z$, such that the points, in the complex plane, which correspond to $1$, $z$, and $z^{2019}$ form an equilateral triangle. Thanks in advance. So far I got $$\frac{z^{2019}-1}{z-1}-\cos(\pi/3)-i\sin(\pi/3)=0$$ and…
Tutan Kamon
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I am puzzled with which one is right.

I am puzzled with which one is right.If my work is wrong.please give me a right explanation in detail.
park ning
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prove that : $ i = \sqrt {-1}\ $

i have a pretty nasty question. i was glancing through a few olympiad papers and stumbled upon this question: prove that $ i = \sqrt {-1}\ $. i tried the conventional methods namely euler's formula but could not figure what to do next. how do you…
Sri Krishna
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How does one go about solving $arg(\frac{z-2i}{z-6}) = \frac{1}{2}\pi$

This should give $$\frac{z-2i}{z-6} = bi$$ but solving that gives me $$z = \frac{-2b +6b^2-6bi +2i}{1+b^2}$$ and substituting $z$ for $x + yi$ gives me $x = \frac{-2b +6b^2}{1+b^2}$ and $y=\frac{-6b +2}{1+b^2}$ And I have no clue how to continue…
ReefG
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Prove $\Re\left(\frac{1-e^{i(n+1)\theta}}{1-e^{i\theta}}\right) =\frac{1}{2}+\frac{\sin{\left(n+\frac{1}{2}\theta\right)}}{2\sin{\frac{\theta}{2}}}$

With $\text{Re}(\frac{1-e^{i(n+1)\theta}}{1-e^{i\theta}})$, I expanded it with polar form and found the real component to be $$\frac{1}{2}+\frac{\cos(n\theta)-\cos((n+1)\theta)}{2{\sin}^2\left(\frac{\theta}{2}\right)}$$ where the numerator was…
smaillis
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Inequality on sum and difference of two unimodular complex numbers

Given two complex numbers $z,w$ with unit modulus (i.e., $ |z|=|w|=1$), which of the following statements will always be correct? a.) $|z+w|\lt\sqrt2$ and $|z-w|\lt\sqrt2$ b.) $|z+w|\le\sqrt2$ and $|z-w|\ge\sqrt2$ c.) $|z+w|\ge\sqrt2$ or…
Broly-29
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Determine set of points in complex plane

Determine set {$w=\frac{2}{3-z}:z=2+iy$, $y \in \mathbb{R}$} in complex plane. I've tried with putting $z$ in denominator and rationalizing. $w=\frac{2}{1-iy}\cdot\frac{1+iy}{1+iy}=\frac{2+2iy}{1+y^2}=\frac{2}{1+y^2}(1+iy)$, The solution says that…
Theo Gray
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Solve for $x$, $(1+2i)^{(2x+6)}=(-11-2i)^{(x+1)} $

$$(1+2i)^{(2x+6)}=(-11-2i)^{(x+1)} $$ Solve for $x$ My Work: Take ln at the both…
Jale'de jaled
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Why does that fact that $a_n \equiv 3 \pmod 5$ and $b_n\equiv 1 \pmod 5$ imply $1/\pi(\arctan(1/3))$ is irrational

So since I've started this multi-part question I've learned: $(3+i)^n = a_n+ib_n$ $a_{n+1} = 3a_n-b_n$ $b_{n+1} = 3b_n+a_n$ $a_n \equiv 3 \pmod 5$ $b_n \equiv 1 \pmod 5$ Now I am asked why the fact that, for $n\geq 1$, $a_n \equiv 3 \pmod 5$ and…
KTF
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Let z be a complex number such that $\frac{z-i}{z-1}$ is purely imaginary. Then the minimum value of $|z-(2+2 i)|$

Let z be a complex number such that $\frac{z-i}{z-1}$ is purely imaginary. Then the minimum value of $|z-(2+2 i)|$ is ? My approach:- $$ \begin{array}{l} \\ \left|z-z^{\prime}\right| \geqslant| \ |z|-\left|z^{\prime}\right|| \end{array} $$ where,…
user791682
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Real part and imaginary part of a complex number of $i^{1/4}$

Find the real and imaginary parts of $$i^{1/4}.$$ It seems to me that the real part is $0,$ because it does not appear and the imaginary part is $$1^{1/4},$$ but it seems too simple for it to be fine. Does the exponent affect anything?
Jorge Salvador Martnez Villafa
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As w moves between $-i$ and $i$ on $w = \frac {1+zi}{z+i}$, prove z moves across positive imaginary axis

I'm having trouble with this: As w moves between $-i$ and $i$ on $w = \frac {1+zi}{z+i}$, prove z moves across positive imaginary axis. I've got the two cartesian equations: $u = \frac {2x}{x^2 + (y+1)^2}$ and $v = \frac {x^2 + y^2 -1}{x^2 +…
Noobcoder
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Complex Modulus

Thanks everyone that answered my last question! I had one more question for a different concept. If the modulus of a complex number $z = |z| = \sqrt{(a^2 + b^2)}$,where $a$ and $b$ are the real and imaginary constants of $z$, are you allowed to use…
HokieFan7
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