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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find real part $z_1 / z_2$ if $|z_1+z_2|=|z_1-z_2|$ and $z_2 \neq 0$

Question: Find the real part $z_1 / z_2$ if absolute value $|z_1+z_2|=|z_1-z_2|$. I thought $z_1= a +bi$ and $z_2= c + di$ then $z_1 + z_2= (a+c) + (b+d)i $ and $z_1-z_2= (a-c)+(b-d)i$ when computing the modulus and squaring both sides I end up…
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$\cos\left(\frac{\pi}{5}\right)$ using De Moivre's Theorem

This is an Exercise 3.2.5 from Beardon's Algebra and Geometry: Show that $\cos\left(\frac{\pi}{5}\right)=\frac{\lambda}{2}$, where $\lambda$ = $\frac{1+\sqrt{5}}{2}$ (the Golden Ratio). [Hint: As $\cos(5\theta) = 1$, where $\theta =…
fr_
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If $z$ is a complex number satisfying the equation $|z+i|+|z−i|=8 $ then maximum value of $|z|$ is?

If $z$ is a complex number satisfying the equation $|z+i|+|z−i|=8 $ then maximum value of $|z|$ is ? I took $z$ as a point p on the graph and drew lines connecting it to $i$ and $-i$. I assumed $z=x+iy$. Therefore x and y should be maximum. If x and…
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Solve Complex Equation $z^3 = 4\bar{z}$

I'm trying to solve for all z values where $z^3 = 4\bar{z}$. I tried using $z^3 = |z|(\cos(3\theta)+i\sin(3\theta)$ and that $|z| = \sqrt{x^2+y^2}$ so: $$z^3 = \sqrt{x^2+y^2}(\cos(3\theta)+\sin(3\theta))$$ and $$4\bar z = 4x-4iy =…
KTF
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Absolute value of the sum of two complex numbers squared

I have two complex numbers in the form $$U_{1} = Ae^{j \phi}$$ and $$ U_{2} = Be^{j \omega}$$ where $j$ is the imaginary unit. What is the expansion of $$|U_{1} + U_{2}|^{2}\;?$$ My initial attempt was to use the FOIL method, and that yielded $$(U_1…
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Using complex numbers to find reflections

if the equation of the curve of the reflection of ellipse $ \frac{(x-4)^2}{16} + \frac{ (y-3)^2}{9} = 1$ about the line $x-y-2=0$ is $16x^2 + 9y^2 + k_1 x -36 y+k_2 =0 $ , then $ \frac{k_1 +k_2}{33}$ =? So, I thought of this method using complex…
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$|z+2|=|z|-2$; Represent on an Argand Diagram

Represent on an Argand Diagram the set given by the equation $|z+2|=|z|-2.$ My attempt: Apparently the answer is $x\leq 0$ $(z = x + yi)$ and $y = 0$, based on the idea that $-x = \sqrt{(x^2 + y^2)}$, but I am struggling to derive this. I originally…
Jfry
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Problem using existing index laws with complex numbers?

I am new to working with complex numbers and am confused about using existing methods of working with indicies. Consider that: \begin{equation}\begin{aligned} & x = -1\\ &(\sqrt{x})^{3}=-i\\ \end{aligned}\end{equation} I am perfectly comfortable…
ajax2112
  • 257
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Find locus of $S$ denoting set of complex numbers $\frac{z+1}{z-3}$, where $z$ varies over set of $|z|=1$.

Question: Let $S$ denote the set of all complex numbers of the form $\frac{z+1}{z-3}$, where $z$ varies over the set of all complex numbers with $|z|=1.$ Find the locus of the points in set $S$. My approach: Let $z=x+iy$, with $x,y\in\mathbb{R}$…
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If $1, \omega_1,\omega_2...\omega_6$ are $7^{th}$ roots of unity, then find the value of $Im(\omega_1+\omega_2+\omega_4)$

$\omega_1$ and $\omega_6$ are conjugate pairs. The same applies for for $\omega_2$ and $\omega_5$ and so on. So $$\omega_1+\omega_2+\omega_4=a+ib$$ $$\omega_6+\omega_5+\omega_3=a-ib$$ $$\implies -1=2a$$ $$\implies a=-\frac 12$$ What should I do…
Aditya
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Solve $\cos(z) =3/4+i/4$

I need to solve the complex trinometric equation $$\cos(z) =\frac{3}{4}+\frac{i}{4} $$ What I've done so far is: Using the cos formula I got $e^{iz} +e^{-iz} =\frac{3}{2}+\frac{i}{2}$ Making $t=e^{iz} $ we have…
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How many complex numbers $z$ are there such that $|z|=1$ and $z^{5040} - z^{720}$ is a real number?

My attempt: If $z^{5040} - z^{720}$ is real, that means that their imaginary parts are equal. $e^{5040i\theta} - e^{720i\theta} = k$ , where $k$ is a real number $\sin{5040\theta} = \sin{720\theta} $ Let $u=720\theta$ $\sin 7u = \sin u$ By graphing…
helpme
  • 661
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Expand $(2i-2)^{38}$ using de Moivre's formula

I've been trying to figure out the way to solve this for a while now, and I'm hoping someone could point me in the right direction to find the answer (or show me how to solve this). The problem I'm having is with this equation: $(2i-2)^{38}$ and I…
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Solve $z^5 + 16\bar z = 0$ for $z\in \mathbb{C}$

Solve $z^5 + 16\bar z = 0$ for $z\in \mathbb{C}$. Need some help figuring out this problem.
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4 answers

How to solve complex number equation: $z^5=\bar z$?

I'm trying to solve this complex number equation: $$z^5=\bar z$$ As far as I understand, every complex number $z$ should be written in trigonometric form: $$r^5(\cos(5\phi) + i\sin(5\phi)) = r(\cos(-\phi) + i\sin(-\phi))$$ Unfortunately, at this…
Artem
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