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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Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$

Why the equation of an arbitrary straight line in complex plane is $zz_o + \bar z \bar z_0 = D$ where D $\in R$ I understand that a vertical straight line can be defined by the equation $z+\bar z= D$ because suppose $z =x+yi$ then $\bar z = x-yi$ …
Parting
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How to solve this complex number question?

$i=\sqrt{-1}$ and $a,b$ and $c$ are positive integers and $$c = (a+ib)^3-191i$$ is given. Find $c$. I expanded the equation that is given and wrote that $$ c = a^3+3ia^2b-3ab^2-ib^3-191i \\ $$ Since $c$ is a positive…
user373239
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Show: $\cos \left( \frac{ 3\pi }{ 8 } \right) = \frac{1}{\sqrt{ 4 + 2 \sqrt{2} }}$

I'm having trouble showing that: $$\cos\left(\frac{3\pi}{8}\right)=\frac{1}{\sqrt{4+2\sqrt2}}$$ The previous parts of the question required me to find the modulus and argument of $z+i$ where $z=\operatorname{cis{\theta}}$. Hence, I found the modulus…
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High School Explanation to Why is $i^i$ multi valued if $i$ is apparently a constant?

High School Explanation to Why is $i^i$ multi valued if $i$ is apparently a constant? Answer should necessarily be understood by a high school student and should not invoke complex formulas. Arguments which may beautifully explain the concept bu t…
Agile_Eagle
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Prove $Re (z) = \frac{z + z^* }{2}$ and $Im (z) = \frac{z − z^*}{2i}$

In texts on complex numbers I often see an exercise that asks to prove the following: $$Re (z) = \frac{z + z^*}{2}$$ $$Im (z) = \frac{z − z^*}{2i}$$ where $z = x + iy$ and $z^* = x - iy$ I understand the meaning of complex numbers, but can't seem…
user2254532
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If $z_1^2+z_2^2$ is real, $z_1(z_1^2-3z_2^2)=2$, and $z_2(3z_1^2-z_2^2)=11$, then find $(z_1^2+z_2^2)^2$

If $z_1$ and $z_2$ are complex numbers such that $z_1^2+z_2^2 \in\mathbb R$ and $$z_1(z_1^2-3z_2^2)=2,\qquad z_2(3z_1^2-z_2^2)=11,$$ then find the value of $(z_1^2+z_2^2)^2$. Given answer is $25$. I have tried many things but I am not getting the…
Ananya
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Software to plot complex numbers in Argand diagram

I'm looking for a software or an online resources that allows me to plot complex number inequalities in the Argand diagram similar to this one. Please, any help is appreciated. Thanks, Arif
Arif
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Why does it make sense to describe complex numbers through powers of $e$?

I'm studying in-depth complex numbers but I keep wondering why it makes sense to describe complex numbers through powers of $e$.
TripleA
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Why is $z = \frac{a}{b}$ a singularity of $ \frac{1}{z|az-b|^2} $?

Could someone explain why $z = \frac{a}{b}$ is a singularity of $ \frac{1}{z|az-b|^2} $? Am I mistaken something?
IgNite
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How to go about solving $((1+iz)/(1-iz))^4 = 1/2 + i\sqrt3/2$?

I have problem solving this equation: $$ \left(\frac{1+iz}{1-iz}\right)^4 = \frac12 + i {\sqrt{3}\over 2} $$ I know how to solve equations that are like: $$ w^4 = \frac12 + i {\sqrt{3}\over 2} $$ And I have solved it to: $$ w =…
Curtain
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How to calculate $(3+4i)\cdot(1+i)$

I have recently read an article on imaginary numbers. It was very interesting, but left me with the above question. It had the answer in the question, it was $-1+7i$. But how do I calculate this?
imulsion
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Can all complex expressions be simplified to the form $a+jb$?

Are there any complex expressions that cannot be simplified to the form $a+jb$, where a and b are real numbers? For example, $$\frac{1}{j}=0+j(-1),\hspace{0.5cm}e^j=\cos(1)+j\sin(1),\hspace{0.5cm}\sin(j)=0+j\frac{e^2-1}{2e}$$ From what I understand,…
Dan
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Prove this three complex are $z_{1}=z_{2}=z_{3}$

When I deal with a geometric problem, get the following algebraic problems: Assmue that $$H(p,q)=\dfrac{\omega p}{\omega-1+a(\omega p-q)},a>0$$ where $\omega^3=1,\omega\neq 1$.…
math110
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Are complex numbers and $(x,y)$ coordinates two sides of the same coin?

I'm learning that imaginary numbers are just a way of representing a rotation. So, are imaginary numbers and multiple variable numbers $(x,y,z,...)$ just two different tools for representing numbers in multiple dimensions? Or are imaginary numbers…
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Show that $z^{-1} = \frac{\bar z}{|z|^2}$

I'm stuck on this question, I have a feeling the answer is very straightforward but I just can't figure it out. Question: Considering $z= x + iy$, show that: $$z^{-1} = \frac{\bar z}{|z|^2}$$ So far this is what I have: $\bar z=x-iy$ and $|z|^2=…
Enrico S
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