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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clarification for the solution of $(z^3=|z|^2 +4)$!?

$(z-2)(z^2+z+2)=0$ so the solutions are $$z=2;$$$$z^2+z+2=0;$$ $$z=-(1+-\sqrt{-7})/2$$ I removed the absolute value for imposing a solution in the field of real, but only because I saw it on wolfram alpha, now I ask you! I can always impose a…
malloc
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compare complex number with 0?

im reading the book "what is mathematics" and find the questions. Calculate $\sqrt{5+12i}$ i followed the hint and wrote the equation $\sqrt{5+12i} = x+yi$ and solved it with these results : x=3,y=2 x=-3,y=-2 so here my intuition told me…
Detective King
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If $z,w \in {\mathbb C}\setminus\{0\}$, prove that $zw$ is real if and only if $\exists{k}\in {\mathbb R}$ such that $w=k\bar{z}$.

Help me please, I can't resolve it. I don't have any idea. If $z,w \in {\mathbb C}\setminus\{0\}$, prove that $zw$ is real if and only if $\exists{k}\in {\mathbb R}$ such that $w=k\bar{z}$.
Salvattore
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Why is the modulus of a complex number $a^2+b^2$?

Why is the modulus not $\sqrt{a^2-b^2}$? Carrying out standard multiplication this would be the result-why is this not the case? I know viewing the complex plane you can easily define the sum as being the distance to the points, but what meaning…
user82004
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Hyperbolic functions identity problem

We know the identity $\tanh(ix)=i \tanh(x)$. My question: is it true that $\tanh^{-1}(ix) = i \tanh^{-1}(x)$ ? If not then is there a similar identity for arctangents? I think there might not be but would like to know how to find one.
Milan Amrut Joshi
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Solve the system of equations $x+y+z=1, x y z=1,|x|=|y|=|z|=1, x, y, z \in \mathbb{C}$

Solve the system of equations $x+y+z=1, x y z=1,|x|=|y|=|z|=1, x, y, z \in \mathbb{C}$ I tried with polar representation, letting $x=\exp{(ia)}$,$y=\exp{(ib)}$, $z=\exp{(ic)}$, with this I got $a+b+c=2 \pi n$, so the three numbers are essentially…
math_learner
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How exactly does Euler notation work for complex numbers?

When textbooks say $e^{i\theta} = \cos\theta + i\sin\theta$, is the $e$ actually Euler's number, 2.71828..., or is this purely a symbolic equivalent notation where the $e$ does not represent this number? If it is the actual number, why is it used…
John
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Determine the fourth roots of -16

Determine the fourth roots of -16 in the form $x +iy$ where $x$ and $y$ are not trigonometric functions. I do not even know what they really want from me in this question. My initial thought wass: $\sqrt[4]{-16} = 0 + 2i = 2i$ but that seems overly…
Leon
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$\mathcal M = \{z \in \mathbb C | |z|=r \}$ , where $r \in \mathbb R , r >0$

The statement of the problem : Consider the set $\mathcal M = \{ z \in \mathbb C | |z|=r\} $ , where $r \in \mathbb R , r >0$ a) Prove that there exists $a,b \in \mathcal M,a \neq b$ such that $a+b \in \mathcal M$. b) Find the values ​​of the…
Last X
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Finding solutions for Real and Imaginary parts of a Complex Number.

This is my starting equation for complex number, $ \hat{k} $ $$ \hat{k}^2 = \mu \epsilon \omega - \mu \sigma \omega^2 i $$ I assumed $ \hat{k} = \alpha + \beta i $ where $ \alpha , \beta \in \mathbb{R} $ I then derived the following: $$ \alpha^4 -…
Rahul Silva
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Sketch a set in the complex plane

How to sketch this set in the complex plane? $$S=\{ z\in C: Re[(4-i)z] > Re[(-5+7i)(4+6i)] \}$$ Where: S - set Re[x+yi] - real part of the complex number C - complex numbers z - complex number I solved the Real parts inside the brackets and…
Arsenii Zarudniuk
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Relationship between circlular $i^n$ property and that $e^i{\theta}$ traces the unit circle.

That $e^{i\theta}$ traces a circle $\theta \in R$ has been well discussed elsewhere. However, I was always curious with it's relation to the following property of $i$: $i^0$ = 1 $i^1$ = i $i^2$ = -1 $i^3$ = -i $i^4$ = 1 These properties seem like…
Joseph
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Simpyfying the expression $(1 + \cos \alpha + i\sin \alpha)^k + (1 + \cos \alpha - i\sin \alpha)^k$

I came across the following question the other day and was wondering if my solution is the correct solution. Here is the question. Prove that $$ (1 + \cos \alpha + i\sin \alpha)^k + (1 + \cos \alpha - i\sin \alpha)^k = 2^{k+1} \cos \left(…
Stephan
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How to calculate integer exponent of complex number with integer parts to not get real values

I'm trying to implement the exponentiation of numbers in my Scheme interpreter. I'm wondering how one Scheme implementation (Kawa) implemented this equation: $(10+10i)^{10}$ Kawa give this output: $+320000000000i$ My implementation (same as few…
jcubic
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How to get to the first step of solving $z^2=i$?

How do I get from $z^2=i$ to $z=x+iy$? Is it a rule you use to solve the equation in general or specifically for this equation? (I don not understand the step from $z^2=i$ to $z=x+iy$) Thank you!
Tony
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