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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How do complex number exponents actually work?

I know Euler's formula and how to take complex exponents, but in it it's $e$ to an imaginary angle, not a number, it seems. From my understanding pi itself is not an angle, but $\pi$ radians is. And since cosine can only take in an angle, or at…
Pineapple Fish
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Is there a proof that the distance from $0$ to $i$ in the complex plane is $1$?

I was just wondering how did people know that the distance between $0$ and $i$ is $1$ in the complex plane, did they just assume this, is it just an axiom, or is there a proof behind it or a reason for it?
Mohannad El-Nesr
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Complex Numbers in Polar Form raised to a power

$$\left ( 1 + \cos \frac{π}{6} + i \sin\frac{π}{6} \right )^6$$ For a question like this, the first thing I would think of using is the DeMoivre's Theorem, however, with the entire real part of it being (1+cos(π/6)), I'm not sure if the rule will…
lessaint
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Finding all $z \in \Bbb C$, expressed in the form $z = a + ib$, with $a, b \in \Bbb R$, satisfying equation $z^3 = -i$

I'm trying to find all $z \in \Bbb C$, expressed in the form $z = a + ib$, with $a, b \in \Bbb R$, satisfying equation $z^3 = -i$. I've figured out that $$(a + ib)^3 = -i \Longleftrightarrow a + ib = \sqrt[3]{-i} \Longleftrightarrow a + ib = -i$$…
898989
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Are complex numbers the set of values of a tower of imaginary units?

This question raised from this one. Solving $i^i=x$ we get $x=e^{(i \frac{\pi}{2} + i2k\pi)i}=e^{-\frac{\pi}{2} - 2k\pi}$ $(k \in \mathbb{Z})$, than what about the values of $x = i^{i^{i^{i^{.^{.^{.{^ \infty}}}}}}}$? Is there an argument to show…
Immanuel Weihnachten
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Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number

Prove $\ \frac{z^{2010} - \bar z^{2010}}{1+z\bar z}$ is imaginary number. I understand that if $\ z = (a+bi) $ then $\ z - \bar z = 2bi $ and the denominator $\ 1+z\bar z $ is $\ 1+|z|^2 $ and therefore it is a real number. so need to prove the…
bm1125
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Express the following complex numbers in the form of $a+bi$

I'm struggling with the following exercises: I tried to use the reasoning as follows: $$(a+bi)^n=(re^{\theta i})^n=r^ne^{\theta in}=r^n(\cos(\theta n)+i\sin(\theta n))$$ So for the first one I did: $$2^{1/6}(\cos(-\frac{\pi}{3} \cdot …
Arnau
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Plots & complex numbers

I'm learning Complex Numbers by myself. I've already understood basics concepts, but now I have problems with plots/graphs. For example, I don't know, how to plot: (1) $|z-1|+|z+1|=4$ (2) $|z+1| - \Im z \leq 1$ (3) $\Re\left(\frac{1-z}{1+z}\right) =…
user8505
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If $aZ^2 + bZ + 1 = 0$ and $|a|=0.5$. What is the value of $|ab' - b|$?

If $aZ^2 + bZ + 1 = 0$ where a,b,Z are complex numbers ; $|a|=\frac 12$ and have a root $\alpha$ such that $ |\alpha|=1$, then what is the value of $|ab' - b|$? (I use $Q'$ to represent the conjugate of $Q$) Since $\alpha$ is a root. I might…
SmarthBansal
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Sum of three complex numbers

We have three complex numbers $z_1, z_2, z_3$ such that $|z_1|=|z_3|=|z_3|=1$ and $z_1+z_2+z_3=1$. Find $$S={z_1}^{2013}+{z_2}^{2013}+{z_3}^{2013}$$ I would like only a hint. I probably know everything elementary about complex numbers so just give…
asd11
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Sketch, on a single Argand diagram, the loci given by $| z − \sqrt 3 − i | = 2$ and $\arg (z) = 16 \pi$

Sketch, on a single Argand diagram, the loci given by (i) $| z − \sqrt 3 − i | = 2$ (ii) $\arg (z) = 16 \pi$ Does anyone know a good way of how to explain solving a problem similar or the same as this? The part $| z − \sqrt3 − i | = 2$ is just a…
user1806497
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Determine all complex numbers satisfying the equation

The equation: $$e^{iz} = 4\cos(z) + 3$$ This is what I have tried so far: $$e^{iz} = 4\left(\frac{e^{iz}+e^{-iz}}{2}\right)+3$$ $$e^{iz} = 2e^{iz}+2^{-iz}+3$$ $$e^{2iz} = 2e^{2iz}+2+3e^{iz}$$ $$e^{2iz}+3e^{iz}+2=0$$ Now, I substitute $e^{iz}$ with…
B.E
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When $|f|$ and $\arg(f)$ are analytic?

Let $f:ℂ→ℂ$ be an analytic function. Define $|f|$ and $\arg(f)$ be the modulus and the argument of $f$. Generally, $|f|$ and $\arg(f)$ are not analytic. My question is about the cases where this happen: $|f|$ and $\arg(f)$ are analytic.
Safwane
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Interpretation of complex number in 1d.

In order to solve the equation $$x^2 + 1 = 0$$ we introduce the imaginary number $i$. But to interpret the complex numbers we think of them as laying in the complex plane, which is a 2d plane. It turns out that instead of introducing the imaginary…
Natanael
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In the equation: $z = (\sqrt3+i)^n$ determine the values of $n$ for which $z$ is a real number.

Question : In the equation: $z = (\sqrt3+i)^n$ determine the values of $n$ for which $z$ is a real number. Working: $z = (\sqrt3+i)^n $ $z^\frac1n = (\sqrt3+i)$ $z^\frac1n = (2cis(\frac\pi6 + 2k\pi))$ I'm not quite if my working is correct, its just…
Plus Twenty
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