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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Is the modulus of i^n 1 for all n?

As the tile says, is $\left | i^{n} \right | = 1$ for all real values of n?
LazySloth13
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need help with complex numbers

$(1-\sqrt3i)(1-i)^2 + 1 + \sqrt3i = ?$ My result is: $-2i - 2\sqrt3 + 1 + \sqrt3i$ Is this the right result or am I doing something wrong? Thank you.
Kurama
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Problem converting to polar form in proof

I wonder if anyone has an idea about how to write $$ \prod_{\substack{j=0\\ j\neq k}}^{n-1} ( e^{\frac{2\pi ik}{n}} - e^{\frac{2\pi ij}{n}})=n,\qquad k=0,1,...,n-1,\; j=0,1,....n-1$$ in a "general" polar form, i.e. $$e^{\frac{2\pi ik}{n}} -…
Sofia Fredriksson
  • 31
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Using the polar form of $1 + i$ and $\sqrt3 + i$ to deduce $\cos (\frac{\pi}{12}), \sin(\frac{\pi}{12})$

I have been beating my head against the following problem and would like a gentle nudge in the right direction. The question states, by writing $1 + i$ and $\sqrt3 + i$ in polar form, deduce that $$\cos (\frac{\pi}{12}) = \frac{\sqrt3 + 1}{2\sqrt2},…
Matt Martineau
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Why would you define $i$ as $i^2=-1$, and why not as $i=\sqrt{-1}$?

I'm not sure if I recall this correctly, but I thought there was a reason why you shouldn't write $i=\sqrt{-1}$. And if this is not true, then I wonder: Why would you define $i$ as $i^2=-1$, why wouldn't you define it as $i=\sqrt{-1}$. I was…
Kasper
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Complex number question conjugate

Could somebody help me please? I've got part one solved. 1) Solve the equation: $z^3=i$ I can do this bit: $$ z = \exp \left( \frac{i\pi}{6} + \frac{2k\pi}{3} \right) $$ so $$ z = \exp \left( \frac{i\pi}{6} \right) \qquad \text{or} \qquad z =…
user131215
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How find this value $\left|\frac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\right|$

let three complex $z_{1},z_{2},z_{3}$ such $$z_{1}+z_{2}+z_{3}\neq 0,|z_{1}|=|z_{2}|=|z_{3}|=1$$ Find this value $$\left|\dfrac{z_{1}z_{2}+z_{1}z_{3}+z_{2}z_{3}}{z_{1}+z_{2}+z_{3}}\right|$$ My idea:if $z_{1},z_{2},z_{3}$ is real numbers,and such…
user94270
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How to show $i^{-1} = -i$?

How can I show $i^{-1} = -i$, where $i$ is the imaginary unit? Here's what I've tried: $i^{-1} = (-1)^{-1/2} = \dots ?$
jaynp
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Find the $z\in \Bbb C$ such that $\bar z=z^3$

Here is what I've done. Let $z\in \Bbb C$ be of the form $a+ib$, lets assume in this case that $a,b\not=0$. We have that $\bar z=a-ib$, and $z^3=a^3-3ab^2+i(3a^2b-b^3)$, so to find the correct $z$, it has to satisfy: $$\begin{cases} a=…
Ana Galois
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Symbol for reflecting complex numbers over the imaginary axis

A bar or star symbol is used for reflecting in the real axis (i.e. complex conjugate). Is there a commonly (or not so commonly) used symbol for inverting the sign of the real part (i.e. reflecting in the Im-axis) ? I'm thinking of a situation in…
Mulu
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Find all roots of $z^2=3-4i$.

Find all roots of $z^2=3-4i$. $z^2=3-4i$ $z^2+4i-3=0$ But how do I go on from here?
Mr Croutini
  • 1,158
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Complex numbers - system of equations

Let $z$ and $w$ be complex numbers such that $|2z - w| = 25$, $|z + 2w| = 5$, and $|z + w| = 2$. Find $|z|$. Any tips where to start? Is there a better way then just squaring both sides and solving then replugging them in?
Freedom
  • 303
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Prove $(1-\epsilon )\cdots(1-\epsilon ^{n-1})=n $ where $\epsilon = \exp(\frac{2\pi i}{n}), n\in \mathbb{N}, n \neq {0,1}$

I have a problem with proving the following: $$(1-\epsilon )\cdots(1-\epsilon ^{n-1})=n,$$ where $$\epsilon = \exp\left(\frac{2\pi i}{n}\right), n\in \mathbb{N}, n \neq {0,1}.$$ I tried to use fact that $$x^n-1=(x-1)\left(x-\exp\left(\frac{2\pi…
fdhd
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Why $0$ is the accumulation point of the set $\Omega=\{\frac{1}{n}+\frac{i^n}{n} \mid n\in N / \{0\} \}$ ??

Why $0$ is the accumulation point of the set $\Omega=\{\frac{1}{n}+\frac{i^n}{n} \mid n\in N / \{0\} \}$ ?? I try to solve it graphing the set $\Omega$ and I saw that the real part of the set when $n$ goes to $0$ it tends to infinity, then there is…
Salvattore
  • 551
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4 answers

Inequality $|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$

I have a problem to prove this inequality $|z_1+z_2|^2 \le (1+|z_1|^2)(1+|z_2|^2)$ $\forall (z_1, z_2)\in \mathbb{C}$. I tried to take the right hand set and subtract the lfs and after simplification I got this: $1+(ax)^2+(by)^2…
pourjour
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