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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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De Moivre's use to a complex number

I have this question which I'm stuck on, here's the question and what I did. Find the smallest positive integer m such that $\left(\sqrt{3}+i\right)^m=\left(\sqrt{3}-i\right)^m$. I expanded out each side using De Moivre's,…
user145591
  • 169
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How to show that $e^{-(-2)^\frac{2}{3}}$ is complex.

I am baffled with showing that $e^{-(-2)^\frac{2}{3}}$ is complex. My understanding is: $$ e^{-(-2)^\frac{2}{3}}= e^{-(4)^\frac{1}{3}}$$ since ${-(4)^\frac{1}{3}} $ is negative real number, so $ e^{-(4)^\frac{1}{3}}$ is always a positive real…
kaka
  • 1,896
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Relation with sum of modulus of complex numbers

Let $z_{1}, z_{2}, ..., z_{n}$ be nonzero complex numbers, with $z_{k}=p_{k}\exp(i\theta_{k})$, where $p_{k}$ is a positive real number and $\theta_{k}$ real. Can you help me prove that $\left | \sum_{k=1}^{n}z_{k} \right…
user20010
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Does $2^i$ exist, if so how do I calculate its value?

I was working on the question, to show that $i^i$ is a real number. That was however straight forward, $$i = e^{i\frac{\pi }{2}}$$ so $$i^i = (e^{i\frac{\pi }{2}})^{i}= e^{-\frac{\pi }{2}}$$ So I thought I'd similarly work out the value of $2^i$…
IRSAgent
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Complex Numbers Closed under division?

My question is very simple that is: Is Complex number closed under division? Can we consider this 0+0i as complex numbers? or it is not a complex number.
user165385
  • 71
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Irrational Roots of Complex Numbers

Given a complex number $e^{ix}$, the nth root can be computed using Euler's formula: $$e^{i(x + 2k\pi)/n} = \cos((x+2k\pi)/n)+i \sin((x+2k \pi)/n).$$ If $n$ is an irrational number, can $k$ vary from negative infinity to infinity without repeating…
tjent
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Distributing powers on complex numbers

Can I not distribute powers on complex numbers as I do with real numbers? For example: Consider $$\left(\frac{1 + i}{1-i}\right)^n = 1$$ Distributing powers as in real numbers: $$(1+i)^n = (1 - i)^n$$ Taking log both sides and eliminating $n$ I am…
Gummy bears
  • 3,408
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Find modulus of $z$ given modulus of $(z-3w)/(3-z\overline{w})$

Question: (23) If $z_1$, $z_2$ are complex numbers such that $\left|\dfrac{z_1-3z_2}{3-z_1\overline{z}_2}\right|=1$ and $|z_2|\neq 1$, then find $|z_1|$. How would I attempt this question? I tried using values for $z_1$ and $\overline{z}_2$ but it…
Gummy bears
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Which one is correct for $\sqrt{-16} \times \sqrt{-1}$? $4$ or $-4$?

As we can find in order to evaluate $\sqrt{-16} \times \sqrt{-1}$, we can do it in two ways. FIRST \begin{align*} \sqrt{-16} \times \sqrt{-1} &= \sqrt{(-16) \times (-1)}\\ &= \sqrt{16}\\ &=4 \end{align*} SECOND \begin{align*} …
Masroor
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What are the three cube roots of -1?

What are the three cube roots of -1? Not sure if this is a trick question, But I have been asked this. one of the ansers is -1, what are the other 2?
mcc
3
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2 answers

Roots of complex numbers

It is known that exists formula for geting a square root of complex number without use of De Moivre formula. Will be interesting if we can find the cubic roots of complex numbers without using De Moivre formula.
Adi Dani
  • 16,949
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Complexes question $(z-1) \over (z+1)$

Given $z \ne -1$. Prove that $(z-1) \over (z+1)$ is an Imaginary number if and only if $|z| = 1$. I tried computing $(z-1) \over (z+1)$ by multiplying like that: $(z-1) \over (z+1)$$(z-1) \over (z-1)$ and then getting: $(z-1)^2$ but if $z = 1$…
Ilan Aizelman WS
  • 4,128
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Solve $z^4+16=0$ where $z$ is a complex number

The following exercise is related to complex numbers so $z$ is a complex number. Can you please check whether I solved correctly the exercise. $$z^4+16=0$$ $$z^4=16i^2$$ $$z^2=4i$$ I transformed the complex number $4i$ into the trigonometric form,…
wonderingdev
  • 1,761
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Solve: $(\cos x+i\sin x)(\cos 2x+i\sin 2x)(\cos 5x+i\sin 5x)={i+1\over \sqrt 2 }$

Can you please give me a hint for the following exercise: $$(\cos x+i\sin x)(\cos 2x+i\sin 2x)(\cos 5x+i\sin 5x)={i+1\over \sqrt 2 }$$ Thank you!
wonderingdev
  • 1,761
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3 answers

Finding the least value for points in a locus

The question goes like this: On an Argand diagram, sketch the locus representing complex numbers $z$ satisfying $|z+i|=1$ and the locus representing the complex numbers $w$ satisfying $\arg(w-2)=\dfrac{3π}{4}$. Find the least value of $|z-w|$ for…
George
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