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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What is the difference between $\arg(z)$ and $\operatorname{Arg}(z)$, where $z=a+bi$

What is the difference between the $\arg(z)$ and the $\operatorname{Arg}(z)$, where $z$ is a complex number of the form $a+bi$, for example: $z = -2 - 2i$ The angle from the positive x-axis to the vector would be $5π/4$ Does that mean that the…
JLL
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Could anyone explain this curious solution to $7^{2x}= 2^x$

I typed the following on wolfram alpha today : $7^{2x} = 2^x$ and found this as a solution besides $x=0$: $x = \dfrac{2\pi i n}{\log2 - 2\log7}$ where $log$ has a base of $e$ and $n$ is any integer. I am scratching my head wondering where they got…
poli-sci
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The roots of the equation $z^n=(1+z)^n$...

The complex roots of the equation $$z^{n}=(1+z)^{n}$$ $A.$ are vertices of a regular polygon $B.$ lie on a circle $C.$ are collinear $D.$ none of these Don't know where to start.. Please help.! NOTE: $z$ is a complex number.
Apurv
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Complex numbers equation problem

I've been having some trouble with this complex question (not my best topic), and I was wondering if I could get any hints or explainations on how to do it. Prove that all the roots of the equation…
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What is the significance of the expression $f(z_1,z_2) = \dfrac{z_1 - z_2}{1+ z_1 z_2 }$?

Across a number of independent problem books involving complex numbers, I have seen the following function pop up $$f(z_1,z_2) = \dfrac{z_1 - z_2}{1+ z_1 z_2 }$$ where $z_1, z_2 \in \mathbb{C}$ and $z_1z_2 \neq -1$. It appears to have some nice…
Trogdor
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Why isn't $\frac{\cos(\alpha) + i\sin(\alpha)}{\cos(\alpha)-i\sin(\alpha)} = 1 + i\sin(2\alpha)$?

I realize that the equality from the title is wrong, but I do not understand why. A correct equality would be $\frac{\cos(\alpha) + i\sin(\alpha)}{\cos(\alpha)-i\sin(\alpha)} = \cos(2\alpha)+i\sin(2\alpha)$ by multiplying with the conjugate of the…
J__n
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Motivation behind the definition of multiplication in $ \mathbb{R}^2 $

I was reading Titu Andreescu’s book on complex numbers, Complex Numbers A to Z. In the book, the author defines multiplication in $ \mathbb{ R }^2 $ as, $ (a,b)( c,d) = (ac - bd, ad + bc) $ I understand how this definition relates to multiplication…
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Finding the number of complex numbers satisfying $|z|=\text{max}\{|z-1|,|z+1|\}$

Find the number of complex numbers satisfying $|z|=\text{max}\{|z-1|,|z+1|\}$ My Attempt: Let $z=x+iy$, so, $$\sqrt{x^2+y^2}=\text{max}\{\sqrt{(x-1)^2+y^2},\sqrt{(x+1)^2+y^2}\}$$ Case I: $\sqrt{x^2+y^2}=\sqrt{(x-1)^2+y^2}\implies \pm x=x-1\implies…
aarbee
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Complex number related problem

Let $z_1,z_2,z_3$ be complex numbers such that $|z_1|=|z_2|=|z_3|=|z_1+z_2+z_3|=2$ and $|z_1–z_2| =|z_1–z_3|$,$(z_2 \ne z_3)$, then the value of $|z_1+z_2||z_1+z_3|$ is_______ My solution is as follow ${z_1} = 2{e^{i{\theta _1}}};{z_2} =…
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Evaluating $(1+x)(1+x^2)\ldots (1+x^{p-1})$ at pth roots of unity for odd primes p

As part of a combinatorics question I was able to simplify it into evaluating $(1+x)(1+x^2)\ldots (1+x^{p-1})$ at pth roots of unity for odd primes p. I have guessed and checked with a computer that this product is always $1$ except for when we use…
Karan Elangovan
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Cube roots of the complex numbers 1+i?

I cant get any good reference in my books regarding cube of complex numbers. Please help me find cube roots of the Complex number i+1??
hemant
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Complex Number Roots

When I am solving to find the root of a complex number what exactly am I finding? Does it relate somehow to the complex plane? What would be it's geometrical representation if it has one?
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What do the colours on a Mandelbrot Set image represent?

From my understanding of the Mandelbrot Set, it represents the set of complex numbers 'c' for which the function: f(z) = z^2 + c that doesn't diverge. The set of these numbers is coloured black while everything else is white. I believe it is also…
Atiksh
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If $\omega$ is a cube root of unity $\not = 1$ then find the minimum value of $|a+b\omega +c\omega^2|$, where $a,b,c$ are integers but not all equal.

Let $z=a+b\omega + c\omega^2$ $$z=a+b\omega -c (1+\omega)$$ $$z=a-c+\omega (b-c)$$ Therefore $$|a-c+\omega (b-c)| \ge ||a-c|-|b-c||$$ How should I proceed?
Aditya
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Prove that $\alpha=\beta=\gamma$ where $\alpha,\beta, \gamma$ are complex numbers.

Let $x,y,z$ be non-zero real numbers. Suppose $\alpha,\beta, \gamma$ are complex numbers such that $|\alpha|=|\beta|=|\gamma|=1$. If $x+y+z=0=\alpha x+\beta y+\gamma z,$ then prove that $\alpha=\beta=\gamma$. I was trying to use the property that…