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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Using complex solutions in a factorisation

I'm working through an assignment, and have become stuck understanding the question... In part (a) I am asked to solve the equation: $z^5 = -1$ I have done this, so I now have a set of solutions: $z_0, z_1, z_2, z_3, z_4$ (one $\in \mathbb{R}$ the…
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How to prove the formula for the argument of a complex number?

$\arg(x + iy) = 2 \cdot \arctan(\dfrac{y}{x + r})$ there is always the mark, this is derived from the 'Half-angle formula' How can I come from $\tan(\phi) = \tan(\phi + k \pi) = \dfrac{y}{x}$ to $\phi = \arg(x + iy)$?
cis
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Number of complex numbers such that $z^{80} = 1$ and other properties

Let $$A = \left \{ z \in \mathbb{C} : \Re z > 0, \Im z < 0, z^{80} = 1 \right \}$$ Then, the number of elements in $A$ is $19$, $20$, $21$, or $22$? I just started studying complex numbers and I've come across this question. I really don't know…
rubik
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Complex numbers and their modulus

Can we cancel the modulus on complex numbers? For example: If we have $$|x + iy| = |n + im|$$ can we simply ignore the modulus on both sides? Or is that a false assumption?
Gummy bears
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Complex variable algebra mishap

One question on a problem set was the following: Show that $x^2 - y^2 = 1$ can be rewritten as $z^2 + \bar{z}^2 = 2$. (With $z = x + iy$) So I started working from the first expression based on the algebraic definition of $Re$ and $Im$ to see…
user3002473
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Cartesian $-10i$ to Polar form

I am trying to convert the following problem to polar form: $$z=-j10.$$ Using this equation, where $|z|=r=\sqrt{x^2+y^2}$ and $\arg z=\theta=\arctan(y/x).$ $$\eqalign{z&=|z|e^{j\arg z}\\ &=re^{j\theta}\\&=r\angle\theta.}$$ I determined that x = 0…
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Product of $n$ complex numbers in rectangular form.

Given a complex number $z_j$ such that $$z_j\in\{a_1+b_1 i,\ a_2+b_2i, \ ...\ ,a_n+b_ni\}$$ is there formula for calculating $$z_1 \cdot z_2 \cdot \dots \cdot z_n =\prod_j z_j?$$ For two complex numbers you use the distributive law,…
user124862
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Complex number equality

To Prove: $\displaystyle (\cos\theta +i\sin\theta)^4(\sin\theta-i\cos\theta)=\cos 8\theta+i\sin 8\theta$ My Attempt: $\displaystyle (\cos4\theta +i\sin4\theta)(\sin\theta-i\cos\theta)=\sin5\theta+i\cos5\theta$ Now what ?
square_one
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Finding Complex Number $z$ in $\frac{z+2i}{z-2i}=\frac{7-6i}{5}$

What I did: Cross Multiply, try to expand out the mod and args, but they all seem to lead to dead end (probably I am not seeing something)
Jiew Meng
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Is it true that "there is no such thing as the square root of minus one"?

Is the statement "there is no such thing as the square root of minus one" a true statement? It seems to me that we need to be careful about the word "the" as it appears in the statement. If we see it as implying uniqueness, then surely the statement…
Hammerite
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Sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1 \}$

My question is to sketch the set $\{ z \in \mathbb{C} | \left|\frac{z-i}{z+i}\right|<1\}$ in the complex plane. I substituted $z$ for $a+bi$, but did not get…
ahorn
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Solve for x if $z$ is a complex number such that $z^2+z+1=0$

I was given a task to solve this equation for $x$: $$\frac{x-1}{x+1}=z\frac{1+i}{1-i}$$ for a complex number $z$ such that $z^2+z+1=0$. Solving this for $x$ is trivial but simplifying solution using the given condition is what's bothering me.…
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Other complex systems

My question would be very short. As we all know, there are complex, quaternion number systems, which are based on multiplication and roots. So, my question is... Is there any other complex number systems that are NOT based on number roots? For…
Martynas Riauka
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Solving simultaneous equations with complex coefficients using real methods

My circuits analysis textbook teases that there's a way to convert a set of n complex equations into a set of 2n real equations, which can then be solved using any calculator that can solve real simultaneous equations. That is, no capability with…
Greg Bell
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Complex roots of a complex number

I know how to find the roots to the equation $z^n=w$, for $n \in \mathbb{R\setminus\{ 0\}}$ (by writing $w$ as $re^{i(\theta+2k\pi)}$), and taking the nth root of both sides, which I'm perfectly happy with, since $r^{1/n}$ is easy to find (with a…
beep-boop
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