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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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Why do we say the root of $-1$ doesn't exist when $(-1)^{3/2}$ squared gives $-1$?

$$ ((-1)^{3/2})^2=(-1) ^3=(-1)(-1)(-1)=-1 $$ I haven't done any further research nothing came up on google for this, my calculator said non real but all the working checks out, I checked the exponent rules again and again before this post edit: I…
Arsh Dixit
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Do all complex roots have to have a conjucate base?

I was working my way through a complex number worksheet with the promblem (3+2i)/z^2=1 and found a solution that is 1.817+0.550i but my other solution is -1.817-0.550i. These are not complex conjugates of each other which why I am confused, is this…
vince rabito
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Is numerical equivalence in the complex numbers actually an equivalence relation?

It seems that either numerical equivalence is not an equivalence relation or complex numbers do not have an inverse. We know that $-1 = i \cdot i$ We also know that $-1 = -i \cdot -i$ If we assume "=" to be an equivalence relation we can get $i…
objectivesea
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If $9z + 2/z$ is a real number, find the value of $zz^*$

I would like to seek some guidance with the following question. Let $z$ be a complex number with $Im(z)$ not equal to $0$. If $9z + 2/z$ is a real number, find the value of $zz^*$ My solutions are as follow Suppose $z = x + iy$ I would have the…
X-men
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How is this frequency function calculated?

I am trying to understand the solutions in my book and came across this: What I dont understand is how they get the last result and where all the $\theta$s and $\pi$s come from.
Clone
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$i^5=i$ so $\operatorname{Ln}(i^5)$ must be equal to $\operatorname{Ln}i$ but why ...

$\newcommand{\Ln}{\operatorname{Ln}}$We know that $i^5=i$ So we must be have $\Ln(i^5)=\Ln i$ Now, according to the "$\Ln$ Law" we must be have: $\Ln(i^5)=5\Ln i$ We will also know that $\Ln i = i\pi/2$ and $5\Ln i=5i\pi/2$ So, why $i\pi/2$ is not…
majid azimi
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What is the value of $\text{arg}(z) + \text{arg}(\bar{z})$ for a complex number?

What is the value of $\text{arg}(z) + \text{arg}(\bar{z})$ for a complex number? According to me the answer should be $0$, by basic $\tan^{-1}$ logic and also since $$arg(z) + \text{arg}(\bar{z}) = \text{arg}(z\bar{z}) = \text{arg}(x^2 + y^2) =…
marks_404
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Two (non compatible) ways of describing the locus $\{ z(\overline{z}+2)=3: z \in \mathbb{C}\}$ Which one is wrong?

Problem: Describe the locus $\mathcal{L}$ determined by the equation $$z(\overline{z}+2)=3, z \in \mathbb{C}.$$ My argument: Let $z=x+iy$. We compute as follows: $$(x+iy)(x-iy+2)=3;$$ $$(x^2+2x+y^2)+(2y)i=3;$$ equaling the real and imaginary parts,…
Amelian
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Why complex numbers are so fundamental and complete (in a sense)?

By "fundamental", I mean that it appears frequently in sciences and engineering. It seems that without complex numbers, we wouldn't be able to describe a lot of stuffs such as electromegnetic waves, quantum states. Of course, we can always use some…
Neo
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Argument of $z^3$ in Argand Plane

If $z=re^{i\theta}$ and $\theta\in (-\frac{\pi}{2},0)$, then argument of $z^3$ lies between ______. My approach is as follow $z^3=r^3e^{i3\theta}$ $arg(z^3)=3\theta$ Hence $3\theta\in (-\frac{3\pi}{2},0)$ or $3\theta\in (0,\frac{\pi}{2})$ but this…
Samar Imam Zaidi
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A cop-out argument by Feynman in his fabled Lectures on Physics

In the Algebra chapter of the Feynman Lectures on Physics, Feynman introduces complex powers: Thus $$10^{(r+is)}=10^r10^{is}\tag{22.5}$$ But $10^r$ we already know how to compute, and we can always multiply anything by anything else; therefore the…
Ad N Khan
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Finding the solutions of an equation of the form $f(z)=0$

That's how I understand it: usually, for finding the solutions of an equation $f(z)=0$, we try to convert this expression to a "simpler one" from which we can deduce which values can satisfy it; thus at the end of the procedure we arrive to an…
Amelian
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If $|z-1|=1$, find $\arg(z)$

If $|z-1|=1$, find $\arg(z)$ Putting $z=r(\cos\theta+i\sin\theta)$ in $|z-1|=1$, $$|r\cos\theta-1+ir\sin\theta|=1\\\implies r^2\cos^2\theta+1-2r\cos\theta+r^2\sin^2\theta=1\\\implies r=2\cos\theta$$ Putting $r$ in $z$,…
aarbee
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Suspicious Squares and Magnitudes of Complex Numbers

Problem: Let $z_1,$ $z_2,$ $z_3$ be complex numbers such that $|z_1| = 1,$ $|z_2| = 2,$ $|z_3| = 3,$ and $$|9z_1 z_2 + 4z_1 z_3 + z_2 z_3| = 12.$$Find $|z_1 + z_2 + z_3|.$ Progress: I received this problem yesterday as a challenge, but I cannot seem…
random guy 2000
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Expressing Complex Number in terms of its conjugate

Given a complex number $z$ , is it possible to express its conjugate $\bar z$ in terms of $z$ using only operations of addition , subtraction , multiplication , division and exponentiation on $z$ as a whole . In other words , expressing $z$ as $x +…
Vivek
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