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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.

19229 questions
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Find the common roots of the equations $x^{12}-1=0$ and $x^4+x^2+1=0$

OPTIONS A) $\pm \omega$ B) $\pm \omega^2$ C) $\pm \omega, \pm \omega^2$ A very easy question obviously, but I have a few doubts. From the second equation, it is obvious that $x^2$ is imaginary cube root of 1, generally represented by $\omega$ and…
Aditya
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How do I show $\arccos(4)$ in form $a+bi$?

How do I show $\arccos(4)$ is ${2\pi k}$ $\pm$ $i\operatorname{arcosh}(4)$? I am getting $\pm$ $iln(4+\sqrt{15}$) When I just use $\cos(z)$ =$\cosh(iz)$, I get: $-i\operatorname{arcosh}(4)$. I’m lost! If someone could type out the full proof, it'd…
Noobcoder
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Equality of fractions in $\mathbb{C}$

In $\mathbb{R}$, $$\frac{a}{b} = \frac{c}{d} \iff ad = bc.$$ In $\mathbb{C}$, division is a bit less clear-cut. Nonetheless, if we have complex numbers $z_1, \ldots, z_4$, can we still assert that $$\frac{z_1}{z_2} = \frac{z_3}{z_4} \iff z_1 z_4 =…
John P.
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Why does it not matter if we define $i$ as $\sqrt{-1}$ or as $-\sqrt{-1}$?

I'm self learning complex numbers and the definition of complex numbers that I use is as follows: The set $\mathbb{C}$ of complex numbers is $\mathbb{R}^2$ endowed with some sense of addition and multiplication defined as $(a,b) + (c,d) = (a+c,…
William
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Finding all complex solutions $z$ of an equation involving $z$ and its conjugate $\bar z$

I need to find all complex solutions of the equation: $2z + 2i\bar z = 0.$ This is what I have done so far: Let $z = x + yi$ and $\bar z = x - yi$ and then substitute into the equation. $2(x + yi) + 2i(x - yi) = 0$ $2x + 2yi + 2xi - 2yi^2 = 0$ $2x +…
ANM
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Embedding of $\mathbb{R}$ in $\mathbb{C}$

The natural inclusion of $\mathbb{R}$ in $\mathbb{C}$ is the mapping $$f: \mathbb{R} \to \mathbb{C}, \; x \mapsto x + 0i.$$ Given this, is it completely accurate to say that $x \in \mathbb{C}$? Or would we rather say that we can identity $x$ with…
John P.
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Complex numbers - finding minimum value

For all complex numbers $z_1,z_2$ satisfying $|z_1|=12$ and $|z_2-3-4i|=5$ , find the minimum value of $|z_1-z_2|$ Can we go like this : Let $z_1 = x +iy$ therefore $|z_1| = \sqrt{x_1^2+y_1^2}$ and $z_2 = x_2+ iy_2$ $|z_2-3-4i| = \sqrt{(x_2-3)^2+…
Sachin
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Complex conjugate of an involved expression

I understand the the complex conjugate of, say, $z:=\exp({a+ib})$ is $z:=\exp({a-ib})$. However , I have a composite expression and I'm not sure how to attack taking it's complex conjugate. Say $z:=i\exp({ib}) / ({a + ic})$ I would be tempted to…
Francky_V
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is $1^z=1$ for all complex values of $z$?

i would like to see if $1^z=1$ is valid for all complex variable $z$,first of all you can rewrite above equation as $1^{a+b*i}=e^0$ here i think that instead of $+$ sign, we may take take complex conjugate form or $-$ sign.from above…
dato datuashvili
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Complex numbers and function

What domain of the z-plane is represented by $$|z+2|+|z-2|\lt 4$$ Please give me a hint... Can I use the formula ? $$|z_1|+|z_2|\geqslant |z_1+z_2|$$
Ankita Pal
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Find all numbers z (if any) such that $e^{6z}=2i$.

I'm not really sure how I should go about solving this. This is a question from the section before logs, so we're supposed to solve it with the knowledge we have of exponential, trig, and hyperbolic functions and not logs. I know that $e^z=1$ iff…
Not2Scary
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if $w$ is a complex number, how to show that $w^\left(1/2\right)$ has 2 roots? except the case for $w=0$

I am looking for a convincing argument to show that if $w$ is a complex number, $w^{\frac 12}$ has 2 different roots.
math_student
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Find all possible values of $z$ such that $\cos z=\frac{3}{4}+\frac{i}{4}$.

Find all possible values of $z$ such that $\cos z=\frac{3}{4}+\frac{i}{4}$. Here is my attempt: $\cos…
user482939
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Solve the equation in complex plane

I am having a problem with a complex equation: $$z^7 - 2iz^4 - iz^3 - 2 = 0.$$ I do not know where to begin. I tried to multiply out the complex part by the $z$ substituting it with $x+iy$. I also tried converting into polar form and solving. I know…
Djboboch
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Is the standard definition of $i$ enough to uniquely describe it?

Ahlfors states "From elementary algebra the reader is acquainted with the imaginary unit $i$ with the property $i^2 = -1$." (Complex Analysis, Lars Ahlfors, page 1) Kreyszig (Advanced Engineering Mathematics) first defines complex numbers “as an…
user
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