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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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how to solve circle dividing equation (complex numbers)

I have a equation that should divide a circle in even parts. As I found its called circle-dividing equation. I'v found same information how to solve a equation which has a form like this: $$z^6 = 1$$ one of the first steps was to transform the…
Karl Adler
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complex number congjugate criteria

Is it true that if a complex number $z_2$ times $z_1$ is the square of norm of $z_1$, then $z_2$ is the conjugate of $z_1$? $z_2 = \bar{z_1} \Leftrightarrow z_1z_2 = \|z_1\|^2?$ It occurs to me to be true: Let $z_1 = r_1e^{i\theta_1}, z_2 =…
1LiterTears
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finding a solution for $m$ given $(1+i)z^2-2mz+m-2=0$

Given the equation: $(1+i)z^2-2mz+m-2=0$, while $z$ is complex and $m$ is a parameter. For which values of $m$ the equation has one solution? So my idea was to use: $b^2-4ac=0$ for $ax^2+bx+c=0$ But it leads to difficult computation which i could…
davon
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How to prove $\text{Ln}\,z_1^{z_2}=z_2\,\text{Ln}\,z_1$ or $(e^{z_1})^{z_2}=e^{z_1z_2}$?

I am trying to prove $\text{Ln}\,z_1^{z_2}=z_2\,\text{Ln}\,z_1$. If I know that $(e^{z_1})^{z_2}=e^{z_1z_2}$, then it would be: Let $z_1=e^w$, then $z_1^{z_2}=(e^w)^{z_2}=e^{wz_2}$. Take Ln on both side, hence…
Gary
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Polar form (complex number)

Write in the form $x+jy=\sqrt{32} (cos(\frac{\pi}{4}) + jsin(\frac{\pi}{4}))$. I'm confused with what the question is asking for and my book doesn't give any examples, help would be much appreciated :)
sarah ryan
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$z^2=-3-3i$ solve for $z$

Use De Moivre's theorem to solve the equation $z^2=-3-3i$. (Give your answers in polar form) Can you please explain why there are two answers? I cannot seem to understand why. By the way, the answers…
cla1n
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express $\mathrm{e}^{(2+i \pi/2)}$ in form $a + bi$

I'm just starting out into Complex numbers, polar and exponential form etc... I can happily convert numbers such as $\mathrm{e}^{i \pi/2}$ but I'm a little stumped with how to handle the extra + 2 which appears in $\mathrm{e}^{(2+i \pi/2)}$. Can…
paar
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Prove that $\frac{\text{x+y}}{\text{r$^2$+xy}}\in \mathbb{R}$

Let two complex numbers $\rm x$ and $\rm y$, satisfy $\rm |x|=|y|=r>0$ and $\rm r^2+xy \neq 0$. Prove that $$\frac{x+y}{r^2+xy}\in \mathbb{R}$$
Iloveyou
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every complex number can be put in this form?

I am reading a proof of a theorem and I see this: "if we put $F(x)$ in this form: $F(x)= r e^{i\theta}$" $F(x)$ takes values in $\mathbb{C}$, so the question is: In $\mathbb{C}$, every element can be put in that form?
Maths Student
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Express $(1-i)^{11}$ in cartesian form.

Express $(1-i)^{11}$ in cartesian form. Apart from expanding the expression, I don't know how to do this. I've looked at the solution and still don't understand how/why it has been done.
Mr Croutini
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Complex Number Homework

How do you solve this question without a calculator? $z^2 = 4- 3i$. Find $z$. I know how to find the answer to this question using de Moivre's theorem with a calculator. What I do is I start out by finding the angle of $z^2$ by finding…
user110069
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Can someone point out the mistake here?

Let $z_1,z_2 \in \mathbb{C}$. Write $z_1=e^{it_1}$ and $z_2=e^{it_2}$. (Let their modulus be 1). Then $$|z_1-z_2|^2=(z_1-z_2) \overline{(z_1-z_2)}=(z_1-z_2)(\overline{z_1} - \overline{z_2})=(e^{it_1}-e^{it_2})(e^{-it_1}-e^{-it_2})$$ But it seems…
ireallydonknow
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Solve using polar form

$x^4-x^3+x^2-x+1=0$. This equation has to be solved in complex numbers using polar form. I tried grouping the power of x terms but it is not working. Any other idea of solving the equation?
Vishnu
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Algebra of the complex plane

Suppose $a$,$b$,$c$,$d$ be the points of set of all complex no$(C)$. with $c$ not equal to $0$ and $ad$ not equal to $bc$.$f$ be a function such that $f(z)$=$(az+b)$/$(cz+d)$. how to prove that $f$ defines a bijection between $C$-{d/c} and…
liesel
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For which values ​​of $z$ the inequality $|e^{z-1}|<2$ holds

I want to find for which values of $z$ the following inquality holds $$|e^{z-1}|<2$$ what I tried to do is: $$|e^{z-1}|=|e^{x-1+y\mathbb{i}}|<2$$ $$=e^{x-1}\cdot(\cos(y) + \mathbb{i} \sin(y))$$ OR another thing I tried: $$z-1=\ln(2) \rightarrow…
Ofir Attia
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