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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Determining locus of a complex number

If $|z+\bar{z}| =|z-\bar{z}|$ then determine the locus of $z$. This is how I attempted it , The given statement implies that $z$ is equidistant from -$\bar{z}$ and $\bar{z}$ so it lies on the perpendicular bisector of $z$ and $\bar{z}$ which is a…
Aditi
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Is $i$ equal to $\sqrt{-1}$?

Some of my math teachers said that $i=\sqrt{-1}$ is the wrong definition for $i$ and that the correct definition is $i^2=-1$. If the second definition if true, then does it mean $\sqrt{-1}= ±i$? Which of the two definitions is true?
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Motivation for complex numbers

I understand that the complex numbers have a geometric representation as ordered pairs of real numbers. From what I understand complex numbers are very useful for working in two dimensions mathematically. The fact that $i^2 = -1$ follows from the…
Joe
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I don't understand $\sqrt{-9i}$.

I try to visualise it on a graph, where x is real numbers and y is the imaginary numbers. $\sqrt{9} = (3,0)$ and $(-3,0)$. $\sqrt{-9} = \sqrt{-1} \times \sqrt{9} = (0,3) $ and $(0,-3)$. $\sqrt{9i}$ = $\sqrt{-9i}$ = Basically, I have some trouble…
dom
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Can we determine if a complex number is greater than another?

Is it possible to determine if one complex number is greater than another? Or as the question implies is there an "order" to complex numbers (like 1 is before 2 in the real numbers)? I thought that would could simply use the modulus to determine if…
frog1944
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Simplifying sum of powers of conjugate pairs

The result of summing a conjugate pair of numbers each raised to the power $n$: $$ (a + bi)^n + (a - bi)^n $$ Produces a real number where $a + bi$ is a complex number. Given the result is real, is there a simplified way to express the above…
keith
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Proving that the roots of a complex equation lie within a circle

Prove that, for integral values of $n\ge 1$, all the roots of the equation $$nz^n=1+z+z^2+...+z^n$$ lie within the circle $\vert z\vert=\frac{n}{n-1}$ Taking modulus on both sides, $$n\vert z\vert^n=\vert1+z+z^2+...+z^n\vert$$ Using triangle…
Aditya Dev
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Find a maximum of complex function

I am trying to find a simple method that does not use the tools of advanced differential calculus to find following maximum, whose existence is justified by the compactness of the close ball $\Delta$ of $\mathbb C$ and continuity of the…
Mohamed
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Finding the (complex) solutions $z$ to $z^4 = w$

What are possible solutions to $$z^4 = w$$ where $z \in \mathbb{C} $ and $w \in \mathbb{R}$? Here is my attempt: Write both numbers in polar form: $r^4(\cos 4 \theta + i\sin 4 \theta) = w(\cos 0 + i \sin 0) $. Working with the conventions of $r…
snoram
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How does one prove $\sqrt{2}|z|\geq |\operatorname{Re}z|+|\operatorname{Im}z|$ holds for all $z\in \mathbb{C}?$

This is Exercise EP $14$ from Fernandez and Bernardes's book Introdução às Funções de uma Variável Complexa (in Portuguese). The authors ask us to prove that the inequality $$\sqrt{2}|z|\geq |\operatorname{Re}z|+|\operatorname{Im}z|$$ holds for all…
user23505
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Geometric/Simpler proof for the following complex numbers problem

I wonder if there is a geometric proof or a short proof of the following: let $z_1,z_2,z_3$ be three complex numbers of modulus $r$. prove that the number $$ \frac{r^4+z_1z_2+z_2z_3+z_3z_1}{z_1+z_2+z_3+z_1z_2z_3} $$ is also of modulus $r$. I…
Beni Bogosel
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Equality of complex numbers

I'm currently reading some notes on Complex Numbers and came across this 'proof' regarding the equality of complex numbers. Claim: Two complex numbers $a+bi$ and $c+di$ are equal iff $a=b$ and $c=d$, where $a,b,c,d$ are real numbers. Now this is an…
Trogdor
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Do "complex percentages" exist?

Well, the origin of this question is a little bit strange. I dreamed - with a book called "Percentages and complex numbers. When I woke up, I thought: "Is this real?" So I started thinking: 1% of 100 = 1 3% of 100 = 3 And more: i% of 100 =…
Ian Mateus
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How can one calculate $i^{i^{i^{i^{.^{.^.}}}}}$?

I was reading an article that said about the value of $i^i$, and I thought to do the following $$\left(i^{i^{i^{i^{.^{.^.}}}}}\right)$$i.e, infinite powers of $ i $.
benjamin_ee
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Solve $z^4 + 4 = 0$

I'm trying to solve it by using its polar form, but then I get $$ \begin{align*} z^4 &= (\rho e^{i\phi})^4 = \rho^4 e^{4i\phi}\\ &= -4 = -4 e^{0i}\\ \end{align*} $$ From the definition of equality of complex numbers, $\rho^4 = -4$ and $4\phi = 0…
user1002327
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