Questions involving complex numbers, that is numbers of the form $a+bi$ where $i^2=-1$ and $a,b\in\mathbb{R}$.

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10
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1answer
236 views

Interpret to a complex plane!

$\newcommand{\Re}{\operatorname{Re}}\newcommand{\Im}{\operatorname{Im}}$The question is: Interpret $$ \Re z + \Im z = 1 $$ geometrically in the complex plane. Writing $z = x + yi$, the condition ...
10
votes
1answer
209 views

For which complex $a,\,b,\,c$ does $(a^b)^c=a^{bc}$ hold?

Wolfram Mathematica simplifies $(a^b)^c$ to $a^{bc}$ only for positive real $a, b$ and $c$. See W|A output. I've previously been struggling to understand why does $\dfrac{\log(a^b)}{\log(a)}=b$ and ...
10
votes
1answer
248 views

Complex Exponential False “Proof” That All Integers Are $0$

The following false "proof" is attributed to Thomas Clausen in 1827, and was stated on page 79 of Nahin's An Imaginary Tale. $e^{i2\pi n}=1$ for all integers $n$. So \begin{align*} ee^{i2\pi ...
9
votes
8answers
2k views

What's the thing with $\sqrt{-1} = i$

What's the thing with $\sqrt{-1} = i$? Do they really teach this in the US? It makes very little sense, because $-i$ is also a square root of $-1$, and the choice of which root to label as $i$ is ...
9
votes
9answers
218 views

How to solve $z^3 + \overline z = 0$ [duplicate]

I need to solve this: $$z^3 + \overline z = 0$$ how should I manage the 0? I know that a complex number is in this form: ...
9
votes
5answers
1k views

Alternative to imaginary numbers?

In this video, starting at 3:45 the professor says There are some superb papers written that discount the idea that we should ever use j (imaginary unit) on the grounds that it conceals some ...
9
votes
3answers
862 views

How to raise a complex number to the power of another complex number?

How do I calculate the outcome of taking one complex number to the power of another, ie $\displaystyle {(a + bi)}^{(c + di)}$?
9
votes
6answers
580 views

Has anyone talked themselves into understanding Euler's identity a bit?

What does the ratio of the circumference of a circle to its diameter have to do with the base of the natural logarithm and $\sqrt{-1}$?
9
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2answers
1k views

Summation of complex numbers

This is a series problem where the terms are complex numbers. I am looking for a better approach to solving this problem. If $\displaystyle z = \frac{1+i}{\sqrt2}$, Evaluate $1 + z + z^2 + ... + ...
9
votes
4answers
247 views

Real part of $(1+2i)^n$

Is it true that for all $n\in \mathbb{N}$, $n\ge 2$ we have $$|\textrm{Re}((1+2i)^n)|>1?$$ I do know de Moivre's Theorem. I do not know how to show that $|\sqrt{5}^n\cos(n\arccos\left ( ...
9
votes
1answer
5k views

Is L'Hopitals rule applicable to complex functions?

I have a question about something I'm wondering about. I've read somewhere that L'Hopitals rule can also be applied to complex functions, when they are analytic. So if have for instance: $$ \lim_{z ...
9
votes
2answers
618 views

How to calculate $i^i$ [duplicate]

I've been struggling with this problem, actually I was doing a program in python and did 1j ** 1j(complex numbers) (In python ...
9
votes
1answer
966 views

How to choose a proper contour for a contour integral?

When analyzing real integrals with contour integrals, how does one choose a proper contour integral? Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
9
votes
2answers
254 views

Maximum of $\frac{\sin z}{z}$ in the closed unit disc.

I have some trouble with the following question: Let $$f(z)=\frac{\sin z}{z},\quad\text{for }z\in\mathbb{C}.$$ What is the maximum of $f$ in the closed unit disc ...
9
votes
3answers
277 views

How to make $\log x^a = a\log x$ work using multivalued complex approach

The following identity holds for all $a$ and $x$ using the principal branch: $$ \log x^a = a\log x + 2\pi i \left\lfloor \pi-\Im (a\log x) \over 2\pi \right\rfloor $$ e.g. for $a=2$, ...
9
votes
2answers
448 views

If the sequence $(a_n b_n)$ converges and $a_n \to 0$, when does $(b_n)$ converge too?

Given that the sequence $(a_n b_n)$ converges and $a_n \to 0$, are there conditions which can be placed on $(b_n)$ and/or $(a_n b_n)$ so that $(b_n)$ converges as well?
9
votes
2answers
94 views

On discriminants and nature of an equation's roots?

Edited: All equations in the post are assumed to have all real coefficients and are minimal polynomials. While trying to ascertain if the Brioschi quintic $B(x)=x^5-10cx^3+45c^2x-c^2=0$ could ever ...
9
votes
2answers
92 views

When does $(e^a)^b = e^{ab}$ hold?

For a complex number $A$ and a real number $B$, when does the well-known formula $(e^A)^B = e^{AB}$ fail? Or does it hold at all for complex A? Since $e^{2\pi i} = 1$, if this formula holds for ...
9
votes
3answers
138 views

Simple way to estimate the root of $x^5-x^ 4+2x^3+x^2+x+1=0$

How to give a mathematical proof that for all complex roots of $x^5-x^ 4+2x^3+x^2+x+1=0$, their real part is smaller than 1, and there is at least one root whose real part is larger than 0. If ...
9
votes
1answer
537 views

How does quater-imaginary (and other imaginary/complex bases) work?

So I've been working on a simple base-conversion program, and having given it the ability to convert from decimal to any base $> 1$ or $< 0$, as well as the $p$-adic (bijective, I think?) bases, ...
9
votes
0answers
69 views

An Inequality with complex numbers and $1/\pi$ [duplicate]

Let $\displaystyle \{z_1,z_2, \ldots, z_n\}$ be $n$ complex numbers such that: $\displaystyle \sum\limits_{k=1}^n|z_k| = 1$ Then we have to show that, there is a subset $S$ of $\{1,2,\ldots,n\}$, ...
8
votes
5answers
1k views

Taking the square root of an imaginary number

We know that when we take the square root of a negative real number, it's realness "splits open" and an "imaginary" dimension is introduced (characterized by the presence of iota). The question is, ...
8
votes
6answers
921 views

Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + … + r^{n-1} = 0$.

Show that if $r$ is an nth root of $1$ and $r\ne1$, then $1 + r + r^2 + ... + r^{n-1} = 0$. I think I can represent all the roots of 1 as follows: $r = 1^{\frac{1}{n}} ( \frac{\cos{2\pi k}}{n} + ...
8
votes
7answers
2k views

How to solve $e^{ix} = i$

I am taking an on-line course and the following homework problem was posed: $$e^{ix} = i$$ I have no idea how to solve this problem. I have never dealt with solving equations that have imaginary ...
8
votes
4answers
435 views

Is it correct to say that $\mathbb{R}$ has fewer elements than $\mathbb{C}$ if both are infinite?

My math teacher said that. I disagreed, but he said that I was wrong. But I'm not convinced - is it really right? Please notice that I'm not talking about $\mathbb{R}$ $⊂$ $\mathbb{C}$, but ...
8
votes
9answers
4k views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
8
votes
3answers
461 views

Is there a problem in defining a complex number by $ z = x+iy$?

The field $\mathbb{C} $ of complex numbers is well-defined by the Hamilton axioms of addition and product between complex numbers, i.e., a complex number $z$ is a ordered pair of real numbers $(x,y)$, ...
8
votes
3answers
574 views

How to prove $|z_1-z_2| \geq |z_1|-|z_2|$ in other way than this?

How to prove $|z_1-z_2| \geq |z_1|-|z_2|$ in other way than this? I mean I tried to find on the internet but could not find. I ask for more straighforward way than the proof that is presented for item ...
8
votes
3answers
487 views

Proof for law of complex exponents using only differential equation

I just read that an elegant proof exists that the law of exponents also holds for complex numbers ($a,b,z$ all complex): $$e^{a+b}=e^ae^b,$$ which only uses the definition that $$y=e^{zt}$$ is a ...
8
votes
4answers
487 views

Why does the imaginary number $i$ satisfy $i\times 0=0$?

Why does the imaginary number $i$ satisfy $i\times 0=0$? I mean, we don't really know what $i$ is. How could we be sure about that? I think there's a reason behind why mathematicians decided that.
8
votes
5answers
603 views

Is there any way to represent an imaginary number?

Is there any way to represent an imaginary number? Like the square root of -1? Is there any possible way to do this? Sorry if you think this is a dumb question. I am a 7th grade student in ...
8
votes
5answers
353 views

Does there exist a field $K$ such that $\mathbb R \subsetneq K \subsetneq \mathbb C$?

I'm thinking of unions of $\mathbb R$ with some subset of $\mathbb C$ but am not sure how to approach this without ending up with all of $\mathbb C$. Doe anyone have any suggestions?
8
votes
2answers
3k views

The set of real numbers is a subset of the set of complex numbers?

So, I was taught that $\mathbb{Z}\subseteq\mathbb{Q}\subseteq\mathbb{R}$. But since the set of complex numbers is by definition $$\mathbb{C}=\{a+bi:a,b\in\mathbb{R}\},$$ doesn't this mean ...
8
votes
3answers
138 views

If $f(z)=\cfrac{z+1}{z-1}$ , then find $f^{1991}(2+i)$

If $f(z)=\cfrac{z+1}{z-1}$ , then find $f^{1991}(2+i)$ Forgive me if the question is too short but really I don't know how to do this one. That's what I have done so far: ...
8
votes
5answers
661 views

What's the importance of a formula for the real and imaginary parts of a complex number?

I've learned that $$\bbox[8px,border:1px solid black]{\operatorname{Re}(z)= \frac{z+\overline{z}}{2} \qquad \qquad \operatorname{Im}(z)=\frac{z-\overline{z}}{2i}} $$ And that in the number $z=a+bi$, ...
8
votes
2answers
409 views

Why is $\left(e^{2\pi i}\right)^i \neq e^{-2 \pi}$?

Here's my (obviously flawed) proof that $1=e^{-2 \pi}$: $$ 1^i=1\\ e^{2 \pi i} = 1\\ \left(e^{2\pi i}\right)^i = 1^i\\ e^{-2 \pi} = 1 $$ What's the issue? I understand that exponentiation is not ...
8
votes
4answers
279 views

When are we (not) allowed to replace $x$ by $ix$?

It seems to be quite a common manipulation to replace $x$ by $ix$. Every time I see it's being done in a textbook, I blindly trust the author without really understanding when are we allowed to do so ...
8
votes
3answers
246 views

Putting ${n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ in a closed form

As the title says, I'm trying to transform $\displaystyle{n \choose 0} + {n \choose 5} + {n \choose 10} + \cdots + {n \choose 5k} + \cdots$ into a closed form. My work: $\displaystyle\left(1 + ...
8
votes
4answers
4k views

Non-integer powers of negative numbers

Roots behave strangely over complex numbers. Given this, how do non-integer powers behave over negative numbers? More specifically: Can we define fractional powers such as $(-2)^{-1.5}$? Can we ...
8
votes
4answers
110 views

What is $(-1)^{3/2}$?

Which one is the right way? $$(-1)^{\frac32}=(-1)^{1+\frac12}=-1\times \sqrt{-1}=-i$$ Or, $$(-1)^{\frac32}= \left((-1)^{\frac12}\right)^{3}= i^3=-i$$ Or, ...
8
votes
2answers
464 views

Are there any elegant methods to classify of the Gaussian primes?

Out of curiosity, are there any relatively quick classifications of all the Gaussian primes, the primes in $\mathbb{Z}[i]$? I found a classification here, but the process comes off as rather tedious. ...
8
votes
2answers
407 views

How to show that $\overline{zw}=\overline{z}\,\overline{w}$?

I thought about first multiplying two complex which aren't in the conjugate form: $$zw=a c+i a d+i b c-b d$$ Then multiply two complex conjugates: $$\overline{z}\,\overline{w}=a c\color{red}{-}i a ...
8
votes
4answers
99 views

$z^n=(z+1)^n=1$, show that $n$ is divisible by $6$.

we are given $z^n=(z+1)^n=1$, $z$ complex number. we want to prove that $n$ is divisible by $6$. I showed that $|z|=|z+1|=1$. Hence $z$ is on the intersection of two unit circles, one centered at ...
8
votes
2answers
349 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
8
votes
3answers
154 views

What is the right treatment for $0^i$?

I need to calculate a limit of a complex expression (had it in a physics research) that contains a term $(r-b)^p$ for $r\rightarrow b+$ where $r,b$ are reals, and $p$ is complex, let's suppose for ...
8
votes
2answers
470 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
8
votes
3answers
186 views

How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality?

Problem: How do you find the maximum value of $|z^2 - 2iz+1|$ given that $|z|=3$, using triangle inequality? My attempt: $$|z^2 - 2iz+1|\le|z|^2+2|i||z|+1$$ $$\implies |z^2 - 2iz+1|\le16$$ ...
8
votes
3answers
2k views

How does this equality on vertices in the complex plane imply they are vertices of an equilateral triangle?

I've read that if the complex numbers $a_1$, $a_2$ and $a_3$ are the vertices of a triangle in the complex plane such that $$ a_1^2+a_2^2+a_3^2=a_1a_2+a_2a_3+a_1a_3 $$ then the vertices are actually ...
8
votes
1answer
129 views

Roots of iterations of polynomials

Let $f \in \Bbb Q[X]$ a polynomial, and let denote by $f^n$ the composition $\underbrace{f \circ \cdots \circ f}_{n \text{ times }}$. Let $R(f^n) \subset \Bbb C$ the roots of $f^n$. I'm interested in ...
8
votes
1answer
111 views

Complex Numbers and their relationship with higher Mathematics

Let $z_1, z_2, \cdots, z_n$ be complex numbers satisfying $$|z_1|+|z_2|+\cdots +|z_n|=1.$$ Prove that there is a non-empty subset of $\{z_1,z_2,\cdots,z_n\}$ the sum of whose elements has modulus at ...