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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11
votes
9answers
948 views

Solve the equation $z^3=z+\overline{z}$

I have been trying to solve an equation $z^3=z+\overline{z}$, where $\overline{z}=a-bi$ if $z=a+bi$. But I cant find any clues on how to move forward on that one. Please help.
11
votes
4answers
324 views

Does $\sin(x+iy) = x+iy$ have infinitely many solutions?

How to prove that $\sin(x+iy) = x+iy$ has infinitely many solutions? I know how to prove that $\sin(x) = x$ has only one solution, but I do not know how to extend this to complex analysis.
11
votes
3answers
606 views

Prove that the complex expression is real

Let $|z_1|=\dots=|z_n|=1$ on the complex plane. Prove that: $$ \left(1+\frac{z_2}{z_1}\right) \left(1+\frac{z_3}{z_2}\right) \dots \left(1+\frac{z_n}{z_{n-1}}\right) \left(1+\frac{z_1}{z_n}\right) ...
11
votes
6answers
872 views

Complex roots of $z^3 + \bar{z} = 0$

I'm trying to find the complex roots of $z^3 + \bar{z} = 0$ using De Moivre. Some suggested multiplying both sides by z first, but that seems wrong to me as it would add a root ( and I wouldn't know ...
11
votes
4answers
2k views

Why is it that Complex Numbers are algebraically closed?

I find it curious that Complex Numbers give enough flexibility to be algebraically closed, where the reals, rational numbers do not. For the reals it is easy to see that they cannot be used to solve ...
11
votes
3answers
766 views

How to raise -1 to non-integer powers

How do you calculate $(-1)^x$ where $x$ is some real number. For example, what is $(-1)^{\sqrt{5}}$. This question came as I was trying to computer $e^{i\pi a}$ where $a$ is irrational.
11
votes
4answers
2k views

Equation of the complex locus: $|z-1|=2|z +1|$

This question requires finding the Cartesian equation for the locus: $|z-1| = 2|z+1|$ that is, where the modulus of $z -1$ is twice the modulus of $z+1$ I've solved this problem algebraically ...
11
votes
3answers
146 views

Prove that an expression is zero for all sets of distinct $a_1, \dotsc, a_n\in\mathbb{C}$

A while ago one of my professors gave the class a problem "to think about when lying on the beach." Well, I've been on the beach several times since then to no avail and my curiosity has finally ...
11
votes
2answers
475 views

A property of roots of the truncated series for $\sin(x)$

Let $p_n(x) = \sum\limits_{k=0}^n \frac{(-1)^kx^{2k+1}}{(2k+1)!}$ In other words, $p_n$ is the polynomial made of the first $n$ terms of the Taylor expansion of $\sin(x)$ around $x = 0$. ...
11
votes
1answer
631 views

Why is $i$ called “imaginary”?

I was reading this question, and, after reading the responses, I felt like I had a much better understanding about how they're just another type of number definition. Why, then, are they called ...
10
votes
6answers
2k views

Solve $z^4+1=0$ algebraically

I know the result and how to solve it using trigonometry and De Moivre. However, given that the complex number $z$ can be rewritten as $a+bi$, how can I solve it algebraically?
10
votes
8answers
1k views

For complex $z$, $|z| = 1 \implies \text{Re}\left(\frac{1-z}{1+z}\right) = 0$

If $|z|=1$, show that: $$\mathrm{Re}\left(\frac{1 - z}{1 + z}\right) = 0$$ I reasoned that for $z = x + iy$, $\sqrt{x^2 + y^2} = 1\implies x^2 + y ^2 = 1$ and figured the real part would be: ...
10
votes
4answers
603 views

Which step in this process allows me to erroneously conclude that $i = 1$

I was playing around with imaginary numbers and exponents and came up with this: $$ i = \sqrt{-1} $$ $$ \sqrt{-1} = (-1)^{1/2} $$ $$ (-1)^{1/2} = (-1)^{2/4} $$ $$ (-1)^{2/4} = ((-1)^{2})^{1/4} ...
10
votes
4answers
911 views

Confused about complex numbers

I am confused about something: \begin{eqnarray} (e^{2 i \pi})^{0.5} = (e^{2 i \pi \cdot 0.5})= e^{i \pi}=-1 \end{eqnarray} but \begin{eqnarray} e^{2 i \pi}=1~ and~ 1^{0.5}=1 \end{eqnarray} ...
10
votes
4answers
5k views

How do you integrate imaginary numbers?

How would you find, for instance, $\int_{0}^{4} i\> x dx$? Can you just treat $i$ as a constant, or do you have to do something more sophisticated? Thanks!
10
votes
5answers
646 views

imaginary numbers - how can I understand them - especially as they occur in 'roots' of polynomials?

In another question here, about roots of equations being imaginary, the accepted answer said something interesting about "imaginary" (as a technical word in math) not meaning "not real". I ...
10
votes
2answers
489 views

Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$

1) Suppose $a \in \mathbb{R}$, and $\exists n \in \mathbb{N}$, that $a^n \in \mathbb{Q}$, and $(a + 1)^n \in \mathbb{Q}$. Prove: Is it true that $a \in \mathbb{Q}$? 2) Suppose $a \in \mathbb{C}$, ...
10
votes
4answers
1k views

How to show that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ indirectly?

I found this amazingly beautiful identity here. How to prove that $A^3+B^3+C^3 - 3ABC = (A+B+C)(A+B\omega+C\omega^2)(A+B\omega^2+C\omega)$ without directly multiplying the factors? (I've already ...
10
votes
4answers
4k views

Do the real numbers and the complex numbers have the same cardinality?

So it's easy to show that the rationals and the integers have the same size, using everyone's favorite spiral-around-the-grid. Can the approach be extended to say that the set of complex numbers has ...
10
votes
3answers
316 views

square root of $1/2 + \sqrt3/2?$

Playing with Maple, I noticed that it gives the square root of $c = 1+\frac{\sqrt3}{2}$ as equal to $a = \frac{1}{2}+\frac{\sqrt3}{2}$. Indeed it checks out. But I got curious: how can I find that ...
10
votes
3answers
637 views

Why this proof $0=1$ is wrong?(breakfast joke)

We have $$e^{2\pi i n}=1$$ So we have $$e^{2\pi in+1}=e$$ which implies $$(e^{2\pi in+1})^{2\pi in+1}=e^{2\pi in+1}=e$$ Thus we have $$e^{-4\pi^{2}n^{2}+4\pi in+1}=e$$ This implies ...
10
votes
2answers
81 views

Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$

Let $z_1,\dots,z_n\in\mathbb{C}$ such that $|z_p|\le1$ for every $p\in\{1,\dots,n\}$. Prove that exist $e_1,\dots,e_n\in\{-1,1\}$ such that $|e_1z_1+{\dots}+e_nz_n|\le\sqrt2$. I have firstly ...
10
votes
1answer
466 views

Is $\mathbb{C}^*$ modulo the roots of unity isomorphic to $\mathbb{R}^+$?

A student came to me showing a question from his exam in basic group theory, in which they are asked to prove that $\mathbb{C}^*$ modulo the subgroup of roots of unity is isomorphic to $\mathbb{R}^+$ ...
10
votes
6answers
730 views

If $A,B,C,D$ are complex numbers on the unit circle with $A+B+C+D=0$, then they form a rectangle

Let $A, B, C, D$ be points on a unit circle. Prove that if $A+B+C+D=0$, then $A,B,C,D$ make a rectangle. (Use complex numbers.) How do I prove this? I tried to use the dot product of 2 adjacent ...
10
votes
1answer
228 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
2answers
303 views

History of the matrix representation of complex numbers

It is well-known to many that $\mathbb{C}$ can be represented by matrices of the form $\left[ \begin{array}{cc} a & b \\ -b & a \end{array} \right]$. For example, see this question or this ...
10
votes
1answer
2k views

Intuition for complex eigenvalues

The eigenvalues of a rotation matrix are complex numbers. I understand that they cannot be real numbers because when you rotate something no direction stays the same. My question What is the ...
9
votes
4answers
2k views

Is this theorem already in existence?

I came up with this theorem (shown below) a few months ago, and I haven't been able to find anything like it on the web. This theorem will give you the quadratic expression that results from ...
9
votes
6answers
6k views

How can you find the cubed roots of $i$?

I am trying to figure out what the three possibilities of $z$ are such that $$ z^3=i $$ but I am stuck on how to proceed. I tried algebraically but ran into rather tedious polynomials. Could you ...
9
votes
7answers
263 views

Is $\mathbb{C}$ equal to $\mathbb{R}^2$?

Complex numbers are usually formally defined as pairs of real numbers. Although there are operations on $\mathbb{C}$, such as complex multiplication, which are not found in operations usually applied ...
9
votes
2answers
2k views

Determinant of an $n\times n$ complex matrix as an $2n\times 2n$ real determinant

If $A$ is an $n\times n$ complex matrix. Is it possible to write $\vert \det A\vert^2$ as a $2n\times 2n$ matrix with blocks containing the real and imaginary parts of $A$? I remember seeing such a ...
9
votes
6answers
501 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
votes
2answers
141 views

There exist $x_{1},x_{2},\cdots,x_{k}$ such two inequality $|x_{1}+x_{2}+\cdots+x_{k}|\ge 1$

Edit: This problem 1 is a 2014 Sydney mathematics competition problem (8th grade). It seems difficult to solve. Show that: There exist complex numbers $x_{1},x_{2},\cdots,x_{k}(k\ge 2)$ such ...
9
votes
1answer
402 views

i^i^i^i^… Is there a pattern? [duplicate]

I was messing around with $i$ and I (haha) noticed that certain progressions arise when I keep on raising $i$ to $i$ to $i$ and so forth. Though, I am not really quite sure what is going on (and I ...
9
votes
2answers
193 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
2answers
423 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
86 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
1answer
150 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 ...
9
votes
1answer
185 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 ...
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
780 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
4answers
426 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
3answers
442 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
481 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
452 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
3answers
746 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 $\displaystyle {(a + bi)}^{(c + di)}$?
8
votes
9answers
185 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: ...
8
votes
4answers
420 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
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 ...
8
votes
5answers
568 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 ...