Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.

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19
votes
3answers
319 views

Does my definition of double complex noncommutative numbers make any sense?

I wanted to factorize $a^2+b^2+c^2$ into two factors in a similar way to $$a^2+b^2 = (a+ib)(a-ib)$$ This doesn't seem to be possible using real or complex numbers. However I came up with the following ...
19
votes
4answers
717 views

How would you explain why $e^{i\pi}+1=0$ to a middle school student?

Hi I was asked by a friends child who is in middle school why $e^{i\pi}+1=0$. Now I couldn't think of a way to explain it so he would understand. Albert Einstein once said “If you can't explain it ...
19
votes
3answers
796 views

Sine of a Complex Number

While I know that $\sin(x)=2$ has no real solution, I tried seeing if it has a complex solution. That equality is equal to $$e^{2ix}-4ie^{ix}-1=0$$ Taking a quadratic in $e^{ix}$ I got ...
19
votes
2answers
456 views

What is the precise definition of $i$?

This may seem like an extraordinarily trivial question and yet it has completely confounded me. The technical definition of $i$ is $$i^2=-1$$ But there are two numbers which fulfill this ...
18
votes
4answers
800 views

Find all roots of $\,(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$

The question is to find all complex roots of $$(x + 1)(x + 2)(x + 3)^2(x + 4)(x + 5) = 360$$ and it is meant to be solved by hand. Is there any quick way to solve this using some trick that I'm not ...
18
votes
7answers
2k views

How can one intuitively think about quaternions?

Quaternions came up while I was interning not too long ago and it seemed like no one really know how they worked. While eventually certain people were tracked down and were able to help with the ...
17
votes
5answers
2k views

How fundamental is the fundamental theorem of algebra?

Despite its name, its often claimed that the fundamental theorem of algebra (which shows that the Complex numbers are algebraically closed - this is not to be confused with the claim that a polynomial ...
17
votes
4answers
760 views

Inequality with Complex Numbers

Consider the following problem: Prove that for every set of complex numbers $\{z_i\}$, with $i$ ranging from one to $n$, there is a subset $J$ such that $$\left|\sum_{j\in J} z_j\right|\ge ...
16
votes
8answers
4k views

Why is the complex number $z=a+bi$ equivalent to the matrix form $\left(\begin{smallmatrix}a &-b\\b&a\end{smallmatrix}\right)$ [duplicate]

Possible Duplicate: Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
16
votes
3answers
10k views

Is the square root of a negative number defined?

I have been in a debate over 9gag with this new comic: "The Origins" And I thought, "haha, that's funny, because I know $i = \sqrt{-1}$". And then, this comment cast a doubt: There is no such ...
16
votes
3answers
629 views

Projection of tetrahedron to complex plane

It is widely known that: distinct points $a,b,c$ in the complex plane form equilateral triangle iff $ (a+b+c)^{2}=3(a^{2}+b^{2}+c^{2}). $ New to me is this fact: let $a,b,c,d$ be the images of ...
15
votes
7answers
841 views

How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$?

Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$? Calculating the complex roots and then the result could be an answer I suppose, but ...
15
votes
9answers
1k views

Is there an interval notation for complex numbers?

Just as $$\{x \in \mathbb{R}: a \leq x \leq b\}$$ can be written in the more-compact form $[a,b],$ is there an analogous notation for $$\{z \in \mathbb{C}:z=x+yi, x \in[a,b], y \in[c,d]\} \quad ?$$ ...
15
votes
4answers
694 views

Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$

I was reviewing some matrices and found this interesting if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
15
votes
2answers
1k views

Wild automorphisms of the complex numbers

I read about so called "wild" automorphisms of the field of complex numbers (i.e. not the identity nor the complex conjugation). I suppose they must be rather weird and I wonder whether someone could ...
15
votes
2answers
506 views

What is $i$ exponentiated to itself $i$ times?

I was just wondering about this. I searched about it on the net and found that it is called tetration and after this comes pentation and then hexation and so on so forth. I don't really understand ...
14
votes
6answers
1k views

inequality involving complex exponential

Is it true that $$|e^{ix}-e^{iy}|\leq |x-y|$$ for $x,y\in\mathbb{R}$? I can't figure it out. I tried looking at the series for exponential but it did not help. Could someone offer a hint?
14
votes
2answers
271 views

Complex numbers, polynomials

Let $a$ be complex number such that $a^5 + a + 1 = 0$. What are possible values of $a^2(a - 1)$? I have tried to find $a$. Is there any way to find it?
13
votes
5answers
2k views

Solving $(z+1)^5 = z^5$

The question says to solve this equation: $(z+1)^5 = z^5$ I did. Just want to find out if I did it properly and if my run-around logic makes sense. First I begin my writing the equations as: $$ ...
13
votes
4answers
4k views

Do “imaginary” and “complex” angles exist?

During some experimentation with sines and cosines, its inverses, and complex numbers, I came across these results that I found quite interesting: $ \sin ^ {-1} ( 2 ) \approx 1.57 - 1.32 i $ $ \sin ...
13
votes
2answers
792 views

Prove that a polynomial has at least one nonreal complex root

Prove that the polynomial below has at least one nonreal complex root $$x^5+\frac{x^4}2+ \frac{x^3}3+\frac{x^2}4+\frac x{24}+\frac 1{120}$$ I have tried to prove that there exist $k\in \Bbb R$, such ...
13
votes
2answers
689 views

What's the name for the property of a function $f$ that means $f(f(x))=x$?

I can think of several examples of functions such that twice application of the function is equivalent to no application of it. Additive inverse Multiplicative inverse Fourier transform Complex ...
13
votes
5answers
900 views

What's bad about calling $i$ “the square root of -1”?

I vaguely recall a teacher telling me that he dislikes introducing the imaginary unit $i$ as "the square root of $-1$", but I can't remember why. Is there a lack of rigour in the statement, or is it a ...
13
votes
8answers
21k views

“Where” exactly are complex numbers used “in the real world”?

I've always enjoyed solving problems in the complex world during my undergrad. However, I've always wondered where are they used and for what? In my domain (computer science) I've rarely seen it be ...
13
votes
3answers
9k views

What is the “standard basis” for fields of complex numbers?

What is the "standard basis" for fields of complex numbers? For example, what is the standard basis for $\Bbb C^2$ (two-tuples of the form: $(a + bi, c + di)$)? I know the standard for $\Bbb R^2$ is ...
13
votes
5answers
838 views

Can I keep adding more dimensions to complex numbers?

I know about the concept of the complex plane, and I was amazed to find out that you're basically rotating numbers around this plane by multiplying by i, but, is there a way to jam the third dimension ...
13
votes
1answer
95 views

Simplification of a trilogarithm of a complex argument

Is it possible to simplify the following expression? $$\large\Im\,\operatorname{Li}_3\left(-e^{\xi\,\left(\sqrt3-\sqrt{-1}\right)-\frac{\pi^2}{12\,\xi}\left(\sqrt3+\sqrt{-1}\right)}\right)$$ where ...
13
votes
2answers
180 views

Maximum of $|(z-a_1)\cdots(z-a_n)|$ on the unit circle

Let $a_1,\ldots,a_n$ be points on the unit circle. Let $P(z)=(z-a_1)\cdots(z-a_n)$. The maximum principle or Rouche's theorem can be used to show that there exists a point $b$ on the unit circle such ...
13
votes
3answers
152 views

$\sum_i x_i^n = 0$ for all $n$ implies $x_i = 0$

Here is a statement that seems prima facie obvious, but when I try to prove it, I am lost. Let $x_1 , x_2 \dots x_k$ be complex numbers satisfying: $$x_1 + x_2 \dots + x_k = 0$$ $$x_1^2 + x_2^2 ...
12
votes
2answers
334 views

Does $z^i=i^z$ have any solutions, beside $z=i$?

Does this equation have any solutions: $$\large{z^i=i^z}$$ Putting polar form of $z$ is better for LHS, But rectangular form is suitable for RHS ! What to do? Thanks!
12
votes
5answers
2k views

Understanding imaginary exponents

Greetings! I am trying to understand what it means to have an imaginary number in an exponent. What does $x^{i}$ where $x$ is real mean? I've read a few pages on this issue, and they all seem to ...
12
votes
3answers
7k views

Can a real symmetric matrix have complex eigenvectors?

A Hermitian matrix always has real eigenvalues and real or complex orthogonal eigenvectors. A real symmetric matrix is a special case of Hermitian matrices, so it too has orthogonal eigenvectors and ...
12
votes
2answers
224 views

Is it possible to construct an ordered field which is also algebraically closed?

It is well known that while the real numbers are totally ordered, they are not algebraically closed, and while the complex numbers are algebraically closed, they are not totally ordered. Is it ...
12
votes
3answers
227 views

What's $(-1)^{2/3}\; $?

I know that $\left ( -1 \right )^{2/3}=\left ( \left ( -1 \right )^{2} \right )^{1/3}=1$ But Matlab computes this as $- 0.5 + 0.8660254038i$ a complex number.Why?
12
votes
2answers
481 views

Complex-number inequality $| z_1 z_2 \ldots z_m - 1 | \leq e^{|z_1 - 1| + \ldots + |z_m - 1|} - 1$

Let $z_1, z_2 \ldots z_m$ be complex numbers, $m \in \mathbb{N}$. Can anybody tell me how to prove the following inequality? $| z_1 z_2 \ldots z_m - 1 | \leq e^{|z_1 - 1| + \ldots + |z_m - 1|} - 1$ ...
12
votes
4answers
384 views

Euler's identity: why is the $e$ in $e^{ix}$? What if it were some other constant like $2^{ix}$?

$e^{ix}$ describes a unit circle in polar coordinates on the complex plane, where x is the angle (in radians) counterclockwise of the positive real axis. My intuition behind this is that ...
12
votes
0answers
5k views

What's more common? Re / Im or Fraktur-R / Fraktur-I for real / imaginary part?

Title says it all. What's more common? Is there one to prefere (maybe due to some norm)? This: $\operatorname{\mathfrak{R}} z, \operatorname{\mathfrak{I}} z$ or that: $\operatorname{Re}z, ...
11
votes
8answers
789 views

A field without a canonical square root of $-1$

The following is a question I've been pondering for a while. I was reminded of it by a recent dicussion on the question How to tell $i$ from $-i$? Can you find a field that is abstractly ...
11
votes
5answers
3k views

How to compute $\sqrt{i + 1}$ [duplicate]

Possible Duplicate: How do I get the square root of a complex number? I'm currently playing with complex numbers and I realized that I don't understand how to compute $\sqrt{i + 1}$. My ...
11
votes
7answers
3k views

Is there a formula for $(1+i)^n+(1-i)^n$?

I'm wondering if there is a formula for the value of $(1+i)^n+(1-i)^n$? I calculated the first terms starting with $n=1$ to be, in order, $2$, $0$, $-4$, $-8$, $-8$, $0$, $16$, $\dots$ So it seems ...
11
votes
9answers
907 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
2answers
3k views

Show how to calculate the Riemann zeta function for the first non-trivial zero

I have very little understanding on how complex functions work but was wondering if someone could show what the summation of the zeta function simplifies to when $s$ is the first non-trivial zero of ...
11
votes
6answers
810 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
762 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
140 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
464 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
618 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
1k 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?