Questions involving complex numbers: numbers of the form $a+bi$ where $i^2=-1$.
206
votes
33answers
17k views
Do complex numbers really exist?
Complex numbers involve the square root of negative one, and most non-mathematicians find it hard to accept that such a number is meaningful. In contrast, they feel that real numbers have an obvious ...
43
votes
3answers
1k views
Why is $i! = 0.498015668 - 0.154949828i$?
While moving my laptop the other day, I ended up mashing the keyboard a little, and by pure chance managed to do a google search for i!.
Curiously, Google's ...
40
votes
4answers
1k views
A new imaginary number? $x^c = -x$
Being young, I don't have much experience with imaginary numbers outside of the basic usages of $i$. As I was sitting in my high school math class doing logs, I had an idea of something that would ...
39
votes
2answers
2k views
A new kind of fractal?
http://www.gibney.de/does_anybody_know_this_fractal
Is this some known kind of fractal?
Update: This one got a lot of great feedback from around the net. I summarized it here: ...
38
votes
12answers
2k views
How can I introduce complex numbers to precalculus students?
I teach a precalculus course almost every semester, and over these semesters I've found various things that work quite well. For example, when talking about polynomials and rational functions, in ...
31
votes
6answers
938 views
Why do we negate the imaginary part when conjugating?
For $z=x+iy \in \mathbb C$ we all know the definition for the "conjugate" of $z$, $\bar{z}=x-iy$. Geometrically this is the reflection of $z$ across the $y$ axis.
My question is: couldn't we have ...
29
votes
11answers
2k views
What is $-i$ exactly?
We all know that $i$ doesn't have any sign: it is neither positive nor negative. Then how can people use $-i$ for anything?
Also, we define $i$ a number such that $i^2 = -1$. But it can also be seen ...
28
votes
19answers
2k views
Interesting results easily achieved using complex numbers
I was just looking at a calculus textbook preparing my class for next week on complex numbers. I found it interesting to see as an exercise a way to calculate the usual freshman calculus integrals ...
28
votes
7answers
2k views
Is $|1-i|$ larger than $|1|$?
I am confused about complex numbers. Does $1-i$ lie outside the unit circle? How do I show that this is larger than $|1|$ in absolute value?
26
votes
6answers
2k views
How to tell $i$ from $-i$?
Suppose now we are trying to explain to students who do not know complex numbers, how do we distinguish $i$ and $-i$ to them? They will object that they both squared to $-1$ and thus they are ...
24
votes
14answers
2k views
How do I understand $e^i$ which is so common?
Raising something to an imaginary number is weird, I have a hard time wrapping my head around that.
And e seems even more common and comes up in many situations, such as:
the non-geometric ...
24
votes
7answers
2k views
Do odd imaginary numbers exist?
Is the concept of an odd imaginary number defined/well-defined/used in mathematics? I searched around but couldn't find anything. Thanks!
24
votes
3answers
1k views
23
votes
4answers
2k views
Prove that $i^i$ is a real number
According to WolframAlpha, $i^i=e^{-\pi/2}$ but I don't know how I can prove it.
21
votes
8answers
2k views
What's the precise meaning of imaginary number?
The same to the title,what's the precise meaning of imaginary number? And on the other hand,how can the imaginary number be reflected in Physics?
21
votes
1answer
275 views
Iterated exponent of $i$
WolframAlpha seems to tell me that $e^{e^{e^{e^{e^{e^{e^{e^{e^{e^{e^i}}}}}}}}}} = 1$, see link. Is this just an error or is it for real? Adding one more $e$ to the bottom of the tower gives me the ...
20
votes
9answers
7k views
How to prove Euler's formula: $e^{it}=\cos t +i\sin t$?
Could you provide a proof of Euler's formula: $e^{it}=\cos t +i\sin t$ ?
thanks.
20
votes
5answers
804 views
Is the square root of -1 rational?
This is not a deep question, but if there is a definite answer then here is the place where I will find it.
Is it justified to say that $i =\sqrt{-1}$ is rational?
The origin of this question lies ...
20
votes
2answers
592 views
What do we lose passing from the reals to the complex numbers?
As normed division algebras, when we go from the complex numbers to the quaternions, we lose commutativity. Moving on to the octonions, we lose associativity. Is there some analogous property that we ...
19
votes
8answers
536 views
Infinite powering by $ i$ [duplicate]
Find the value of:
$i^{i^{i^{i^{i^{i^{....\infty}}}}}}$
Simply infinite powering by i's and the limiting value.
Thank you for the help.
19
votes
2answers
667 views
Complex solutions for Fermat-Catalan conjecture
The Fermat-Catalan conjecture is that $a^m + b^n = c^k$ has only a finite number of solutions when $a, b, c$ are positive coprime integers, and $m,n,k$ are positive integers satisfying $\frac{1}{m} + ...
18
votes
16answers
2k views
An example for a calculation where imaginary numbers are used but don't occur in the question or the solution.
In a presentation I will have to give an account of Hilbert's concept of real and ideal mathematics. Hilbert wrote in his treatise "Über das Unendliche" (page 14, second paragraph. Here is an English ...
18
votes
9answers
1k views
$i^2$ why is it $-1$ when you can show it is $1$?
We know $$i^2=-1 $$then why does this happen?
$$
i^2 = \sqrt{-1}\times\sqrt{-1}
$$
$$
=\sqrt{-1\times-1}
$$
$$
=\sqrt{1}
$$
$$
= 1
$$
EDIT: I see this has been dealt with before but at least with ...
18
votes
3answers
245 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 ...
18
votes
2answers
344 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 ...
17
votes
8answers
10k views
what is the square root of i?
If i is the square root of -1, is the square root of i imaginary? Is it used or considered often in mathematics? How is it notated?
16
votes
4answers
589 views
Mandelbrot fractal: How is it possible?
I'm a programmer and have recently played around a bit with rendering Mandelbrot fractals / zooming into them.
What I can't grasp: How can such infinite, complex shapes come out of somewhat 10 lines ...
16
votes
1answer
397 views
What is wrong with this fake proof $e^i = 1$?
$$e^{i} = e^{i2\pi/2\pi} = (e^{2\pi i})^{1/(2\pi)} = 1^{1/(2\pi )} = 1$$
Obviously, one of my algebraic manipulations is not valid.
16
votes
3answers
427 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
5answers
1k 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 ...
14
votes
2answers
456 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 ...
14
votes
2answers
381 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
2answers
809 views
De Moivre's Theorem. Motivation and origins.
I've purchased "A Source Book in Mathematics" some time ago and I'm still baffled by De Moivre's paper on his formula. We all know the famous
$$\{\cos(x) + i \sin(x)\}^n = \cos(nx)+i \sin(nx)$$
but ...
13
votes
9answers
1k 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 ...
13
votes
7answers
1k 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 ...
13
votes
4answers
325 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 ...
13
votes
1answer
358 views
What lies beyond the Sedenions
In the construction of types of numbers, we have the following sequence:
$$\mathbb{R} \subset \mathbb{C} \subset \mathbb{H} \subset \mathbb{O} \subset \mathbb{S}$$
or:
$$2^0 \mathrm{-ions} \subset ...
12
votes
6answers
735 views
How to calculate $z^4 + \frac1{z^4}$ if $z^2 + z + 1 = 0$?
Given that $z^2 + z + 1 = 0$ where $z$ is a complex number, how do I proceed in calculating $z^4 + \dfrac1{z^4}$?
Calculating the complex roots and then the result could be an answer I suppose, but ...
12
votes
4answers
6k views
How do I get the square root of a complex number?
If I'm given a complex number (say $9 + 4i$), how do I calculate its square root?
12
votes
3answers
2k views
Is the square root of a negative number defined?
I have been in a debate over 9gag with this new comic: "The Origins"
And I thought, "haha, that's funny, because I know $i = \sqrt{-1}$".
And then, this comment cast a doubt:
There is no such ...
12
votes
2answers
614 views
What's the name for the property of a function $f$ that means $f(f(x))=x$?
I can think of several examples of functions such that twice application of the function is equivalent to no application of it.
Additive inverse
Multiplicative inverse
Fourier transform
Complex ...
12
votes
5answers
723 views
What's bad about calling $i$ “the square root of -1”?
I vaguely recall a teacher telling me that he dislikes introducing the imaginary unit $i$ as "the square root of $-1$", but I can't remember why. Is there a lack of rigour in the statement, or is it a ...
12
votes
4answers
384 views
Relation of this antisymmetric matrix $r = \left(\begin{smallmatrix}0 &1\\-1&0\end{smallmatrix}\right)$ to $i$
I was reviewing some matrices and found this interesting
if $r = \begin{pmatrix} 0&1\\ -1&0 \end{pmatrix}$ then $rr=-I$, also $$\exp{(\theta r)} = \cos\theta I + \sin\theta r$$ No wonder, the ...
12
votes
2answers
425 views
Inequality with Complex Numbers
Consider the following problem:
Prove that for every set of complex numbers $\{z_i\}$, with $i$ ranging from one to $n$, there is a subset $J$ such that
$$\left|\sum_{j\in J} z_j\right|\ge ...
12
votes
1answer
1k views
Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
Using n_th root of unity
$(e^{\frac{2ki\pi}{n}})^{n} = 1$
Prove that
$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$
12
votes
2answers
418 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
3answers
115 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 ...
11
votes
8answers
699 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
2answers
950 views
-1 is not 1, so where is the mistake?
I know there must be something unmathematical in the following but I don't know where it is:
\begin{align}
\sqrt{-1} &= i \\ \\
\frac1{\sqrt{-1}} &= \frac1i \\ \\
\frac{\sqrt1}{\sqrt{-1}} ...
11
votes
4answers
279 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 ...
