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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379
votes
21answers
65k views

What are imaginary numbers?

At school I really struggled to understand the concept of imaginary numbers. My teacher told us that an imaginary number is a number which has something to do with the square root of -1. When I ...
379
votes
6answers
26k views

Why is $1 - \frac{1}{1 - \frac{1}{1 - \ldots}}$ not real?

So we all know that the continued fraction containing all $1$s... $$ x = 1 + \frac{1}{1 + \frac{1}{1 + \ldots}} $$ yields the golden ratio $x = \phi$, which can easily be proven by rewriting it as ...
348
votes
35answers
42k 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 ...
97
votes
11answers
6k views

Is there a domain “larger” than (i.e., a supserset of) the complex number domain?

I've been teaching my 10yo son some (for me, anyway) pretty advanced mathematics recently and he stumped me with a question. The background is this. In the domain of natural numbers, addition and ...
76
votes
7answers
4k views

Why are There No “Triernions”? [duplicate]

Since there are complex numbers (2 dimensions) and quaternions (4 dimensions), it follows intuitively that there ought to be something in between for 3 dimensions ("triernions"). Yet no one uses ...
68
votes
9answers
4k views

Why $\sqrt{-1 \times {-1}} \neq \sqrt{-1}^2$?

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}} ...
64
votes
25answers
7k views

Easy example why complex numbers are cool

I am looking for an example explainable to someone only knowing high school mathematics why complex numbers are necessary. The best example would be possible to explain rigourously and also be clearly ...
60
votes
11answers
40k 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.
59
votes
3answers
3k 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 ...
54
votes
4answers
3k 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 ...
54
votes
15answers
6k 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 ...
51
votes
17answers
5k 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 ...
51
votes
4answers
786 views

Integers $n$ such that $i(i+1)(i+2) \cdots (i+n)$ is real or pure imaginary

A couple of days ago I happened to come across [1], where the curious fact that $i(i-1)(i-2)(i-3)=-10$ appears ($i$ is the imaginary unit). This led me to the following question: Problem 1: Is $3$ ...
48
votes
19answers
6k 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 ...
47
votes
5answers
13k 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.
47
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: ...
45
votes
8answers
5k views

Why is there never a proof that extending the reals to the complex numbers will not cause contradictions?

The number $i$ is essentially defined for a property we would like to have - to then lead to taking square roots of negative reals, to solve any polynomial, etc. But there is never a proof this cannot ...
43
votes
1answer
457 views

How many non-rational complex numbers $x$ have the property that $x^n$ and $(x+1)^n$ are rational?

In this answer, I proved the following statement: For each positive integer $n$, there are finitely many non-rational complex numbers $x$ such that $x^n$ and $(x+1)^n$ are rational. These complex ...
42
votes
8answers
91k 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?
42
votes
10answers
4k views

What's the difference between $\mathbb{R}^2$ and the complex plane?

I haven't taken any complex analysis course yet, but now I have this question that relates to it. Let's have a look at a very simple example. Suppose $x,y$ and $z$ are the Cartesian coordinates and ...
42
votes
5answers
2k 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 ...
40
votes
10answers
4k views

Is “$a + 0i$” in every way equal to just “$a$”?

I'm having a little argument with my friend. He says that "$a + 0i$" is, in every way, absolutely equal to "$a$" (e.g.: $2 + 0i = 2$). I say this is practically the case, so in every calculation you ...
39
votes
2answers
2k views

Is $i = \sqrt{e^{\pi\sqrt{e^{\pi\sqrt\ldots}}}}$?

I was messing with the identity $e^{i\pi}=-1$ and I got that $i = \sqrt{e^{\pi\sqrt{e^{\pi\sqrt\ldots}}}}$ and on. I plugged it in to a calculator and it was infinite. It grew very fast. Does that ...
38
votes
9answers
4k 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?
38
votes
11answers
39k views

What is $\sqrt{i}$?

If $i=\sqrt{-1}$, is $\large\sqrt{i}$ imaginary? Is it used or considered often in mathematics? How is it notated?
38
votes
3answers
2k views

What is the value of $1^i$?

What is the value of $1^i$? $\,$
37
votes
10answers
3k views

Why $\sqrt{-1 \times -1} \neq \sqrt{-1}^2$? [duplicate]

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 ...
34
votes
3answers
2k views

Fibonacci infinite sum resulting in $\pi$

I found the following identity. While trying to prove it, I found some things that I don’t quite understand: $$\frac{\pi}{4}=\sqrt{5} \sum_{n=0}^{\infty} \frac{(-1)^n F_{2n+1}}{(2n+1) ...
31
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 ...
30
votes
3answers
3k 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 ...
30
votes
1answer
635 views

Distribution of roots of complex polynomials

I generated random quadratic and cubic polynomials with coefficients in $\mathbb{C}$ uniformly distributed in the unit disk $|z| \le 1$. The distribution of the roots of 10000 of these polynomials are ...
29
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 ...
29
votes
1answer
359 views

A spiralling sequence based on integer divisors. Has anyone noticed this before?

Firstly, please excuse the informal style of my explanation, as I am not a mathematician, although I am aware that this can be explained in more formal terms. I have mapped integers to points on a ...
28
votes
7answers
3k views

What does it mean to divide a complex number by another complex number?

Suppose I have: $w=2+3i$ and $x=1+2i$. What does it really mean to divide $w$ by $x$? EDIT: I am sorry that I did not tell my question precisely. (What you all told me turned out to be already known ...
28
votes
8answers
2k views

Why is $1/i$ equal to $-i$?

When I entered the value $$\frac{1}{i}$$ in my calculator, I received the answer as $-i$ whereas I was expecting the answer as $i^{-1}$. Even google calculator shows the same answer (Click here to ...
28
votes
7answers
4k 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!
28
votes
7answers
3k views

Can a complex number be prime?

I've been pondering over this question since a very long time. If a complex number can be prime then which parts of the complex number needs to be prime for the whole complex number to be prime.
28
votes
1answer
1k 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.
27
votes
5answers
1k views
27
votes
1answer
426 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 ...
26
votes
9answers
2k views

What does it mean to represent a number in term of a $2\times2$ matrix?

Today my friend showed me that the imaginary number can be represented in term of a matrix $$i = \pmatrix{0&-1\\1&0}$$ This was very very confusing for me because I have never thought of it ...
26
votes
7answers
7k views

Can a complex number ever be considered 'bigger' or 'smaller' than a real number, or vice versa?

I've always had this doubt. It's perfectly reasonable to say that, for example, 9 is bigger than 2. But does it ever make sense to compare a real number and a complex/imaginary one? For example, ...
26
votes
9answers
40k views

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

I've always enjoyed solving problems in the complex numbers 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 ...
26
votes
3answers
5k views

Prove that $\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$

Using $\text{n}^{\text{th}}$ root of unity $$\large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
25
votes
5answers
2k 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 ...
25
votes
2answers
970 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 ...
24
votes
3answers
17k 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 ...
23
votes
9answers
3k views

$1/i=i$. I must be wrong but why? [duplicate]

$$\frac{1}{i} = \frac{1}{\sqrt{-1}} = \frac{\sqrt{1}}{\sqrt{-1}} = \sqrt{\frac{1}{-1}} = \sqrt{-1} = i$$ I know this is wrong, but why? I often see people making simplifications such as ...
23
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?
23
votes
8answers
900 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.