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

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35
votes
6answers
3k 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}} ...
26
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 ...
300
votes
35answers
33k 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 ...
38
votes
10answers
20k 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.
21
votes
3answers
4k views

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

Using n_th root of unity $$\Large\left(e^{\frac{2ki\pi}{n}}\right)^{n} = 1$$ Prove that $$\prod_{k=1}^{n-1}\sin\frac{k \pi}{n} = \frac{n}{2^{n-1}}$$
28
votes
7answers
3k 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!
34
votes
3answers
2k views

What is the value of $1^i$?

What is the value of $1^i$? $\,$
24
votes
7answers
50k 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?
5
votes
3answers
2k views

Proving $\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$

I am being asked to prove that $$\sum\limits_{k=0}^{n}\cos(kx)=\frac{1}{2}+\frac{\sin(\frac{2n+1}{2}x)}{2\sin(x/2)}$$ I have some progress made, but I am stuck and could use some help. What I did: ...
321
votes
21answers
58k 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 ...
28
votes
10answers
29k 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?
14
votes
8answers
22k 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 ...
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 ...
31
votes
4answers
7k 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.
15
votes
4answers
726 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 ...
6
votes
1answer
1k views

Simple Complex Number Problem: $1 = -1$ [duplicate]

Possible Duplicate: -1 is not 1, so where is the mistake? I'm trying to understand the exact point of failure in the following reasoning: \begin{equation*} 1 = \sqrt{1} = \sqrt{(-1)(-1)} = ...
8
votes
3answers
738 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 $\displaystyle {(a + bi)}^{(c + di)}$?
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 ...
6
votes
2answers
732 views

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$

Prove if $|z| < 1$ and $ |w| < 1$, then $|1-zw^*| \neq 0$ and $| {{z-w} \over {1-zw^*}}| < 1$Given that $|1-zw^*|^2 - |z-w|^2 = (1-|z|^2)(1-|w|^2)$I think the first part can be proven by ...
7
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 ...
4
votes
4answers
341 views

Complex power of a complex number

Can someone explain to me, step by step, how to calculate all infinite values of, say, $(1+i)^{3+4i}$? I know how to calculate the principal value, but not how to get all infinite values...and I'm ...
22
votes
3answers
2k 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
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 ...
7
votes
4answers
3k 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 ...
2
votes
4answers
149 views

Values for $(1+i)^{2/3}$

This question might seem easier than I'm making it seem. But how many values are there for $(1+i)^{2/3}$? Do I let $z=(1+i)^{2/3}$ so that $z^3=2i$? I'm asked to write each in polar coordinates and in ...
0
votes
1answer
194 views

For what values $\alpha$ for complex z $\ln(z^{\alpha}) = \alpha \ln(z)$?

For example, when $\alpha = 2$, $\ln(z^{2}) \neq 2\ln(z)$, because argument z is determined up to constant $2 \pi k$. So $$ \ln(z^{2}) = \ln(z) + \ln(z) = \ln(z_{k_{1}}) + \ln(z_{k_{2}}) \neq ...
39
votes
19answers
5k 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 ...
18
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 $r = \left(\begin{smallmatrix}0 &amp;1\\-1&amp;0\end{smallmatrix}\right)$ to $i$ On Wikipedia, it says that: Matrix ...
2
votes
2answers
390 views

Division of Complex Numbers

Ahlfors says that once the existence of the quotient $\frac{a}{b}$ has been proven, its value can be found by calculating $\frac{a}{b} \cdot \frac{\bar b}{\bar b}$. Why doesn't this manipulation show ...
37
votes
10answers
2k 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 ...
4
votes
5answers
919 views

How to combine complex powers?

Regarding this thread, it is not possible to combine complex powers in the usual way: $$ (x^y)^z = x^{yz} $$ There was mention of multi-valued functions, is there some theory that makes this all ...
10
votes
4answers
590 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} ...
5
votes
4answers
281 views

Evaluate $\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$ using complex analysis

How do I compute $$\int_0^\infty \frac{(\log x)^2}{1+x^2} dx$$ What I am doing is take $$f(z)=\frac{(\log z)^2}{1+z^2}$$ and calculating $\text{Res}(f,z=i) = \dfrac{d}{dz} \dfrac{(\log ...
17
votes
4answers
763 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 ...
4
votes
7answers
3k views

Simple applications of complex numbers

I've been helping a high school student with his complex number homework (algebra, de Moivre's formula, etc.), and we came across the question of the "usefulness" of "imaginary" numbers - If there ...
3
votes
5answers
452 views

Fourier Analysis

I am interested in Fourier Analysis. But I don't get why the coefficients are choosen that way and why the Fourier series will converge to a given function? Can someone provide me simple information ...
3
votes
1answer
327 views
0
votes
5answers
420 views

Simple doubt about complex numbers

The question itself is simple, but I'm weak in math, and I'm training a lot every day to be the best I can, so, working on complex numbers, I got stuck on a simple multiplication: $\sqrt{3i} * 2i$ ...
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 ...
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: ...
7
votes
2answers
2k views

Prove that the zeros of an analytic function are finite and isolated

Let us assume that the zeros of $f = \{Z_1,\ldots,Z_n,a\}$ are infinite and converge towards $a$. The book which I am reading says that any neighborhood of $a$ will contain infinite zeros. Since $f$ ...
5
votes
3answers
880 views

Complex conjugate of $z$ without knowing $z=x+i y$

Is it possible to determine (and if so, how) the complex conjugate $\bar{z}$ of $z$, if you don't already know that $z = x + i y$? I think you can use $\log(z)$ to get the angle, and therefore the ...
0
votes
3answers
148 views

Compute the Integral

Compute the integral. $$\int_{-\infty}^\infty \frac{x^4}{1+x^8} \, dx$$ The answer at the back of the book is $$\frac{\pi}{4\sin(\frac{3\pi}{8})}$$
5
votes
3answers
1k views

Comparing real and complex numbers

If I'm correct, a complex number can be interpreted as a set in the following manner: $$ \forall x, y \in \mathbb{R}, x + yi = \{(x,\ y)\}.\ \mathbf{(1)} $$ My question is, is it technically ...
5
votes
2answers
333 views

General question on relation between infinite series and complex numbers

This is a strictly preliminary question. I hope to elicit some discussion/s which will lead to a more acceptable form for the question on this site. I'm trying to understand how the study of the ...
2
votes
1answer
67 views

Trying to evaluate $\prod_{k=1}^{n-1}(1-e^{2k\pi i/n})$ for my complex analysis homework

For my complex analysis homework, I am trying to show that the integral of the real function $1/(1+x^n)$, for integer $n\ge2$, along the positive real line is $$\int_0^{\infty}\frac{dx}{1+x^n} = ...
1
vote
4answers
141 views

Express $(1-i)^{11}$ in cartesian form.

Express $(1-i)^{11}$ in cartesian form. Apart from expanding the expression, I don't know how to do this. I've looked at the solution and still don't understand how/why it has been done.
1
vote
1answer
147 views

Integral of series with complex exponentials

Suppose that $f\in L^2(\mathbb{R}/2\pi\mathbb{Z})$ takes the form $$f(\theta)=\sum_{n=1}^\infty a_ne^{in\theta}.$$ The function $$F(z)=\sum_{n=1}^\infty a_nz^n$$ converges in $|z|<1$. How can I ...
50
votes
15answers
4k 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 ...
24
votes
9answers
1k 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 ...