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

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7
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2answers
426 views

How to teach a High school student that complex numbers cannot be totally ordered?

I once again need your precious knowledge! I am not sure which is the best pedagogic way to teach a High school student about why complex numbers cannot be totally ordered. When I was in High school ...
7
votes
1answer
323 views

Is there a “good” way to visualize complex vectors?

We often represent complex numbers as vectors in $\mathbb{R}^2$ with $x$ being the real axis and $y$ being the imaginary axis. We often represent 2-dimensional vectors over $\mathbb{R}$ in a similar ...
7
votes
1answer
93 views

Why does $\sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )+{1\over2}-{1\over3} = \gamma$?

How could one prove that $$x = \sum_{k=2}^\infty k \left( \sum_{j=2^k}^{2^{k+1}-1} \frac{(-1)^j}{j} \right )$$ is such that $x+{1\over2}-{1\over3} = \gamma$ ? I am having problems just calculating ...
7
votes
5answers
249 views

I don't understand $\sqrt{-9i}$.

I try to visualise it on a graph, where x is real numbers and y is the imaginary numbers. $\sqrt{9} = (3,0)$ and $(-3,0)$. $\sqrt{-9} = \sqrt{-1} \times \sqrt{9} = (0,3) $ and $(0,-3)$. ...
7
votes
2answers
270 views

How to show a complex number inequality

A classmate consulted me this problem, after a few moment's thought I found it was difficult, so I wish to try my luck here. Let $z_1,z_2,z_3,z_4\in \mathbb{C}$ such that ...
7
votes
1answer
80 views

Visualization of complex functions

How can I find the image of a rectangle $$a<\Re(z)<b \\c<\Im(z)<d$$ under the function $z^3$ , I tried breaking up $z$ into $x+iy$ and cubing it. I got the real and imaginary parts ...
7
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2answers
685 views

Simplest examples of real world situations that can be elegantly represented with complex numbers

Mathematics could be defined as the study of formally defined abstractions. These abstractions may or may not be useful for describing real world phenomenon. Indeed, Physics could be defined as the ...
7
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1answer
155 views

Is there a complex variant of Möbius' function?

When you're dealing with arithmetic functions, you might have come across the classical Möbius' function $$ \mu(n)=\begin{cases} (-1)^{\omega(n)}=(-1)^{\Omega(n)} &\mbox{if }\; \omega(n) = ...
7
votes
1answer
281 views

How does quater-imaginary (and other imaginary/complex bases) work?

So I've been working on a simple base-conversion program, and having given it the ability to convert from decimal to any base $> 1$ or $< 0$, as well as the $p$-adic (bijective, I think?) bases, ...
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0answers
45 views

Cauchy representation and branch point order

This question is about the nature of branch points which arise in certain Cauchy-integral representations of functions of a single complex argument, $z$. The main application is for dispersion ...
7
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0answers
3k 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, ...
6
votes
5answers
216 views

How do i find $(1+i)^{100}?$

How do I find $(1+i)^{100}$ without expanding $(1+i)$ 100 times? Is there a quicker way to do this? The hint was to find the modulus and argument of $1+i$ which I've got as $\sqrt{2}$ and $\pi/4$ ...
6
votes
6answers
534 views

Extending the set of complex numbers

Mathematics as a science became richer when Cantor invented the real numbers. Then scientists wanted to solve equations which were not solvable in the real numbers so they invented the complex ...
6
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6answers
3k 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 ...
6
votes
8answers
580 views

Positive and negative complex numbers?

Can there be such a thing as positive and negative complex numbers? Why or why not? What about positive or negative imaginary numbers? It seems very tempting to say $+5i$ is a positive number ...
6
votes
5answers
250 views

Does there exist a field $K$ such that $\mathbb R \subsetneq K \subsetneq \mathbb C$?

I'm thinking of unions of $\mathbb R$ with some subset of $\mathbb C$ but am not sure how to approach this without ending up with all of $\mathbb C$. Doe anyone have any suggestions?
6
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3answers
242 views

4 dimensional numbers

I've tought using split complex and complex numbers toghether for building a 3 dimensional space (related to my previous question). I then found out using both together, we can have trouble on the ...
6
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5answers
496 views

Proving complex numbers

Let $z_1,z_2$ be two complex numbers such that $z_1 + z_2$ and $z_1\dot\ z_2$ are each negative real numbers. Prove that $z_1$ and $z_2$ must be real numbers. My attempt at a solution ...
6
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5answers
147 views

Strong characterization of $\mathbb C$ with respect to $\mathbb R$

$\mathbb R^2$ is not a field, but $2$-tuple arithmetic rules like $(a,b)(c,d)=(ac-bd,ad+bc)$ coupled with $\mathbb R^2$ make it a field, but are there other rules than $(a,b)(c,d)=(ac-db,ad+bc)$ ...
6
votes
3answers
468 views

An application of Vandermonde determinant

Let $\lambda_1,\ldots,\lambda_n$ be complex numbers such that for each positive integer $k\geq 0$, $$\sum_{i=1}^n \lambda_i^k=0.$$ Here I am supposed to show that $\lambda_i=0$ for each $i\in ...
6
votes
3answers
667 views

What's the Difference Between a Vector and an Hypercomplex Number?

What's the difference between a vector and an hypercomplex number? For instance a 4-vector and a quaternion. They seem to share many properties. Perhaps this question could be put more generally as: ...
6
votes
1answer
721 views

Complex roots of $z^6 + z^3 + 1 = 0$

The equation I'm trying to solve is $f(z) = 0$ where $$f(z) = z^6 + z^3 + 1$$ I already tried the following: randomly throwing in complex numbers and real numbers, rational root theorem, banging my ...
6
votes
6answers
219 views

What is $(-1)^{\frac{2}{3}}$?

Following from this question, I came up with another interesting question: What is $(-1)^{\frac{2}{3}}$? Wolfram alpha says it equals to some weird complex number (-0.5 +0.866... i), but when I try ...
6
votes
6answers
812 views

Are there any calculus/complex numbers/etc proofs of the pythagorean theorem?

I have been looking for proofs for the pythagorean theorem that don't use area calculation but calculus, complex numbers or any other interesting ways to proof it. I would love to see any interesting ...
6
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3answers
5k 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 ...
6
votes
2answers
161 views

Why isn't $\int\sin(ix)~dx$ equal to $i\cos(ix)+C$ ?

I was playing around with imaginary numbers, and I tried to solve $$\int\sin(ix)~dx$$ and ended up getting $$i\cos(ix)+C$$ But apparently the answer is $$i\cosh(x)+C$$ I was just wondering, is this ...
6
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2answers
219 views

A ‘strong’ form of the Fundamental Theorem of Algebra

Let $ n \in \mathbb{N} $ and $ a_{0},\ldots,a_{n-1} \in \mathbb{C} $ be constants. By the Fundamental Theorem of Algebra, the polynomial $$ p(z) := z^{n} + \sum_{k=0}^{n-1} a_{k} z^{k} \in ...
6
votes
3answers
215 views

If $\theta\in\mathbb{Q}$, is it true that $(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$?

Is the following true if $\theta\in\mathbb{Q}$? $$(\cos \theta + i \sin \theta)^\alpha = \cos(\alpha\theta) + i \sin(\alpha\theta)$$ Is it true if $\alpha\in\mathbb{R}$? In each case, prove or give a ...
6
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2answers
113 views

Finding non-negative integers $m$ such that $(1 + \sqrt{-2})^m$ has real part $\pm 1$.

I believe that the integers $m$ with $(1+\sqrt{-2})^m$ having real part $\pm 1$ are $0, 1, 2$ and $5$, but I'm having trouble proving it. Write $$a_m = \Re((1+\sqrt{-2})^m) = \frac{(1 + \sqrt{-2})^m ...
6
votes
3answers
99 views

What is the value of $\ln \left(e^{2 \pi i}\right)$

I know that $$e^{2 \pi i} = 1$$ so by taking the natural logarithm on both sides $$\ln \left(e^{2 \pi i}\right)=\ln (1)=0$$ however, why isn't this $2 \pi i$ as expected? Is it beacuse logarithms ...
6
votes
2answers
612 views

is a one-by-one-matrix just a number (scalar)?

I was wondering. Clearly, we cannot multiply a (1x1)-matrix with a (4x3)-matrix; However, we can multiply a scalar with a matrix. This suggests a difference. On the other hand, I was, for example, in ...
6
votes
1answer
308 views

Why are primitive roots of unity the only solution to these equations?

I was led by this question to the following problem: Find $n$ complex numbers $\lambda_1\dots\lambda_n\in\mathbb{C}$ that satisfy $$\begin{align} \sum_i\lambda_i & =0\\ \sum_i\lambda_i^2 ...
6
votes
2answers
269 views

How did they simplify this expression involving roots of unity?

Suppose $\omega$ is a primitive seventh root of unity. I would like to find as simple an expression as possible for $$ \sum_{j=0}^6 (1 + \omega^j)^n. $$ The book I am looking at gives $$ 2^n ...
6
votes
2answers
332 views

Two points on circle resulting in 5 equal regions

What values of $Z_1$ and $Z_2$ make the five regions of the unit circle, shown below, equal in area? $\overline{Z_1}$ and $\overline{Z_2}$ are conjugates of $Z_1$ and $Z_2$; in other words they lie ...
6
votes
3answers
288 views

How can people understand complex numbers and similar mathematical concepts?

In mathematics, how does something like complex numbers apply to the real world? Why do complex numbers exist? How can we comprehend addition of complex numbers? For example, addition of natural ...
6
votes
1answer
68 views

Prove that $\cos(z)$ and $\sin(z)$ are surjective over the complex numbers. [duplicate]

I have an exercise that says: (a) Prove that $\cos(z)$ and $\sin(z)$ are surjective functions from $\mathbb C \to \mathbb C$. (b) Find the solutions of the equation $\cos(z)=\dfrac{5}{4}$. As far ...
6
votes
1answer
119 views

Do there exist equations that cannot be solved in $\mathbb{C}$, but can be solved in $\mathbb{H}$?

Excluding polynomials (whose solutions are covered by the Fundamental Theorem of Algebra), do there exist any univariable equations that cannot be solved in the complex numbers, but can be solved ...
6
votes
4answers
92 views

$\frac{1}{1-e^{\frac{ik\pi}{n+1}}}+\frac{1}{1-e^{-\frac{ik\pi}{n+1}}}=1$?

I'm working on an assignment where part of it is showing that $S_k=0$ for even $k$ and $S_k=1$ for odd $k$, where $$S_k:=\sum_{j=0}^{n}\cos(k\pi x_j)= \frac{1}{2}\sum_{j=0}^{n}(e^{ik\pi ...
6
votes
3answers
177 views

Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $

I'm trying to find $$\sum _{k=1}^{n} k\cos(k\theta)\qquad\text{and}\qquad\sum _{k=1}^{n} k\sin(k\theta)$$ I tried working with complex numbers, defining $z=\cos(\theta)+ i \sin(\theta)$ and using ...
6
votes
1answer
128 views

determining if a complex number is a root of unity

How would you determine if $a+ib$ is a $n$th root of unity for some unknown $n$? Obviously the modulus of $a+ib$ must be $1$. But you also need to determine if the $a+ib$ is located at the vertex ...
6
votes
1answer
380 views

How to choose a proper contour for a contour integral?

When analyzing real integrals with contour integrals, how does one choose a proper contour integral? Many cases can be solved by integrating around the top half of a circle with radius of infinity ...
6
votes
8answers
219 views

Am I wrong in thinking that $e^{i \pi} = -1$ is hardly remarkable?

I believe my trouble is that the identity, $e^{i \pi} = -1$, comes down to the definition of the exponentiation of $i$, which seems rather obscure to me. This is my current understanding of ...
6
votes
2answers
229 views

Why is $x^2$ +$ y^2$ = 1, where $x$ and $ y$ are complex numbers, a sphere?

I've heard $x^2 + y^2$ = 1, where $x$, $y$ are complex numbers, is supposed to be a sphere with two points removed, or also a cylinder. The problem is I've been trying to wrap my head around this for ...
6
votes
3answers
14k views

How to get principal argument of complex number from complex plane?

I am just starting to learn calculus and the concepts of radians. Something that is confusing me is how my textbook is getting the principal argument ($\arg z$) from the complex plane. i.e. for the ...
6
votes
3answers
371 views

Determining $|z-1|$ when $z=\cos\theta +i\sin\theta$ and $\theta$ is acute

As the question indicates we are supposed to find the modulus of z-1. When trying to solve the problem I drew a diagram which you can see below: The book I am working in solved a similar problem ...
6
votes
2answers
393 views

Total ordering on complex numbers

Show that there doesn't exist a relation $\succ$ between complex numbers such that (i) For any two complex numbers $z,w$, one and only one of the following is true: $z\succ w,w\succ z,$ or ...
6
votes
2answers
170 views

A Complex Inequality

I just need to show that : $$\int_0^{2\pi}\left|{\frac{i(Re^{i\theta})^\lambda}{1+Re^{i\theta}}}\right| d\theta \le \int_0^{2\pi} \frac{R^\lambda}{R-1}d\theta : 0 < \lambda <1 , R>1$$ Is ...
6
votes
2answers
1k views

How do I calculate the equation of a circle given 3 complex numbers?

Given three complex values (for example, $2i, 4, i+3$), how would you calculate the equation of the circle that contains those three points? I know it has something to do with the cross ratio of the ...
6
votes
2answers
116 views

Evaluate $\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}\ \mathrm dx$ by complex methods

find integral $$\int\limits_0^\pi \frac{\sin^2x}{2-\cos x}dx$$ what I had in mind is to use Euler formula, to turn it into a complex integral and change the limits of integration from $ -\pi$ to ...
6
votes
2answers
229 views

Discovery of complex numbers

A popular story about the discovery of the complex numbers goes as follows. Once the formula for the solution of the cubic equation has been discovered its application to the equation $x^3=15x+4$ ...