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

learn more… | top users | synonyms

6
votes
3answers
268 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
votes
5answers
555 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
votes
4answers
154 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
530 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
742 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
1k 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
224 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
931 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
votes
4answers
330 views

$e^{i\theta}$ $=$ $\cos \theta + i \sin \theta$, a definition or theorem?

My question is simply whether the well-known formula $e^{i \theta}$ $=$ $\cos \theta$ $+$ $i \sin \theta$ a definition or there is some proof of the result. It seems to me that the formula is a ...
6
votes
7answers
177 views

why is $\sqrt{-1} = i$ and not $\pm i$? [duplicate]

this is something that came up when working with one of my students today and it has been bothering me since. It is more of a maths question than a pedagogical question so i figured i would ask here ...
6
votes
3answers
364 views

A difficult inequality involving complex numbers

Suppose that $z_1,\ldots,z_n$ are complex numbers with the property that there is some constant $C$ such that $$\big|z_1^r+\cdots+z_n^r\big|\leqslant C$$ for all integers $r\geqslant0$. Show that ...
6
votes
2answers
169 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
votes
2answers
5k views

Multiplying complex numbers in polar form?

Could someone explain why you multiply the lengths and add the angles when multiplying polar coordinates? I tried multiplying the polar forms ($r_1\left(\cos\theta_1 + i\sin\theta_1\right)\cdot ...
6
votes
2answers
242 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
217 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
votes
6answers
232 views

Solve $\cos{z}+\sin{z}=2$

I am trying to solve the question: $\cos{z}+\sin{z}=2$ Where $z \in \mathbb{C}$ I think I know how to solve $\cos{z}+\sin{z}=-1$: $1+2\cos^2{\frac{z}{2}}-1+2\sin ...
6
votes
2answers
146 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
111 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
3answers
333 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
2answers
233 views

$x^2+1=0$ uncountable many solutions [duplicate]

Possible Duplicate: Why are the solutions of polynomial equations so unconstrained over the quaternions? Coudl someone explain me the following: Why should $x^2+1=0$ have uncountable ...
6
votes
2answers
1k 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
327 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
298 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
339 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
1answer
139 views

Are there any arguments against the Riemann hypothesis?

We all know the well known Riemann hypothesis that the zeroes of the Riemann-zeta function have real part $1/2$ seems to hold (as far as I know) for all prime numbers. I was curious if there were any ...
6
votes
3answers
302 views

Help understanding $e^{it}=\cos t+i\sin t$ by way of matrices and vector fields

I was brushing up on my complex arithmetic in preparation for a class in ODE's this semester and I found myself looking at Exercise 2.7.5 in Introduction to Complex Analysis for Engineers by Michael ...
6
votes
4answers
182 views

Can I conjugate a complex number : $\sqrt{a+ib}$?

Can I conjugate a complex number: $\sqrt{a+ib}$ ? Actually my maths school teacher says and argues with each and every student that we can't conjugate $\sqrt{a+ib}$ to $\sqrt{a-ib}$ because ...
6
votes
1answer
126 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
125 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
8answers
230 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
4answers
99 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
190 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
142 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
2answers
291 views

Primitive roots of unity

I am trying to show that, If $$f\left( x\right) =a_{0}+a_{1}x+\ldots +a_{k}x^{k}$$ then $$\dfrac {1} {n}\left\{ f\left( x\right) +f\left( wx\right) +\ldots +f\left( w^{n-1}x\right) \right\} ...
6
votes
2answers
298 views

Does it make sense to compare complex numbers in certain circumstances?

I know that $\mathbb{C}$ is an unordered field and that (strictly non-real) complex numbers cannot be 'compared' in the sense that one is less than/greater than another. However, we can compare real ...
6
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 $f$ ...
6
votes
3answers
415 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
175 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
1answer
378 views

Is a 3D Mandelbrot-esque fractal analogue possible?

I understand that (unlike complex numbers) there's no consistent 3 dimensional number system (even 4D loses some nice properties). Regardless, I'm wondering if there might be a 'trick' to create a 3D ...
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
1answer
117 views

Please help me find a complex number book suitable for me

Its been two weeks since I've joined this site, and I have received wonderful answers to my complex number questions at the shortest time. I am specially very weak in Complex numbers, and I see ...
6
votes
2answers
123 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
292 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 $x^3=15x+4$ ...
6
votes
3answers
134 views

Solving equation $(z-i)^3=(4-i\sqrt{48})z^3$ in $\mathbb C$.

I need to solve the following equation in $\mathbb C$. $$(z-i)^3=(4-i\sqrt{48})z^3.$$ I tried with trigonometric form , but having $z-i$ on the LHS is confusing me, since I get ...
6
votes
4answers
212 views

Solve $\operatorname{Arg} (z-2) - \operatorname{Arg} (z+2) = \frac{\pi}{6}$

I'm trying to solve $$\operatorname{Arg}(z-2) - \operatorname{Arg}(z+2) = \frac{\pi}{6}$$ for $z \in \mathbb{C}$. I know that $$\operatorname{Arg} z_1 - \operatorname{Arg} z_2 = \operatorname{Arg} ...
6
votes
2answers
95 views

Some equation with complex numbers

Given $a,b \in \mathbb{C}$ such that $a^2+b^2=1$, it is clear that $x:=a\bar{a}+b\bar{b}$ is a real number and that $yi:=a\bar{b}-\bar{a}b$ is imaginary (i.e $y$ is real). Moreover, a direct ...
6
votes
1answer
139 views

Axiomatic definition of complex numbers

Trying to build axiomatically the set $\mathbb C$ of complex numbers, my first attempt was to define $\mathbb C$ with three structures: addition, multiplication and conjugate: $\langle\mathbb ...
6
votes
1answer
284 views

Why is the bailout value of the Mandelbrot set 2?

For the past few days I've been studying the Mandelbrot set, and many say that if the iterations of a point stay within a magnitude of 2, the point converges. A very natural question of "why is the ...
6
votes
1answer
622 views

What are the uses of split-complex numbers?

The set of Complex numbers can be used in lots of domains like geometry, vectorial calculations, solving equation with no real solution etc. But what are the uses of split-complex number that can't be ...
6
votes
1answer
186 views

$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$

$$\int_{-\infty}^\infty \exp (a t^3 + b t^2 + c t) \mathrm{d}t,\;\;(a,c)\in\mathbb{I}, \; \; \Re(b)\le0$$ i.e. an oscillation with frequency $3\Im(a)t^2 + 2\Im(b)t + \Im(c)$ and phase $0$, multiplied ...