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

learn more… | top users | synonyms

5
votes
7answers
465 views

What is the value of $i+i^2+i^3+\cdots+i^{23}$? [duplicate]

Can anyone help me with this question and show me a step by step solution please? The imaginary number is $i$ is defined such that $i^2=-1$. What is $i+i^2+i^3+\cdots+i^{23}$?
5
votes
6answers
384 views

Can you explain $(1 + iX/n)^{n}$ without using e, sin, or cos?

In my ongoing strugle to understand $e^{\pi i}$ I managed to narrow down my conceptual difficulty. I'm having intuitive trouble understanding why $(1 + iX/n)^{n}$ is conceptually the same as a ...
5
votes
7answers
464 views

Is $0^0=1$ postulate independent of all other axioms of complex numbers?

This question is inspired by the other question which asked for a proof that $i^i$ is a real number. Many calculators when asked for $0^0$ return 1. I asked a mathematician how to prove that but he ...
5
votes
6answers
2k 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 ...
5
votes
4answers
185 views

Why is it impossible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that $|z_1- z_2|=|z_1-z_3|=|z_2-z_3|=|z_1-z_4|=|z_2-z_4|=|z_3-z_4|$?

A. It is possible to find distinct $z_1,z_2,z_3\in \mathbb C$ such that $|z_1-z_2|=|z_1-z_3|=|z_2-z_3|$. Answer: True B. It is possible to find distinct $z_1,z_2,z_3, z_4\in \mathbb C$ such that ...
5
votes
5answers
2k views

What is the norm of a complex number?

I'm in a number theory class and I'm trying to understand what the norm is... For some complex number $Z = a +bi$, $Z$ times the conjugate of z is equal to $(a^2)+(b^2)$. Most of what I've read about ...
5
votes
1answer
108 views

A case where $z^z = 0$ where $z$ is complex number

Is there any case where $z^z = 0$ where $z$ is complex number? The case excludes the case where $z=0$.
5
votes
3answers
653 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: ...
5
votes
3answers
796 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 ...
5
votes
3answers
921 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
3answers
204 views

$n$th derivative of $e^x \sin x$

Can someone check this for me, please? The exercise is just to find a expression to the nth derivative of $f(x) = e^x \cdot \sin x$. I have done the following: Write $\sin x = \dfrac{e^{ix} - ...
5
votes
2answers
4k 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 ...
5
votes
2answers
248 views

Can a Gaussian integer matrix have an inverse with Gaussian integer entries?

Is there any way to characterize the set of complex matrices with Gaussian integer entries whose inverses also have Gaussian integer entries? I'm aware of the numerous examples of integer matrices ...
5
votes
4answers
153 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 ...
5
votes
5answers
189 views

Find all real values $a$ and $b$ such that $a+ib=i^{i^{i}}$?

Find all real values $a$ and $b$ such that $a+ib=i^{i^{i}}$ ? My Try : using the fact that $z^w=e^{w \log z}$, First I compute $i^i$ $$i^i=e^{i \log i}= e^{i (\log |i| + i arg (i))}=e^{- ...
5
votes
5answers
363 views

Complex numbers and trig identities. I've heard this question is easy but I don't know how. Help?

Using the equally rule $a + bi = c + di$ and trigonometric identities how do I make... $$\cos^3(\theta) - 3\sin^2(\theta)\ \cos(\theta) + 3i\ \sin(\theta)\ \cos^2(\theta) - i\ \sin^3(\theta)= ...
5
votes
1answer
978 views

Using the fifth roots of unity to find the roots of $(z+1)^5=(z-1)^5$

The question I am working on starts of with: Find the five fifth roots of unity and hence solve the following problems I have done that and solved several questions using this, however ...
5
votes
3answers
917 views

Polynomial of degree 4 with real coefficients, two complex roots given.m

Write in the form f(z) = 0, where f(z) is a polynomial of degree 4 with real coefficients, the equation having (3 + i) and (1 + 3i) as two of its roots. Can anyone help me? I'm guessing the two ...
5
votes
4answers
446 views

solve $ z^2 -6z + 25 $ into complex conjugate

I need to solve this :$$ z^2 -6z + 25 = 0$$ My book says 'complete the square' so : 1.$$ (z - 6/2)^2 -36/4 + 25 $$ 2.$$ (z - 3)^2 -9 + 25 $$ 3.$$ (z - 3)^2 + 16 $$ Now how exactly does the above ...
5
votes
2answers
105 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 ...
5
votes
1answer
178 views

Why all composite numbers have this property?

Define $f(n)=\sum\limits_{A \in S} f_{1}(n,A),\ n>2,\ n \in \mathbb{Z}$, where $S$ is the power set of $\{\frac{1}{2},\cdots ,\frac{1}{n-1}\}$. Define $\ f_1(n,\varnothing)=1,\ ...
5
votes
2answers
100 views

Could anyone explain this curious solution to $7^{2x}= 2^x$

I typed the following on wolfram alpha today : $7^{2x} = 2^x$ and found this as a solution besides $x=0$: $x = \dfrac{2\pi i n}{\log2 - 2\log7}$ where $log$ has a base of $e$ and $n$ is any integer. ...
5
votes
3answers
1k 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: ...
5
votes
2answers
5k views

Complex Conjugate of Complex function

I am currently reading Hamming's Numerical Methods for Scientists and Engineers. On pg. 79 he discusses the topic of finding the zeros of a complex analytic function. He then proceeds to discuss ...
5
votes
2answers
225 views

Number of Complex Roots of a Complex Polynomial

This is related to the question I asked regarding finding the complex roots of $z^3+\bar{z}=0$. It turned out that there were 5 complex roots, but because the equation was of degree 3 I was only ...
5
votes
3answers
421 views

Determine the set $\{w:w=\exp(1/z), 0<|z|<r\}$

I can't do this exercise of Conway's Book: For $r>0$ let $A=\{w:w=\exp(1/z), 0<|z|<r\}$, determine the set $A$. Any hints?
5
votes
1answer
113 views

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ .

Find the value of $\sqrt{i+\sqrt{i+\sqrt{i+\dots}}}$ ? How to find if it is convergent or not? Thanks!
5
votes
2answers
82 views

Is $\mathrm{GL}_n(\mathbb C)$ divisible?

A group $G$ (possibly non-abelian) is divisible when for all $k\in \Bbb N$ and $g\in G$ there exists $h\in G$ such that $g=h^k.$ Is the group $\mathrm{GL}_n(\mathbb C)$ divisible? Or more precisely, ...
5
votes
2answers
186 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 ...
5
votes
2answers
200 views

Complex analysis : If $z =re^{i\theta}$, then prove that $|e^{iz}| =e^{-r\sin\theta}$

Problem : If $z =re^{i\theta}$, then prove that $|e^{iz} | =e^{-r\sin\theta}$ My working : $z = re^{i\theta} = r(\cos\theta + i\sin\theta)$ $\Rightarrow iz = ir(\cos\theta +\sin\theta) $ = ...
5
votes
3answers
97 views

$w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$

I am stuck on the following problem: Let $w_1,w_2$ are distinct complex numbers such that $|w_1|=|w_2|=1$ and $w_1+w_2=1$.Then the triangle in the complex plane with $w_1,w_2,-1$ as vertices ...
5
votes
2answers
93 views

If $\left |z-3\right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$

If $\left |z-3 \right |=\left |z+i\right |$, where $z=x+iy$, prove that $3x+y=4$. I have got to the point where I have $\left |z \right |= \sqrt{x^2+(y+1)^2} = \sqrt{(x-3)^2+y^2}$ But really ...
5
votes
2answers
202 views

The roots of the derivative $P'(z)$ of the polynomial $P(z)\in\mathbb C[x]$ lie in the convex hull of the set of roots of $P(z)$.

Assume $S=\{z_1,z_2,...,z_k\}, z_i\in \mathbb C$$, C(S)$ and define $$C(S):=\{z=a_1z_1+a_2z_2+...+a_kz_k | a_i\ge0 ,a_1+a_2+...+a_k=1\}$$ where $$A:=\{z\in \mathbb C:f(z)=0 ...
5
votes
2answers
262 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\} ...
5
votes
2answers
884 views

How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate?

How to solve $z^4 +6z^2 +25 = 0$ into complex conjugate? I started with $$(z^2 + 3)^2 + 16 = 0$$ $$(z^2 + 3)^2 = - 16$$ $$z^2 + 3 = \pm 4i$$ Is this the way to start solving the equation or am ...
5
votes
1answer
66 views

Confused about exponents and imaginary/real answers

I am confused about some exponent behavior. $$(-2)^{7.6} = (-2)^{\frac{76}{10}} = ((-2)^{76})^{\frac{1}{10}} = ((-2)^{\frac{1}{10}})^{76}$$ Is there something wrong in this logic? When I plug the ...
5
votes
5answers
303 views

Prove that $(x^2-x^3)(x^4-x) = \sqrt{5}$, where $x= \cos(2\pi/5)+i\sin(2\pi/5)$

Prove $(x^2-x^3)(x^4-x) = \sqrt{5}$ if $x= \cos(2\pi/5)+i\sin(2\pi/5)$. I have tried it by substituting $x = \exp(2i\pi/5)$ but it is getting complicated.
5
votes
2answers
122 views

Expand $(4+i)(5+3i)$ and show $\pi/4=\arctan{1/4}+\arctan{3/5}$

I can't remember ever having done this before so if someone could help me out that would be great. The question is expand $(4+i)(5+3i)$ and hence show that $\pi/4=\arctan{1/4}+\arctan{3/5}$. ...
5
votes
2answers
1k 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
4answers
165 views

Can a cubic that crosses the x axis at three points have imaginary roots?

I have a cubic polynomial, $x^3-12x+2$ and when I try to find it's roots by hand, I get two complex roots and one real one. Same, if I use Mathematica. But, when I plot the graph, it crosses the ...
5
votes
1answer
483 views

Hermitian/positive definite matrices and their analogues in complex numbers

I've heard a couple of times some people say that in a way, Hermitian matrices are to matrices as real numbers are to complex numbers. I know two examples where this is sort of true: Complex ...
5
votes
1answer
93 views

$\lim_{x\to 2} \, \sqrt{x-2}$

$$\lim_{x\to 2} \, \sqrt{x-2}$$ When you take the right hand limit for this expression, you get $0$. However, if you take the left hand side it gives an imaginary number. However, do you consider ...
5
votes
2answers
221 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 ...
5
votes
1answer
108 views

What indexes do the subgroups of $\mathrm{GL}_n(\Bbb C)$ have?

Let $B_n\subset\mathrm{GL}_n(\Bbb C)$ be the group of invertible upper-triangular matrices. What is the index $[\mathrm{GL}_n(\Bbb C):B_n]?$ (By index I mean the cardinality of a coset space.) In ...
5
votes
1answer
343 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 ...
5
votes
1answer
830 views

Why are imaginary numbers called imaginary numbers

Why do we call imaginary numbers "imaginary numbers"? As far as I can tell, there's nothing really imaginary about them. They exist. They're used all the time. What makes them so "imaginary"?
5
votes
1answer
194 views

Complex numbers system of equations problem with 5 variables

Let $z_0$,$z_1$,$z_2$,$z_3$ and $z_4$ such that $z_i\in C$ that hold: $$(1)|z_0|=|z_1|=|z_2|=|z_3|=|z_4|=1$$ $$(2)z_0+z_1+z_2+z_3+z_4=0$$ $$(3) z_0z_1+ z_1z_2+z_2z_3+z_3z_4+z_4z_0=0$$ Prove that ...
5
votes
1answer
72 views

Region in complex plane with $|1-z|\leq M(1-|z|)$

Let $M>0$. Describe the region in the complex plane such that $|1-z|\leq M(1-|z|)$. To start, I take $M=1$. The inequality becomes $|1-z|\leq 1-|z|$. But by triangle inequality, we have ...
5
votes
1answer
189 views

Good texts in Complex numbers?

I have asked some members on chat about good text to study complex numbers , they recommended for example , "Visual Complex Analysis" by Needham and "complex analysis" by Steins. But, I look for a ...
5
votes
2answers
305 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 ...