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

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0
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1answer
26 views

Using De Moivre's theorem with relation to the argument of a complex number

Given that $Z^4 = 64(\cos\pi+ i\sin\pi)= 64(-1+0i) = -64$ I understand that the argument [$arg(Z^4)$] is $\pi$, now if instead given the form $Z^4 =64(-1+0i)$ and I desired to find the argument ...
0
votes
6answers
71 views

How to find the roots of $-x^3+3x^2-7x+5 = 0$?

I would like to understand how to go about solving something like this, not just get the solution but some kind of methodology (that hopefully makes as much intuitive sense as possible); I honestly ...
0
votes
2answers
28 views

Complex numbers in polar form

If we have two complex numbers, in polar form, as the numerator and denominator of a fraction, and we are asked to write them as a single complex number, is there an easier way to deal with them ...
0
votes
5answers
42 views

imaginary number evaluation

Question Let $z_1 = 1 + i$, $z_2 = 2 - i$, evaluate $$\left | \frac{z_1}{z_2} \right |$$ I have this question! Its to evaluate the fraction ! what I did is the following ...
2
votes
2answers
22 views

Proof of multiplicative inverse for polar complex numbers [duplicate]

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(\cos(\alpha)+i\sin(\alpha))$. I can do ...
1
vote
3answers
36 views

Proof for complex numbers and square root

Use the polar form of complex numbers to show that every complex number $z\neq0$ has two square roots. I know the polar form is $z=r(\cos(\alpha)+i \sin(\alpha))$. I'm just not sure how to do this ...
1
vote
2answers
39 views

Use polar complex numbers to find multiplicative inverse

Use the polar form of complex numbers to show that every complex number $z\neq0$ has multiplicative inverse $z^{-1}$. If $z=a+bi$, then the polar form is $z=r(cos(\alpha))+i(sin(\alpha))$. I can do ...
-3
votes
1answer
22 views

Fourth roots of a certain complex number [on hold]

Find the fourth roots of $81(\cos 320^\circ + i\sin 320^\circ )$. Write the answer in trigonometric form. \begin{array} \text{a.} & 3(\cos 160^\circ + i \sin 160^\circ ); & &3(\cos ...
1
vote
1answer
37 views

Solution to second order differential equation

I'm reading a paper in which the authors solve the following equation: $\frac{d^{2}}{dz^{2}}\hat{p}$($\bf{q}$$,z)$-$q^{2}\hat{p}$($\bf{q}$$,z)$-$\frac{iq_{y}}{(2\pi)^{2}}\delta(z-z_{2})$=0 here ...
1
vote
3answers
20 views

Number of complex numbers such that $z^{80} = 1$ and other properties

Let $$A = \left \{ z \in \mathbb{C} : \Re z > 0, \Im z < 0, z^{80} = 1 \right \}$$ Then, the number of elements in $A$ is $19$, $20$, $21$, or $22$? I just started studying complex ...
0
votes
1answer
18 views
0
votes
1answer
21 views

Cauchy-Riemann and Analytic Functions

Using the Cauchy-Riemann conditions, tell if $f(z) = z^{*}$ is analytic I have tried this: $Z = x + iy$ $f(x + iy) = Z^{*} = x - iy$ $U(x,y) = x$ $V(x,y) = -y$ $U_x = 1$ Deriving respect to $x$ ...
1
vote
3answers
108 views

Why does the imaginary number $i$ satisfy $i\times 0=0$?

Why does the imaginary number $i$ satisfy $i\times 0=0$? I mean, we don't really know what $i$ is. How could we be sure about that? I think there's a reason behind why mathematicians decided that.
3
votes
1answer
38 views

Conformal mapping of the domain bounded by a line segment and a circular arc

I am trying to construct a conformal map from the region $R$ which is the set of points in the complex plane bounded by the segment connecting $i$ and $1$ and the part of the unit circle in the first ...
2
votes
3answers
79 views

Complex equations with no complex solutions?

Are there complex equations that admit no complex solutions, but rather quaternions or hypercomplex solutions, for example, in complete analogy to, say, the equation $x \times x = -1$ when restricted ...
2
votes
0answers
45 views

A hard Conformal Mapping problem

I am trying to construct a conformal map from $R = \{z \in \mathbb{C} : -1 < Re(z) < 1$ and $Im{(z)} > 0\} \cap \{z \in \mathbb{C} : |z| > 1\}$ to the unit disk $\mathbb{D}$. I am really ...
-1
votes
0answers
34 views

Cardinality of the set of complex numbers [duplicate]

Given the continuum hypothesis, does the cardinality of $\mathbb{R}$ ($\aleph_1$) equal the cardinality of the set of complex numbers? If not, what is the cardinality of $\mathbb{C}$? Would it be ...
1
vote
1answer
23 views

Sketching regions is complex plane

When sektching the region $\left|\frac{2z-1}{z+i}\right|$$\geq$1 on the argrand diagram, how should we go about identifying the region, should we take $\left|2z-1\right|\geq\left|z+i\right|$ or ...
1
vote
4answers
29 views

Complex numbers and their modulus

Can we cancel the modulus on complex numbers? For example: If we have $$|x + iy| = |n + im|$$ can we simply ignore the modulus on both sides? Or is that a false assumption?
3
votes
5answers
292 views

Distributing powers on complex numbers

Can I not distribute powers on complex numbers as I do with real numbers? For example: Consider $$\left(\frac{1 + i}{1-i}\right)^n = 1$$ Distributing powers as in real numbers: $$(1+i)^n = (1 - ...
3
votes
4answers
92 views

Find modulus of $z$ given modulus of $(z-3w)/(3-z\overline{w})$

Question: (23) If $z_1$, $z_2$ are complex numbers such that $\left|\dfrac{z_1-3z_2}{3-z_1\overline{z}_2}\right|=1$ and $|z_2|\neq 1$, then find $|z_1|$. How would I attempt this question? I ...
0
votes
3answers
76 views

Solving the equation $(z-2)^{4}+(z+1)^{4}=0$

$(z-2)^{4}+(z+1)^{4}=0$ I tried starting by solving $z^{4}=1$ with the solutions being , $1cis (\frac{n\pi }{2})$, where $n = -1, 0, 1, 2$ I am unsure about how to proceed from here, I tried to ...
1
vote
0answers
61 views

The sum (or difference) of two irrational numbers

So far I that for any irrational number without a real part (that $-n=\overline{n}$) plus/minus any irrational number with the same restrictions equals another irrational number. However, I want to ...
0
votes
10answers
76 views

Complex numbers and their imaginary parts

Question: If $$z = \left(\frac{\sqrt3}{2} + \frac{i}{2}\right)^{107} + \left(\frac{\sqrt3}{2} - \frac{i}{2}\right)^{107} $$ Show that Im(z) = 0 I have no idea how to even start the question. Please ...
0
votes
4answers
50 views

Correct this problem in complex numbers

Prove for all $|z| = 3$, $$\frac{8}{11} \leq \left | \frac{z^2 + 1}{z^2 + 2}\right | \leq \frac{10}{7}.$$ Here is what I did, $$\frac{8}{11}\leq \frac{8}{z^2 + 2}=\frac{|z^2| - 1}{z^2 ...
0
votes
3answers
41 views

For all complex numbers with $|z|=2$, inequality $2\le |z-4|\le 6$ holds

Prove for all $|z| = 2$, $$2 \leq |z - 4| \leq 6$$ I tried $|(x - 4) + iy| = |x^2 + 16 - 8x + y^2| = |20 - 8x|$ I also tried using triangle inequality $2 = |z| = |z - 4 + 4| \leq |z - 4| + 4$ ...
0
votes
1answer
20 views

Euclidean division - For what values of a, does the polynomial g(t) get divided by f(t) in the complex ring

They want to find the values of a where g(t) can be divided by f(t). $f(t) = t^2 + it − ai$ $g(t) = t^4 + (1 − i)t^3 + (1 − 2i)t^2 − 3at − (4 + 2i)a$ Euclidean algorithm: $g(t) = f(t)q(t) + ...
1
vote
2answers
88 views

Polar form of a complex number

Question: Write the polar form of $$\frac{(1+i)^{13}}{(1-i)^7}$$ Well its obviously impractical to expand it and try and solve it. Multiplying the denominator by $(1+i)^7$ will simplify the ...
1
vote
1answer
34 views

Mean value theorem for harmonic

In Problems and Solution in Mathematics by Ta-Tsien, exercise 5123, the mean value theorem is used as: \begin{equation} \text{log} |F(0)| = \frac{1}{2 \pi} \int_0^{2\pi} \text{log}|F(re^{i\theta})| ...
2
votes
2answers
43 views

Is there any subset of Complex numbers that is algebraically closed?

That any polynomial that is allowed to have coefficients from that subset has also a root in that subset
1
vote
0answers
26 views

Complex numbers product and ratio, prove this relation.

Define a table $T$ as follows: $$\text{If}\; n\geq k \; \text{then} \; T(n,k) = (2+3 i) \sum _{i=1}^{n-1} T(n-i,k-1)+(5+7 i) \sum _{i=1}^{n-1} T(n-i,k) \; \text{else} \; T(n,k) = 0$$ Then take rows ...
1
vote
1answer
37 views

Complex variable algebra mishap

One question on a problem set was the following: Show that $x^2 - y^2 = 1$ can be rewritten as $z^2 + \bar{z}^2 = 2$. (With $z = x + iy$) So I started working from the first expression based on ...
1
vote
1answer
45 views

Find the set of $z$ which satisfies the given equation

Let $w \to w^{a}$ be the principal branch of the power function defined for $|\mathrm{Arg}(w)| <\pi$. Find the set of all values of $z\in \mathbb{C}$ such that the following identity holds for ALL ...
2
votes
0answers
43 views

Factorial of Complex Values

Since the gamma function is an analytic continuation of the factorial function, we can find the factorial of complex values. How does one go about doing so? I've looked far and wide on the internet ...
3
votes
5answers
136 views

Which one is correct for $\sqrt{-16} \times \sqrt{-1}$? $4$ or $-4$?

As we can find in order to evaluate $\sqrt{-16} \times \sqrt{-1}$, we can do it in two ways. FIRST \begin{align*} \sqrt{-16} \times \sqrt{-1} &= \sqrt{(-16) \times (-1)}\\ &= ...
11
votes
3answers
109 views

Prove that an expression is zero for all sets of distinct $a_1, \dotsc, a_n\in\mathbb{C}$

A while ago one of my professors gave the class a problem "to think about when lying on the beach." Well, I've been on the beach several times since then to no avail and my curiosity has finally ...
2
votes
3answers
67 views

Real matrices with non-real eigenvalues

I know this covers a lot, so perhaps someone could redirect me to a helpful website. for a) I have no idea where to start on the proof, as I don't understand why this is true. for b) I also have ...
1
vote
0answers
41 views

Covering Space of $\mathbb{C}-\{a,b\}$ via Multivalued Function

Consider the multivalued complex function $f(z)= \sqrt{z-a}+\sqrt{z-b}$, where $a\neq b$, defined in the domain $U=\mathbb{C}-\{a,b\}$. The graph $W$ of $f$ defines a regular covering space $W ...
1
vote
0answers
56 views

Finding the number of elements in $\left(ℤ[i]\right)_m$

If $m$ = $m_1$ + $m_1i$ ∈ $ℤ[i]$, is there a general formula to find the number of elements in $\left(ℤ[i]\right)_m$?
4
votes
2answers
37 views

$\sum_j e^{i\phi_j}$ vs $\sum_j e^{ip\phi_j}$

Let $\phi_j$ be a collection of angles. If $p$ is a positive integer, how is the sum $\sum_je^{i\phi_j}$ related to $\sum_je^{ip\phi_j}$?
2
votes
1answer
44 views

The complex equation

In solving $|z|i +2z =1$, I seem to be constantly getting two solutions while both answer key and Wolfram claim to be only one. What am I doing wrong? Let's share the fun: $(\sqrt{x^2 +y^2}) i +2x ...
2
votes
5answers
66 views

Find $n$ for which $(1+i)^{2n}=(1-i)^{2n}$

Question: Find the values of $n$ for which $$(1+i)^{2n}=(1-i)^{2n}$$ wolfram alpha tells me that the answer should be : $$n=\frac{2i\pi m}{\log(1-i)-\log(1+i)}$$ $$n=-\frac{i(2\pi ...
6
votes
2answers
106 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 ...
0
votes
1answer
41 views

Number theory proof regarding norms

How would you prove that if $x$ is a prime in $ℤ[i] \Longleftrightarrow$ $N(x)$ is a prime in $ℤ$ N(x) represents the norm of x.
2
votes
2answers
97 views

Why is $ i^2 \neq (1 + i)^4$?

Today I read that you can see the number $i$ as the rotation of 90° and therefore i^2 is the rotation of 180° or -1. I also learned that $1+i$ is 45° but if this would be true I should be able to ...
2
votes
1answer
35 views

What does taking the $n^{\text{th}}$ root of a complex number geometrically mean?

What are the geometrical implications of taking the $n^{\text{th}}$ root of a complex number, say $3+4i$. What is the geometrical implication of $\sqrt[n] {3+4i}$ in the complex plane?
1
vote
1answer
53 views

A simple complex inequality

I feel this is not hard, but no way to prove it $|\sqrt{z^2 -4}-z|\le 2$ Any body can help? Thanks! The total statement should be one of the branchs of square root should satisfy this ...
1
vote
3answers
132 views

The Name for $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ Exclusively Algebras?

The Wikipedia page for Normed Division Algebras has been redirected to Normed Algebras and the explanation given is that $\mathbb{R}$, $\mathbb{C}$, $\mathbb{H}$ and $\mathbb{O}$ algebras are not the ...
0
votes
1answer
35 views

Extension to the complex numbers for ex. 12 in ch. 6 of Axler's “Linear Algebra Done Right”

I'm wondering how the answer to Sheldon Axler's exercise 12 of chapter 6 "Linear Algebra Done Right" changes when the underlying field is extended from the reals to the complex numbers. The exercise ...
1
vote
2answers
46 views

Evaluate expression in the form $a+bi$.

So, I have to evaluate $\sqrt{-3}\sqrt{-12}$ into the form $a+bi$. I know that $i^2 = -1$ so $i = \sqrt{-1}$ What I have done is: $$\begin{align}\sqrt{-3}\sqrt{-12} &= \sqrt{3(-1)}\sqrt{12(-1)}\\ ...