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

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What is $i^i$?Imaginary, Real, HyperComplex?

What is $ i^i$, where $i$ is the imaginary unit. Apparently wolfram alpha and google give: $$i^i\approx0.207879576=e^{-\pi/2}$$ But how? Maybe let me try: $$x=i^i\implies x=\exp(i\ln i)$$ ...
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0answers
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Is $\mathrm{col}(\lambda I_n-A)\subseteq \mathrm{col}(B) $ for a complex $\lambda$?

Let $A\in\mathbb{R}^{n\times n}$, let $I_n$ denote the identity matrix of order $n$, and let $ \mathrm{col}$ denote column space. I'm interested in understanding for what values of $\lambda \in ...
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1answer
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Complex Field - Proving $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$

Like the title states, I'm trying to prove that $\left(\frac{z_1}{z_2}\right)^{\star} = \left(\frac{z_1^{\star}}{z_2^{\star}}\right)$ where z is a complex number and z* is its conjugate. I keep ...
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1answer
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Help with proof in complex analysis

I was looking at the proof of the result, the image of $\mathbb R_\infty$ under mobius transformation is a circle. I don't follow how does this step $(a\bar d-\bar bc)w+(b\bar c-\bar ad)\bar w+b\bar ...
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1answer
78 views

Why isn't the identity $\sqrt{ab}$ = $\sqrt{a} \cdot \sqrt{b}$ always true?

If we take $a=b=-1$ the the L.H.S. is $1$ but the R.H.S. is $-1$. Is this identity not applicable for complex numbers? How to prove this and prove that this is not applicable for some complex ...
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3answers
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A tricky complex numbers if and only if proof

For complex numbers $z$ and $w$ prove that $$|z|^2w -|w|^2z = z-w\quad \iff\quad z=w\quad\text{or}\quad z\bar{w}=1.$$ How would you go about solving this problem?
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1answer
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Can you graph equations with a negative discriminant? And how do you plot complex numbers both on a 2D complex plane and a 4D complex plane?

I don't understand the relationship between complex numbers and that way they are graphed. The equation I am working with is $2x^{2} - 6x + 5 = 0$ where my two roots are complex solutions: $x = ...
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1answer
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Can the coefficients used to prove a set of functions is linearly dependent be imaginary?

Example: $\cos x$, $e^{ix}$, $3\sin x$. I can show: $C_1\cos x + C_2 e^{ix} + C_33\sin x = 0$ if $(C_1,C_2,C_3) = (1,-1,i/3)$ But i don't know if $C_3 = i/3$ is a valid coefficient to choose. Can ...
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2answers
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geometric description of set of complex number

A set of complex number: $$S=\{ z\in \Bbb C : |z|=\lambda |z-1|\}$$ what's the geometric description? I try to draw it ... which seems like a circle but cannot find the equation to describe it..
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Factorization of $z^4 +1 = (z^2 - \sqrt 2z+1)(z^2 + \sqrt 2 z+1)$ for complex z

How can I get this equation from LHS to RHS by using the four roots of $z^4 +1 = 0$ are $z=\pm\sqrt{\pm i}$ $$z^4 +1 = (z^2 - \sqrt2 z+1)(z^2 + \sqrt2 z+1)$$
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6answers
615 views

Is the Complex Conjugate the Only Way to Get a Real Number?

Is the complex conjugate of a number (or a real multiple of it) the only complex number which, when multiplied with the original number, gives a real number?
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3answers
60 views

What is the correct notation for flipping $a$ and $b$ values in a complex?

I'm currently doing some experiments on fractals and in one of my equation I need to flip the real and imaginary components of a complex number, such as : $$ z = a + bi $$ Becomes : $$ z = b + ai ...
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2answers
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When does $(e^a)^b = e^{ab}$ hold?

For a complex number $A$ and a real number $B$, when does the well-known formula $(e^A)^B = e^{AB}$ fail? Or does it hold at all for complex A? Since $e^{2\pi i} = 1$, if this formula holds for ...
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3answers
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Triangle Inequality on complex numbers

Problem Let $z= x + iy$, then prove that: $$|x| + |y| \le 2 ^{1/2} |z|$$ Progress I've tried to write $|z|$ as $(x^2 + y^2)^{1/2}$, and to make some algebra after this, but I'm really new at ...
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1answer
45 views

Geometric interpretations of an equality.

I need to prove that for complex numbers $w_1, w_2$ and $w_3$ if: $$\frac{w_2-w_1}{w_3-w_1}=\frac{w_1-w_3}{w_2-w_3}$$ then: $$|w_2-w_1|=|w_3-w_1|=|w_2-w_3|$$ by geometric interpretation of the given ...
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1answer
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Supremum of the set $\{\operatorname{Re}(iz^3+1) : |z|<2\}$

I need to find supremum of the set of all real numbers of the form $\operatorname{Re}(iz^3+1)$ such that $|z|<2$. By the inequality $-|w|\le \operatorname{Re}(w)\le |w|$ we have ...
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2answers
34 views

If $p(z)$ is a monic polynomial then $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$

I need some help with this problem: If $p(z)$ is a monic polynomial of degree $n$ then there is a $b\in\mathbb{C}$ such that $p(z)+b=(z-z_1)(z-z_2)\cdots (z-z_n)$ where $z_1,z_2,\dots,z_n$ are simple ...
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4answers
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Complex number equations

I cannot solve two problems regarding complex equations. 1)Let $z^2+w^2=0$, prove that $$z^{4n+2}+w^{4n+2}=0, n \in \mathbb{N^{*}}$$ What I tried; $$z^2 \cdot z^{4n}+w^2 \cdot w^{4n}=0 \iff ...
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2answers
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For a complex number $c$, how does a plot of $c^{-4},c^{-3}, c^{-2}, c^{-1},c^0, c^1,\dots, c^4$ look like?

I am on the road so can't test it for myself: what would happen if I took a complex number $C = a + bi$ and plotted the following in the complex plane; $$C^{-4}, C^{-3}, C^{-2}, C^{-1}, C^0, ...
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1answer
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Solving system of equations with complex numbers

Equation 1$$ \frac{V_{1}}{5} + \frac{V_{1}-V_{2}}{10+j6} - 10\angle45^\circ = 0 $$ Equation 2 $$ -4V_{1} + \frac{V_{2}-V_{1}}{10+j6} + \frac{V_{3}}{-j2} + \frac{V_{3}}{8+j7} = 0 $$ Equation 3 $$ ...
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1answer
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Complex roots (review) (advise)

I have to find the complex roots and want a review of my procedure to see if is correct A. $$\sqrt{3i}$$ $$\left |z \right |=3 $$ $$phase= 90^{\circ}=\displaystyle\frac{\pi}{2}$$ ...
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2answers
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summing the powers of a complex number

Let $z=e^{\frac{2\pi i}{5}}$, then $1+z+z^2+z^3+5z^4+4z^5+4z^6+4z^7+4z^8+5z^9=?$ I am kind of confused since by drawing a graph, $1+z+z^2+z^3+z^4$ should be zero, but using computational softwares ...
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0answers
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Subring of $\mathbb{Z}[i]$ and an infinite set $X$ such that $\exists x \forall y \in X \,\,x^2\mid y^2$ but $\forall x \forall y \,\,x \not\mid y$

This is a question derived from A subring of the ring of Gaussian integers such that $a^2 \mid b^2$ does not lead to $a\mid b$ in infinitely many such cases. Is there a subring $R$ of Gaussian ...
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1answer
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Finding equations for plane figures using complex coordinates

I have to find conditions defining the following plane figures: Where: $a=3$ and $b=7$ I know that circumference form is: $$\left |z-z_0 \right | =b$$ So, for c. with center $(3,3)$ and radius ...
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the absolute value of $\frac{1}{e^{i\omega t}-1}$

I am told to get the absolute value of $$\frac{1}{e^{i\omega t}-1}$$ I sense that there's something ridiculously simple about this, but I tried working from the fact that if I square it, the absolute ...
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2answers
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Can I conjugate a complex number: $\sqrt{a+ib}$?

Can I find the conjugate of the complex number: $\sqrt{a+ib}$? Actually my maths school teacher says and argues with each and every student that we can't conjugate "square root of $a+ib$" to "square ...
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Complex number cubed [closed]

You are given z^3 = cos(−1/2*Pi )+i*sin(−1/2*Pi) in polar form. This equation has exactly 3 complex roots. What is the set of three principle arguments (in radians) belonging to those three roots? ...
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Weird conformal map problem

Construct a conformal map from the region $\omega$ = open disk of radius 1 centered at 0 minus the closed disk of radius 0.5 centered at 0.5 to $\mathbb{D}$ = disk radius 1 centered at 0. I really ...
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Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$.

Let $f(z) = z + z^2$ and let $V = \displaystyle \{z \in \mathbb{C} : |z| < \frac{1}{2}, \frac{3\pi}{4} < arg\{z\} < \frac{5\pi}{4}\}$. $(a)$ Show that $f(V) \subset V.$ $(b)$ Let $f_n$ be ...
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How to find the principal argument of $z^4$, given $z$?

I am having trouble with a homework question. Let $ z= \cos\left(\frac{3}{4}\pi\right)+i \sin\left(\frac{3}{4}\pi\right)$. What is the principal argument of $z^4$ in radians? Is it undefined? If ...
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3answers
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Why does this work for $ i^{2i} $?

I'm finding the principal value of $$ i^{2i} $$ And I know it's solved like this: $$ (e^{ i\pi /2})^{2i} $$ $$ e^{ i^{2} \pi} $$ $$ e^{- \pi} $$ I understand the process but I don't understand ...
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1answer
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How can I find $x$, $y$ values for $\frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i$

$$ \frac{(1+i)x-2i}{3+i}+\frac{(2-3i)y+i}{3-i}=i $$ I believe the format I need in order to solve this problem should be such that the real parts and imaginary parts are separated, ...
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1answer
55 views

Polar form of the sum of complex numbers $\operatorname{cis} 75 + \operatorname{cis} 83 + \ldots+ \operatorname{cis} 147$

The number $\operatorname{cis} 75 + \operatorname{cis} 83 + \operatorname{cis} 91 +\dots+ \operatorname{cis} 147$ is expressed in the form $r\operatorname{cis}(\theta)$, where $0\leq \theta< ...
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1answer
41 views

Why is a complex number plus infinity equal to infinity?

Why is $$2 + 3 i + \infty = \infty$$ according to Mathematica and Wolfram Alpha? Shouldn't it be: $$2 + 3 i + \infty = \infty + 3 i$$ ? After all: $$2 + 3 i + 10 = 12 + 3 i$$ and not: $$2 + 3 i ...
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How is $ i^{-1} = -i$ and $i^{-3} = i$?

Now I know that with positive powers of $i$ the cycle is: $i , -1 , -i , 1\ldots$ The negative power cycle is: $-i , -1 , i , 1 \ldots$ Can someone explain to me how $\frac 1 {\sqrt{-1}}$ is equal ...
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Roots of Unity: Different Methods

I am aware of two different 'methods' for finding say the cube roots of 1. They are Let $z^3=1$ and $z=R(\cos \theta + i\sin\theta)$, then use DMT and equate terms to find that $R=1$, ...
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Principal angle and Euler form of cube root of unity.

The principal angles of the complex arguments are in the range $-\pi \le x\le \pi $. If so why does the comlex number $\omega =\frac{\left(-1-\sqrt{3}i\right)}{2}$ is represented by ...
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1answer
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Finding complex roots of integer polynomials

How would one find approximates for complex root of polynomial with integer coefficients,I know for example the Newton's method $$x_n=x_{n-1}-\frac{f(x_{n-1})}{f'(x_{n-1})}$$ Anyway is it possible to ...
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2answers
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Failure of De Moivre's Theorem

I know that De Moivre's Theorem does not necessarily work for non-integer powers. The classic counter-example is by considering $\left (\cos \theta + i \sin \theta \right )^n=\cos n\theta + i \sin n ...
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2answers
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Difference between the complex roots of $f(x)$ and $|f(x)|^2$

I suppose a basic question, but it's causing me more problems than I envisioned! I have some polynomial $f(x)$ for which the roots are complex, $x+iy$. How will these roots change if I now take ...
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Calculating $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$.

I need to calculate $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}$ for $\alpha \in \mathbb Q$ and $r \in \mathbb R$. My Attempt: $\sum_{n=0}^\infty (r e^{2 \pi i \alpha})^{n!}=\sum_{n=0}^\infty r ...
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4answers
52 views

Writing the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form

Now I can't finish this problem: Express the complex number $z = 1 - \sin{\alpha} + i\cos{\alpha}$ in trigonometric form, where $0 < \alpha < \frac{\pi}{2}$. So the goal is to determine both ...
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1answer
40 views

Construction of Hyper-Complex Numbers

How does one construct a hyper-complex number multiplication table? For example: Quarternions: ...
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30 views

Equality of complex numbers

I'm currently reading some notes on Complex Numbers and came across this 'proof' regarding the equality of complex numbers. Claim: Two complex numbers $a+bi$ and $c+di$ are equal iff $a=b$ and $c=d$, ...
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rationalize the complex number multiplication rule

For a middle school student without previous knowledge of complex number, how do one introduce the multiplication rules of complex number? i.e., if we have two real number pairs of $(a,b)$ and ...
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25 views

Complex numbers - locii

I have been asked to solve the following and represent the answer graphically: A) $| \arg z - (\pi/4) | < (\pi/2)$ I understand that this means the difference between the argument of $z$ and ...
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2answers
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Is ring of Gaussian rationals in unique factorization domain?

Instead of Gaussian integers, let us think about Gaussian rationals, where $a$ and $b$ in $a+bi$ are rational numbers. Then would ring of Gaussian rationals be in unique factorization domain?
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square of complex numbers

I have this equation from here: but it is not equal to: $$(a + bi)^2 = a^2 + 2abi + (bi)^2.$$ could someone explain me what is the difference between this two calcultion?
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Convergence of infinite series of complex numbers [duplicate]

This has been bugging me for some months since our lecturer, a fields medalist, mentioned that he couldn't solve it when he was our age, yet had had two students submit solutions to it (during our ...
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1answer
47 views

Factoring a complex polynomial

Factorize the polynomial : $$ p(x) = x^{5} - x^{4}+ 4x - 4 $$ In real factors in the lowest degree possible. So in previous questions I have been given at least one rot so that I can factorize it ...