What is Complex algebra: Definition and 22 Discussions

In mathematics, a field of sets is a mathematical structure consisting of a pair



⟨
X
,


F


⟩


{\displaystyle \langle X,{\mathcal {F}}\rangle }
consisting of a set



X


{\displaystyle X}
and a family





F




{\displaystyle {\mathcal {F}}}
of subsets of



X


{\displaystyle X}
called an algebra over



X


{\displaystyle X}
that contains the empty set as an element, and is closed under the operations of taking complements in



X
,


{\displaystyle X,}
finite unions, and finite intersections.
Fields of sets should not be confused with fields in ring theory nor with fields in physics. Similarly the term "algebra over



X


{\displaystyle X}
" is used in the sense of a Boolean algebra and should not be confused with algebras over fields or rings in ring theory.
Fields of sets play an essential role in the representation theory of Boolean algebras. Every Boolean algebra can be represented as a field of sets.

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  1. A

    I ##1^x =2## has complex solutions?

    The Youtuber 'Blackpenredpen' claims that ##1^x=2## has solutions ##x= \frac{-i \ln(2)}{2\pi n}## with ##n \in \mathbb{Z}## and ##n \neq 0##. Somone else in a forum claims that because ##1^x## is not injective there are more solution branches and this solution mixes these branches somehow. Who...
  2. D

    A Are there any hypercomplex time values?

    Are there any mathematical models which operate with physical time as with hypercomplex number? If yes, are there any related experiments?
  3. siddjain

    I Prove Complex Inequality: $(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|)>=\sqrt{2}$

    Prove that $$(|z_1 + z_2| + |z_1 - z_2|)(|z_1| + |z_2|) >= \sqrt{2}$$
  4. R

    Argument of a complex expression

    Problem Statement: What is the correct way of computing the argument of the following equation? Relevant Equations: I am trying to compute the argument ##\Phi## of the equation $$\frac{r-\tau\exp\left(i\varphi\right)}{1-\tau r\exp\left(i\varphi\right)} \tag{1}$$ which using Euler's equation...
  5. Santilopez10

    For what m is the complex number $(\sqrt 3+i)^m$ positive and real?

    Homework Statement Find all $$n \in Z$$, for which $$ (\sqrt 3+i)^n = 2^{n-1} (-1+\sqrt 3 i)$$ Homework Equations $$ (a+b i)^n = |a+b i|^n e^{i n (\theta + 2 \pi k)} $$ The Attempt at a Solution First I convert everything to it`s complex exponential form: $$ 2^n e^{i n (\frac {\pi}{3}+ 2\pi...
  6. Mutatis

    Write ##5-3i## in the polar form ##re^\left(i\theta\right)##

    Homework Statement Write ##5-3i## in the polar form ##re^\left(i\theta\right)##. Homework Equations $$ |z|=\sqrt {a^2+b^2} $$ The Attempt at a Solution First I've found the absolute value of ##z##: $$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$. Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...
  7. Safder Aree

    Contour Integration over Square, Complex Anaylsis

    Homework Statement Show that $$\int_C e^zdz = 0$$ Let C be the perimeter of the square with vertices at the points z = 0, z = 1, z = 1 +i and z = i. Homework Equations $$z = x + iy$$ The Attempt at a Solution I know that if a function is analytic/holomorphic on a domain and the contour lies...
  8. A

    Solving Schrodinger's Equation with a weak Imaginary Potential

    Homework Statement A particle of energy E moves in one dimension in a constant imaginary potential -iV where V << E. a) Find the particle's wavefunction \Psi(x,t) approximating to leading non-vanishing order in the small quantity \frac{V}{E} << 1. b) Calculate the probability current density...
  9. Rectifier

    Solving Complex Equation: $$ \bar{z} = z^n $$

    The problem I would like to solve: $$ \bar{z} = z^n $$ where ##n## is a positive integer. The attempt ## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ## ## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1## ##...
  10. K

    I Need a help in solving an equation (probably differentiation

    I am trying to find out the interference condition between tool and a part. The below attached snapshot is the equation between interference and machine feed. At dy/dx = 0, I will have max. interference, which I intend to find. Except x and y every alphanumeric character in the following...
  11. Adgorn

    B Square root of a negative number in a complex field

    Mod note: Fixed all of the radicals. The expressions inside the radical need to be surrounded with braces -- { } (This question is probably asked a lot but I could not find it so I'll just ask it myself.) Does the square root of negative numbers exist in the complex field? In other words is...
  12. Wi_N

    Complex Roots of Z^n = a + bi: Finding Solutions Using De Moivre's Theorem

    Homework Statement I just can't seem to get the right answer. z^4+80i=0 looking at the complex plane u see the radius=r=80 (obviously) using De Moivre extension: z^n=(r^(1/n))(cos((x/n)+k2pi/n)-isin((x/n)+k2pi/n)z1=((80)^(1/4))(cos(3pi/8)+isin(3pi/8) shouldnt this be a root?z2=...
  13. Wi_N

    Solving Complex Quadratic Equations with Imaginary Coefficients

    ok having major problems. i can easily solve z^2 + pz +a+bi=0 solutions but that extra qiz is really annoying me. z^2 + 3z+4iz-1+5i=0 (z+2i)^2+3z-5+5i=0 z+2i = w, z=w-2i w=-3(w-2i)+5-5i then I am not getting anything sensible for solving x and yi. what am i doing wrong?
  14. Remixex

    Complex algebra problem (roots)

    Homework Statement First off i wasn't sure if i should put this in precalc or here so i just tossed a coin[/B] I must find the roots of the expression z^4 +4=0 (which I've seen repeatedly on the internet) Use it to factorize z^4 +4 into quadratic factors with real coefficients The answer is...
  15. PsychonautQQ

    The complex algebra graded by Z-2

    I'm trying to understand something in my notes here... So if we call the real part of the complex algebra 'even' and the imaginary part 'odd' then this graded algebra is communitive but NOT graded commutative. so ab = ba for all a and b in C. If we call the whole complex algebra 'even' and...
  16. C

    Impedance network and complex algebra

    Homework Statement This is one part of a wider question, I'm only posting the part I'm having trouble with. $$ \begin{align} \text{Given an impedance network } B &= \frac{Z_1 \parallel Z_3}{Z_2 + Z_1 \parallel Z_3} \\ \text{show that: } \frac{1}{B} &= 1 + \frac{R_2}{R_1} + j\frac{\omega CR_2}{1...
  17. J

    Calculating Current Supplied by Voltage Source (Complex algebra)

    I've attached the question that I am referring to. I believe I'm heading in the right direction with this one by stating that: 1/Rt = 1/(6+j8)Ω + 1/(9-j12)Ω But I am confusing myself with my algebra. Any help is appreciated
  18. N

    Nearly impossible complex algebra problem

    Homework Statement https://wiki.math.ntnu.no/lib/exe/fetch.php?hash=d26b1f&media=http%3A%2F%2Fwww.math.ntnu.no%2Femner%2FTMA4115%2F2012v%2Fexams%2Fkont.eng.pdf Assignment 1. "Find all complex numbers s such that Im(-z + i)= (z+i)2" What do I do? Homework Equations The Attempt at a...
  19. P

    Complex Algebra: Trigonometry (tan)

    Homework Statement Let tan(q) with q ε ℂ be defined as the natural extension of tan(x) for real values Find all the values in the complex plane for which |tan(q)| = ∞ Homework Equations Expressing tan(q) as complex exponentials: (e^iq - e^(-iq))/i(e^iq + e^(-iq)) The Attempt...
  20. F

    Simple complex algebra problem

    Homework Statement \frac{\beta}{\alpha + i 2\pi f} = \frac{\alpha \beta}{\alpha^{2}+ 2 \pi f}- i \frac{2 \pi \beta}{\alpha^{2}+ 2 \pi f} Homework Equations complex conjugate: (a + ib) * (a - ib) = a^2 + b^2 The Attempt at a Solution If it matters, this is from a book on Fourier...
  21. R

    Complex Algebra Help: Finding u(x,y) & Proving Result

    Homework Statement I am told: {\frac {{\it du}}{{\it dx}}}=y and {\frac {{\it du}}{{\it dy}}}=x. Need to find u(x,y) which is a real valued function and prove the result. Homework Equations The Attempt at a Solution Well, I think the answer is of the form u(x,y) = xy + c because...
  22. W

    Corresponding Areas in the z and w Planes for Algebraic Equations

    The question reads: "what part of the z-plane corresponds to the interior of the unit circle in the w-plane if a) w = (z-1)/(z+1) b) w = (z-i)/(z+i)" I really am having problems understanding what the question is asking. I don't understand what the w plane is, and in which plane...
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