Complex number Definition and 436 Threads

Problem: Let $z=x+iy$ satisfy $$z^2=z+|z^2|+\frac{2}{|z|^3}$$then the possible values of $x+y$ is A)$-2^{1/4}$ B)$2^{1/4}$ C)$3^{1/4}$ D)$-5^{1/4}$ Attempt: Substituting $z=x+iy$ is definitely not a good idea, it can be solved by substituting but since this is an exam problem, I believe that...

To prove that $$sin^{-1}(ix)=2n\pi\pm i log(\sqrt{1+x^2}+x)$$ I can prove $$sin^{-1}(ix)=2n\pi+ i log(\sqrt{1+x^2}+x)$$ but facing problem to prove $$sin^{-1}(ix)=2n\pi- i log(\sqrt{1+x^2}+x)$$ Help please

How do I solve z^4+16=0 in a+bi form. I don't see how it can be factored so what do I do?

Solve the following equations in the form a +bi. a) z^3-1=0 b) 3z^4+i=1-2i Apparently, the solution for a) is this: z^3=1 z=1 z=a+bi=1 sqrt(a^2+b^2)=1 I don't understand why a+bi=1 is not the final answer. Why do you have to make it into a modulus?

Homework Statement z^3 + (-5+2i)z^2 + (11-5i)z -10+2i =0 has a real root, find all the solutions to this equation. The Attempt at a Solution I have only solved imaginary number polynomials with a given root, but this has no given root, how do I find the real solution? that I can then...

I was thinking in the bus, we have theoretically speaking particles such as tachyons with imaginary number for their mass. These particles if they exist always run at speeds higher than c. My question is has anyone thought of modifying perhaps Lorentz transformations to give us the possiblity...

Homework Statement Wolfram gives a step to step solution for \sqrt[3]{-9+46i} → 27+ 54i -36 - 8i → 27+ 54i -36i^2 + 8i^3 = (3+2i)^3 Can you tell me (where to find) the rules of this derivation? Thanks

Homework Statement Find complex number z from equation Sinz+cosz=4Homework Equations The Attempt at a Solution I rewrite sinz+cosz=4 in form: \frac{(exp[i*z]-exp[-i*z])}{2i}+\frac{(exp[i*z]+exp[-i*z])}{2}=4 then subsitute t=exp[i*z] So I gain t2(1+i)+8ti-1+i=0,I think i can rewrite it in form...

Digital signal processing -- conjugate reciprocal of a complex number what is the difference between conjugate of a complex number and conjugate reciprocal of a complex number i am asking with reference to z transform...Thanyou

Hello guys! Homework Statement The question is like this: If ##z=\frac{a}{b}## and ##\frac{1}{a+b}=\frac{1}{a}+\frac{1}{b}##, find ##z##. The Attempt at a Solution This question is challenging for me because I don't know exactly where to start. The latter condition stated, the sum of...

Homework Statement Find all complex numbers z such that z^6 = -64 Homework Equations I tried it in two ways, once with the sum of cubes (a + b)(a^2 - ab + b^2) as well as turning it into polar form and attempting it that way, z = re^(iθ) The Attempt at a Solution I completely couldn't...

Homework Statement Calculate Z1=5+j10 Z2=10+j8 Z3=10+J5 RL=40 V=100 VTH=VX(Z2/Z1+Z2) ZTH=Z3+(Z1Z2/Z1+Z2) I=VTH/(ZTH+RL) IL=? Homework Equations My question is what calculation method is more accurate: First to convert complex numbers in polar forms, and then calculate or calculate complex...

This is a thread for complex number problems in applied mathematics. 1. Prove that: 1 + cos x + cos 2x + ...cos (n - 1)x = {1 - cos x + cos (n - 1)x - cos nx} / 2 (1 - cos x) = 1/2 + [{sin (n - 1/2)x}/2sin (x/2)]2. If a = cos x + i sin x, b = cos y + i sin y, c = cos z +...

1. $|\frac{i(2+i)^3}{(1-i)^2}|$ Is there any way to complete this without expanding the numerator?2. what is the argument of $ -2\sqrt{3}-2i$ I got $r=4$ then $\cos\theta_1 $ $= \frac{-2}{\sqrt{3}{4}}$ and $-2=4\sin\theta_2$ $\theta_1 = \pi - \frac{\pi}{6} = 5\frac{\pi}{6}$ and $\theta_2 =...

Hello MHB, Solve the following system of linear differential equation $$f'=f-g$$ $$g'=f+g$$ with bounded limit $$f(0)=0$$, $$g(0)=1$$ could anyone check if My answer is correct? Just to make sure I understand correctly! ps we get $$\lambda=1-i$$ and $$\lambda=1+i$$ Regards, $$|\pi\rangle$$

If I'm graphing (3+4i)/25, would the x-point be 3/25 and the y-point be 4i/25?

I have for a while been trying to really understand the notion of a complex number and the construction of the complex number system. My knowledge of mathematics so far is very limited and spans mostly linear algebra (no pun intended), discrete mathematics (where I have yet to see complex...

Homework Statement I am doing a problem of a contour integral where the f(z) is z1/2. I can do most of it, but it asks specifically for the principal root. I have been having troubles finding definitively what the principal root is. Anyplace it appears online it is vague, my book doesn't...

Two complex numbers z1 and z2 are taken such that |z1+z2|=|z1-z2|, and z2 not equal to zero. Prove that z1/z2 is purely imaginary (has no real parts). I started by taking z1=a+bi, and z2=c+di, then z1+z2=a+c+i(b+d) and z1-z2=a-c+i(b-d) |z1+z2|=√(a+c)^2 + (b+d)^2 |z1-z2|=√(a-c)^2 + (b-d)^2...

Homework Statement For my waves class, I have to do this problem. I've previously completed a question like this except there was no phase constant (∏/4) in that question. Express the following in the form x = Re [Ae^i\alphae^iwt x=cos(wt + ∏/4) - sin(wt) Homework Equations euler's...

Homework Statement So on one of my homework assignments I had to find the complex fifth roots of one. Because of Euler's equation the arguments are simply 0, 2pi/5,4pi/5,6pi/5,and 8pi/5. It is easy to see that (e^i2pi/5)^5 = (e^i2pi) = cos(2pi) + isin(2pi) = 1 + 0 = 1 but on my paper I wrote...

Homework Statement Use the supernode technique to find I0 in the circuit Homework Equations nodal analysis, AC analysis The Attempt at a Solution Taking the left essential node as V1 and the right essential node as V2, V1 + 12V = V2 and -V1/(1-1j)Ω +...

Homework Statement z=1 + e^(iθ) calculate z^2 and lzl^2 Homework Equations The Attempt at a Solution for z^2 (1+e^(iθ))(1+e^(iθ)) = 1 + 2e^(iθ) + e^(i2θ).. is that the final answer? i expanded it into cosines and sines as well but that doesn't simplify anymore i don't...

Homework Statement Create a class which contains the following: a)instance variables for the complex number. For x+yi, x and y are the variables. b)constructor to initialize these variables. c)methods to return the real and imaginary parts of the complex number d)method to compute the...

is E = {(-2)^5}^(1/5) a complex number ? my cal gives E=(-2), but this gives 1.61+1.78i http://www.wolframalpha.com/input/?i=%28%28-2%29^%285%29%29^1%2F5 someone please explain this..

Hi, How can we explain this paradox: Thanks in advance.

So I wrote a program that has one main function call two other functions in an equation and then calculates a value. The problem is the numbers I need returned from those two other functions need to stay in their complex form. Using double() I get an error when trying to return that. Is there...

complex number ?? Homework Statement Let ω be the solution to the equation x2+x+1=0 Get the value of ω10+ω5+3=Homework Equations complex numbers?The Attempt at a Solution When I try solving the first equation I hit a complex number which is making me think I am wrong. (x+1/2)2=-3/4 Again if...

z1 = −11+2 i and z2 = −1+13 i are given I need to find the following in the form x + y i. conjugate of z1 = conjugate of z1z2= z1/z2 = a) how would i go about it and b) can someone provide the solutions to the questions if possible

Homework Statement The locus of z satisfying the inequality (z + 2 i) / (2 z + i) < 1 where z = (x + i y) Homework Equations none The Attempt at a Solution After putting the value of z = (x + i y) , I tried to rationalize it but now I am stuck. can somebody explain how to solve it. The...

Hi all, This is a little problem I'm unable to figure out, in CMPLX mode of my calculator 25i*20i/5i=-100i But 25i*20i/(5i) = 100i, Here i=sqrt(-1). What is the difference between these two expressions? Thanks for your help.

Hello! Few weeks ago we started learning complex number. And i have some questions about that because not all i understand and also don't know if my solution is good :) Also don't know if my name of topic is good :) if something goes wrong just say. I'd be grateful My task: |(z+1)/(z-1)|=3 need...

Homework Statement Find both square roots of the following number: -15-8i Homework Equations De Moivre's thm: rn(cos(n\sigma) + i sin(n\sigma) The Attempt at a Solution So to use De Moivre's I have to find the modulus and the argument. actually in this question r =...

I'm given 1-a\cdot e^{-i\cdot 2 \pi f}. The squared absolute value apparently is |1-a\cdot e^{-i\cdot 2 \pi f}|^2=1+a^2-2acos(2 \pi f). Sadly the awnser doesn't show the steps of this derivation. I have tried many times to derive it my self but have not been able to do so. I feel like i...

Homework Statement 4{cos(13∏/6)+isin(13∏/6)} = 4((√3/2)+(i/2)) = 2√3+2i Homework Equations The Attempt at a Solution This is an example from my textbook. The part which I do not understand is how to convert the cos and sin of radians into those fractions. Any help is greatly appreciated.

I have solutions for eigenvalues to be λ1=i-1 = √2 e^i(3∏/4) and λ2=i+1 =√2 e^i(∏/4) How do you go from the i-1 to the next bit for both? Thanks

Find an orthonormal basis for P2(ℂ) with respect to the inner product: <p(x),q(x)> = p(0)q(0) + p(i)q(i) + p(2i)q(2i) the q(x) functions are suppose to be the conjugates I just don't know how to write it on the computer Attempt: This is where I'm having trouble. So usually I'm given...

Find all the values of $$(\sqrt{3}+i)^{1/6}$$. What is its principle value?I have doubt about the second part. We have heard about the principal value of the amplitude of a complex number. But here the principal value of the complex number itself is asked for. Please help

Homework Statement The problem along with its solution are attached as TheProblemAndSolution.jpg. This post's focus is on part (iii), specifically. Homework Equations z (overlined) = z. (according to book) z (double overlined) = z. (according to me) The Attempt at a Solution I...

This question might be elementary: If the norm of two complex numbers is equal, can we deduce that the two complex numbers are equal. I know in ℝ we can just look at this as an absolute value, but what about ℂ? So mainly: let |z| = |w|*|r| can we say → z = w*r ? Thanks

All, I've been beating my head up on this equation, so I thought I'd try here to see if someone could help me. In the equation (sqrt{32}(2x-d))^-4 = (-d/2 - 1/2 )^2 + sqrt{(1+d)^2 + d/4}/4 Where the right hand side is my attempt to solve for d out of the magnitude |z| of a complex...

Homework Statement A textbook of mine asserts that ℝ is a subset of ℂ. The motivation for this is drawn by defining complex addition and multiplication and then showing that these operations on complex numbers of the form (x,0), with x an element of ℝ, are isomorphic to the field ℝ witih...

Hi, What should be the influence of the imaginary part on a complex number? I am asking because I am running a simulation model where the input is a complex number; say z=a+ib Now the problem is that I get the same result when I put a=0 and give some high value to b, as when I do the...

z= (10-j+elogj)2 I tried expanding but it just made it more complicated. help please!

Homework Statement Let w = a + bi be a complex number and let T : C -> C be defined by T(z) = w · z. Considering C as a vector space over R, find the matrix B representing T relative to the basis {1, i} of C. Homework Equations The Attempt at a Solution I think you use...

Hey, So I have a sum of complex numbers that really should be easy, but I'm not getting the right solution. It is with respct to using the Gram Schmidt process U1 = (i, -1, i) U2 = (1,1,0) So I perform the Gram Schmidt with U1 being my initial vector selection and I get: V2 =...

Homework Statement Evaluate {(1+i)^(1-i)} and describe the set{1^x} when x is a real number, distinguish between the cases when x is rational and when x is rational. for now considering the complex number. 2. The attempt at a solution i don't know how to start with,for firest part i just...

Homework Statement Given that, x+yi=3/[2+cos θ+i(sin θ)] , prove that x^2+y^2=4x-3 Homework Equations r^2=x^2+y^2 The Attempt at a Solution since we know that x=r sinθ , y=r cosθ, i multiply r on the right hand side to equal 3r/[2r+x+yi], then multiply its comjugate (2r+x)-yi to...

Homework Statement Let z \in \mathbb{C}. Prove that z^{1/n} can be expressed geometrically as n equally spaced points on the circle x^2 + y^2 = |z|^2, where |z|=|a+bi|=\sqrt{a^2 + b^2}, the modulus of z. Homework Equations // The Attempt at a Solution My problem is that I am...

I have some math problems What is the solution to this equation : z dash(complex conjugate) = z^3 Z is complex number I try to multiply both sides by Z in the left i get Z dash Z => |Z| but i don't see the solution ---- P is primitive 9th root of unity. Calculate the sum 1 + 2P...