Complex number Definition and 436 Threads

Homework Statement I've been recapitulating some lessons we learned in high school 2 years ago for the exams I need to take this year. There was this exercise I couldn't solve in a nice way. z^2005+(1/z^2005) if we know that z^2+z+1=0 Homework Equations I couldn't came up with a...

Homework Statement \left( \frac{2 \sqrt{3}+2i}{1-i} \right)^{50} Homework Equations z = a+ib z = r(cos \phi + isin \phi) r = \sqrt{a^2+b^2} cos \phi = \frac{a}{r} sin \phi = \frac{b}{r} \frac{z_1}{z_2} = \frac{r_1}{r_2}\left( cos(\phi_1-\phi_2) + isin(\phi_1 -...

Hey i have a question: Q. One root of the cubic equation is z^3 + az + 10 = 0 is 1 + 2i. (i). Find the value of the real constant a. (ii). Show all the roots of the equation on an Argand Diagram. (iii). Show that all three roots satisfy the equation |6z - 1| = 13, and show the locus...

Homework Statement Express (see attachment) with a real denominator Homework Equations Not sure if there is really relevant equations to use here. The Attempt at a Solution First I multiply the top and bottom by the exponential. That gives me e^ix/(e^ix-r^2). I think this is...

Homework Statement Attached question Homework Equations The Attempt at a Solution The second part of question is relatively easy, it is the first part of the question where I need help with(using arg zw = arg z + arg w to show arg z^n = n arg z). Also, is the question asking...

Homework Statement Suppose a and b are real numbers, not both 0. Find real numbers c and d, such that \frac{1}{a + bi} = c + di Homework Equations I said that: 1 = (c + di) (a + bi) 1 = (ac - bd) + (bc + ad)i so if b = 0 then \frac{1}{a} = c + di The rationale...

[2(cos(pi/3)+isin(pi/3))]1/2 I simplified it to 21/2(cos(pi/6)+isin(pi/6)), but I have no idea what else to go to. Any tips would be very helpful, thx in advance

Homework Statement Let ω be the complex number e^(2πi/3)， Find the power series for e^z + e^(ωz) + e^((ω^2) z). Homework Equations The Attempt at a Solution I can show that 1+w+w^2=0, don't know if it would help. Could anyone please give me some hints? Any input is appreciated!

Homework Statement I did the question with help, but did not understand why did we multiply e^-i(x/2) How do I know what to multiply for getting the real part of a complex number in exponential form? Homework Equations The Attempt at a Solution

Take, P = 4e^{-j\frac{\pi}{3}} Q = 4-3j R = 2e^{j\frac{\pi}{2}} S = 5 note: I'm using j to be a complex number, it's equivalent to i in mathematics - A: express p q r s in both cartesian (a+ib) and polar (re^itheta) forms B: sketch p q r and s on the complex plane...

Hellow My question is about Complex number. I can solve and calculate complex numbers. But i m still Unsure about the Concept of Complex number. I have read many articles and they say that a symbol i is added to make the solutions possible.\ In communication engineering, as well as...

Homework Statement C=A*e^(-i*wt)*sin(k*x); A,w,t,k,x are real numbers. Find imaginary part. Homework Equations The Attempt at a Solution Im(C)=cos(wt)-i*sin(wt)

Homework Statement Let z,w be complex numbers. Homework Equations Prove there is a real number \alpha < 1 such that \left|\frac{z^7 + z^3 - i}{9} - \frac{w^7 + w^3 - i}{9}\right| \leq \alpha \left|z - w\right| The goal is to show that \displaystyle q(z) = \frac{z^7 + z^3 - i}{9} is a...

I am confused about something, this isn't homework I was just fooling around with complex numbers, and found this: e^{2\pi i}=1 so ln e^{2\pi i}=ln 1=0= 2\pi i Can someone explain this? the 2\pi i=0 part...I must have done something illegal...

Show that a complex number, w exists such that the fifth roots may be expressed as 1, w, w^2, w^3 and w^4I am having trouble understanding what the question is asking of me. Could anyone please help? Thanks.

if a is a complex number which satisfy ia^3+a^2-a+1=0 then find \left | a \right | ? one way is to put a=x+iy any other short way ?

I've struggled for days reading about square roots of complex numbers and I get most of the problems but not this one. I really want to understand what is going on in this problem, hope someone can help! 1. The complex number (C) is C = 1/\sqrt{i*x} . find the two roots of C. The solution...

Find the real part of the complex number: (1-12i)e^{-2+4i} I know that z = a + ib can be rewritten as z = |z|e^{i\theta} but that doesn't help because the coefficient of the e is not a scalar value, rather a complex number. Thanks Tom

How does taking the multiplicative inverse (reciprocal) of a complex number change its magnitude and argument?

Homework Statement Given that (1-\sqrt{3}i)^{n} is real and positive , use de Moivre's Theorem to show that the values of n are terms of arithmetic progression.Homework Equations The Attempt at a Solution I've worked it out . n=3k s.t k element of positive integers.

Homework Statement i)Find the fourth roots of the complex number -1+\sqrt{3}i , giving your answer in the form of re^{i\theta} . ii)Deduce the solution of the equation z^{8}+2z^{4}+4=0 , giving your answer exactly in the form re^{i\theta} .Homework Equations The Attempt at a Solution...

Homework Statement If given two complex numbers z1 and z2 that have arguments \theta and \phi, and moduli r and R respectively, then find an expression for the mod-arg form of z1+z2 Homework Equations z=x+iy=re^{i\theta}=rcis\theta The Attempt at a Solution I can't seem to find a...

Homework Statement Consider: y = f(x) = ax^2+bx+c, where a, b and c are real constants. Prove that y*=f(x*) Homework Equations conjugate of a complex number x=a+jb and x*=a-jb The Attempt at a Solution You can show the answer of f(x*) by substituting (a-jb) for each x in f(x), but...

Homework Statement If z = \frac{a}{b} and \frac{1}{a + b} = \frac{1}{a} + \frac{1}{b}, find z. Homework EquationsI'm pretty sure z is a complex number. The Attempt at a SolutionI have no idea where to start. The teacher did nothing like this in class. I tried something that...

Homework Statement What is the polar form of the complex number 3-4i? Homework Equations z=r*cos(theta)+i*r*sin(theta) The Attempt at a Solution 5(cos(arctan(-4/3))-i*sin(arctan(-4/3))) This is what I thought the correct answer would be, but it was a multiple choice quiz and...

Homework Statement show that \sqrt{1+ja} is equivalent to \pm(1+j)(a/2)^{1/2} with a>>1 Homework Equations Euler's formula? The Attempt at a Solution with a>>1 |z| = \sqrt{(1 + a^{2})} == a lim a-->infinity arctan (a/1) == \pi/2 \sqrt{z} = \sqrt{(ae^{j\pi/2})} \sqrt{z} =...

Homework Statement Find the 3 solutions of ei\pi/3z3=1/(1+i) Homework Equations ei\theta=cos(\theta)+isin(\theta) The Attempt at a Solution i have put i/(1+i) into polar form,1/\sqrt{2} ei\stackrel{\pi}{4} So i get z3 = \stackrel{1}{\sqrt{2}}ei-\pi/12 Then i got stuck...

Homework Statement Find complex number \lambda such that e\lambdat solves \frac{d^{2}y}{dt^{2}} + 4\frac{dy}{dt} + 5y = 0 Express this solution in the form eat(cos(bt) + i sin(bt)) Homework EquationsThe Attempt at a Solution So the first part is fine, using \lambda2 + 4\lambda + 5 = 0 to get...

Sorry i can tell I am being stupid and missing something here. Homework Statement if 7^(3+2i)=re^(i[theta]) find values of the real numbers r and [theta] Homework Equations er^(i[theta])=r(cos[theta]+isin[theta]) The Attempt at a Solution Ok you know that...

Homework Statement Find the minimum value of arg(z) where z satisfies the inequality |z + 3 -2i| </_ 2 Homework Equations Is this working correct? Thank you for help in advance The Attempt at a Solution Z lies on a circle with radius 2 and centre -3,2 arg(z)min = pi - 2...

Complex number help! Homework Statement Find the modulus and principle argument of 1/(-sqrt(3)+i) Homework Equations The Attempt at a Solution I attempted this solution by using the complex conjugate, and as i^2=-1, i eventually ended up with 4 in the denominator. Any suggestions?

Homework Statement (x+iy)=[1/x-iy]+2 Homework Equations The Attempt at a Solution

Let x,y be in the complex plane Say, (1+i)x+ (2+i)y -5 = 0 (3+2i)x + (4+i) -10 = 0 I couldn't solve such a system of equations with neither of casio fx-9750g, hp 50g and TI-89.. Any help would be appreciated. Thanks in advance.

How to use complex number in the alternating current circuit?

Homework Statement Find the principal value of i^2i Homework Equations principal value = Ln r + iArg(z) The Attempt at a Solution dont know how to start..

hi PF 1. (2+2i) First Quadrant 2. (-2+2i) Second Quadrant 3. (-2-2i) Third Quadrant 4. (2-2i) Fourth Quadrant consider (2+1i) then Tan^-1(2/2) which will be 45 degree if we consider (-2+2i) then it will be -45 degree but angle will not -45 degree actually we get answer by adding or...

how to find number (3-i)^{12} ? i know theorem for powers of complex numbers, but i have to know argument of trygonometrical form. and its cos x = 3/4 and sin x = -1/4 and i don't know what to do with that.

Homework Statement I have a motor with a full load output rating of 2.8 kW with an effciency off 70% and at 80% of full load runs with a Power Factor of 0.8 lagging. A second motor has afull load output of 4.2kW with an effciency of 85% and at 80% of full load runs with a PF of 0.75...

4z + z(bar) = 5 + 9i then z = [solve for this] I don't know how to re arrange this equation (5+9i)/4 = z+z(bar)

Homework Statement \frac{(cos60 - isin60)^5 * (cos45 - isin45)^3}{(cos15-isin15)^7} Homework Equations The Attempt at a Solution I have had several tries so far, but simply do not know what to do. Would somebody be so kind and simplify this expression step by step. I...

Homework Statement Sketch 0 \le arg z \le \frac{\pi}{4} (z \not= 0)The Attempt at a Solution I know from my book that his is a punctured disk aka deleted neighborhood only because it says so and because it is in the form of 0 < \mid z - z_0 \mid < \epsilon. I honestly have no clue...

Homework Statement Find all the roots in rectangular coordinates, exhibit them as vertices of certain squares, and point out which is the principal root. The Attempt at a Solution The problem is (-8 -8\sqrt{3}i)^{\frac{1}{4}} and I found the four roots easily to be \pm(\sqrt{3} -...

Homework Statement I'm trying to derive the formula for the multiplicative inverse of a complex number. I say that: z=a+bi z-1=u+vi and zz-1 must be 1 so zz-1=(a+bi)(u+vi) =(au-bv)+(av+bu)i =1 So in order for zz-1=1, (au-bv) must be 1 and (av+bu) must be 0. My teacher has solved this as...

Homework Statement On the first day of Electromagnetism class, the professor gave this problem to us to solve. I never learn about taking derivative of complex number. Can someone give me some hints? his problem was: Given P= 0.5 Re(I*V) I= V/(A+B) A= R+jX , B=Y+ jZ V is...

can some 1 help me tho solve this a complex number, z satisfied the equation |z|=2 draw the locus W which satisfied the equation w=(z+2)/(z-1) thx

When finding the current in a 4+j3 balanced load star connected circuit. with 400V line voltage at 50Hz. Do you find the current by dividing the phase voltage approx. 230V by 4+j3 ohms? and if so. how do you divide normal numbers with complex numbers? your help is appreciated.

z=x+yi determine the values of x and y such that z=\sqrt{3+4i} I'm not even sure where to start with this one, so any help would be greatly appreciated

Homework Statement Find limit as z->infinity of exp(z) where z is complex Homework Equations See above The Attempt at a Solution The solution should be that the limit does not exist, but I don't know why. Any explanations?

Homework Statement Hello! I'm lost on how to start this, I've got formulas given to me from the text, but I have no idea on how to piece everything together. So I need to use established properties of moduli to show that when \left.\left|z_{3}\right|\neq\left|z_{4}\right|, then...

hey guys I've got a problem If z = 3− i and w = 1+3i, evaluate Im (z∗w)/(2z − 3w) my attempt... ill do top first (3+i)(1+3i)=3+9i+i+3i^2=10i? bottom=3-2i-3+9i=7i therefore answer equals 10/7 however answer is 3/13. could someone please help. thanks