Complex numbers Definition and 724 Threads

Hi out there peps, very nice forum! (my first topic) Atm I am dealing with complex numbers, and I've got kinda problem solving this task. Hope for some help. Anyway, it sounds like this. - Name all the roots for the equation e^((pi*z)^2)=i, for which modulus is less than 1. Its...

Homework Statement (-e)^iπ answer is -e^-π2 not sure how to describe this one, but i need to find the roots. Homework Equations (r^n)e^(itheta)n = (r^n)cos(thetan) + isin(thetan) n is an element of the reals The Attempt at a Solution i'm not sure what to do with this, it...

Hello, My prof showed a way of dealing with difficult circuits using complex numbers. But I have no idea what he was on about, and can't find that method in any book. Does someone know what I'm talking about, and can someone point me in the way of some materials for this? Thanks!

Homework Statement Describe the set of all z \in \mathbb{C} such that the series \sum_{n=1}^{\infty} (1-z^2)^n converges Homework Equations Basic analytic techniques. The Attempt at a Solution This is from a graduate complex analysis class, and I just have a feeling my answer is too...

Homework Statement What is the locus given by z(\overline{z}+2)=3 where the overbar means conjugate. Homework Equations The Attempt at a Solution After using z=x + yi and expanding the backets, one gets the equation: x^2+2x+y^2+2iy=3 or (x+1)^2+y^2 +2iy=4 which is a circle crossed with the...

I know that a complex number can be written in form of a+bi and r(cos(theta) + isin(theta)) but I don't understand the the representation of it as r*e^(i * theta) also

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|. How do we know that? is |z+w|=|z-w|?? Note that z and w are complex numbers.

OK, in my book we have an inequality ||z|-|w||\leq|z+w|\leq|z|+|w| then from here it simply states, "Replacing w by -w here shows that ||z|-|w||\leq|z-w|\leq|z|+|w|. How do we know that? is |z+w|=|z-w|?? Note that z and w are complex numbers.

given z, w\inC, and |z|=([conjugate of z]z)1/2 , prove ||z|-|w|| \leq |z-w| \leq |z|+|w| I squared all three terms and ended up with : -2|z||w| \leq |-2zw| \leq 2|z||w| I know this leaves the right 2 equal to each other but i figured if i show that since there exists a z\geqw\geq0, then...

How can M_{2}(\mathbb{C}) be written as a combination of elements of \mathbb{C} and elements of \mathbb{H}?

When do you become introduced to complex numbers? For example, raising functions to the power i or inputting i into trig functions etc... I know they are used widely in physics, but when are you supposed to learn about them? None of the courses at my school up to and including diff Eq mention...

Homework Statement I need help on a little review please. z=2-i What is arg(iz) Homework Equations Well iz= 1+2i The Attempt at a Solution I think this should end up being arg(2i/1) But this doesn't seem to make sense because I am wanting to find an angle here right? I am...

Homework Statement Know: modulus(z) < 3 WTS: |Im(z2 - zbar + 6)| <12 where zbar is the complex conjugate Homework Equations z = x + iy The Attempt at a Solution |Im(z2 - zbar + 6)| = |Im(x2 + 2i*x*y - y2 - x + iy + 6)| = |2xy + y| So I want to show |2xy + y|< 12 I already proved it...

Homework Statement I was trying to do this problem and noticed that I should be able to express SQRT(27) in terms of i and cancle out the i making it more simple I can't seem to remember how to do this thanks [IMG=http://img294.imageshack.us/img294/5396/captureow.jpg][/PLAIN] Uploaded...

Homework Statement Solve the following equation: z^4+z^3+z^2+z+1 = 0 z is a complex number. 2. The attempt at a solution I was trying to factorize it to 1st degree polynomial multiplied by 3rd degree polynomial: (z+a)(z^3+bz^2+cz+1/a) = 0 I discovered that I need to solve 3rd...

I've been working through the book The Story of i(sqrt of -1). It's kinda like a story with a lot of Math. The first 2 chapters deal with cubics and geometry for solving cubics functions. I understand the algebra behind it but I'm getting lost with the big picture. I need a supplemental book or...

What is the best way of introducing complex numbers to engineers who are weak at mathematics? They normally want something tangible or relevant examples.

Let's say that I have two complex numbers, a and b, with different arguments. From a few "experiments" with a computer, I think that there always exists a positive integer n such that -pi/2 <= Arg(a^n) <= pi/2 and pi/2 <= Arg(b^n) <= 3pi/2. In other words, if Arg(a) = thetaA and Arg(b) =...

I'm revising complex numbers and having trouble with this question... Question: Verify that 2 of the roots of the equation: z^3 +2z^2 + z + 2 = 0 are i and -2. Find any remaining roots Attempt at solution: i^3 +2 i^2 + i + 2 = (-1)i + 2(-1) +i + 2 = -i -2 + i +2 =0...

Homework Statement I found when z = 1 the Taylor series expansion for z1/3 by taking 11/3 = 1. What if I was taking 11/3 = omega = e2i pi/3 Homework Equations The Attempt at a Solution

Homework Statement Show that \sqrt{\frac{1} {2} (a + \sqrt {a^2+b^2})} + i \sqrt{\frac{1} {2} (-a + \sqrt {a^2+b^2})}= a+ib Homework Equations The Attempt at a Solution Distributed the i and then the 1/2's in each term which gave: \sqrt{\frac{a} {2} + \frac{ \sqrt...

Homework Statement i) Solve the equation z^3 = \mathbf{i}. (ii) Hence find the possible values for the argument of a complex number w which is such that w^3 = \mathbf{i}(w*)^3. I'm stuck on part ii. Homework Equations The Attempt at a Solution The answer to the equation in...

Hi, I am having some difficulty understanding what is going on with the TI-84 complex number calculation in switching between rectangular and polar coordinates, hopefully someone can clarify this for me. For example, take the term 6<30 (6 angle 30). When I calculate this by hand going from...

I really, really need to know the formula that adds (or subtracts) two complex numbers in polar form, and NOT in rectangular form. I know there is such formula (I saw it in some book), and it's composed of cosines and sines. Please, please don't tell me to convert back to rectangular form...

How to prove the set of complex numbers is uncountable? Let C be the set of all complex numbers, So C={a+bi: a,b belongs to N; i=sqrt(-1)} -------------------------------------------------- set of all real numbers is uncountable open intervals are uncountable...

Homework Statement Let Z and W be complex numbers. If /Z/ and /W/ are rational and /W-Z/ is rational, then /(1/Z)-(1/W)/ is rational. Homework Equations The Attempt at a Solution How do I represent Z and W as rational complex numbers?

I was considering complex numbers (independently) and I came across an interesting question. Are any of these statements true? 1<i 1>i i<-i -1<i -i<i<-1

Homework Statement Show that the solutions of the equation 2sin(z) + cos(z) = isin(z) are given by z = (n\pi-\frac{\pi}{8}) - \frac{1}{4}iln2Homework Equations e^{iz} = cos(z) + isin(z) sinz = \frac{1}{2i}(e^{z}-e^{-z}) z_{1}^{z_{2}} = e^{z_{2}lnz_{1}} lnz = lnr...

Homework Statement Given P(z)=4z^3-2z+1 where z=cost+isint, find the maximum and minimum modulus on the argand diagram for the graph as t moves from 0 to 2\pi. I want to check if my solution is valid, and if there is an easier approach to it because I do somewhat answer the question, but I skip...

Homework Statement One root of the equation x^2 + ax + b = 0 is 4 + 5i. Write down the second root. Homework Equations N/a? The Attempt at a Solution My problem is it's a "write down" question which suggests no working required. This is probably so simple but I just don't...

Homework Statement (2 CIS (pi/6))*(3 CIS (pi/12)) Homework Equations Also what is CIS? I believe it's Cos+i*sin but how do you use it? The Attempt at a Solution i simplified it to 6 CIS (pi/12) How do i turn it into cartesian?

Homework Statement Draw an argand diagram to represent the follwing property: real(z) < abs(z) < real(z)+img(z) Homework Equations z = x+iy; real(z) = x abs(z) = sqrt(x^2 + y^2) img(z) = y The Attempt at a Solution substituting original expression with x, y, and sqrt(x^2 + y^2)...

Hi this isn't homework, just a practice problem I already have the answer too for my waves class: z=sin(wt)+cos(wt) Express this in the from Z=Re[Aej(wt+\alpha)] I know how to express sine in the form of cosine, and cosine in the from of a complex exponential, but I don't know how to do...

When given a complex number z=x+iy and transforming this into its mod-arg form giving rcis\theta where r=\sqrt{(x^2+y^2)} and \theta=arctan(y/x), we are assuming that -\pi/2<\theta<\pi/2. What if however a student is asked to convert the complex number -1-i into mod-arg form? If they just start...

I have a ton of homework with square roots of complex numbers. Like sqrt(2 + 3i) What is the fastest way to break these down into its approximates like 1.67 + 0.895i without using a TI89/Maple/Matlab/Mathmatica.

This result came up in my diff eq class the other day: If i = x^2 then x = [(sqrt(2)/2) + (sqrt(2)/2)i]^2 While there aren't a lot of use for complex numbers in this class, I still feel stupid for not knowing it. Another trick that I'd like to learn about is the "complexifying the...

Homework Statement A man travels 12 kilometres northeast, 20 kilometres 30° west of north and finally 18 kilometres 60° south of west. Determine his position with respect to his starting point. Homework Equations Using complex numbers z = a + ib |z|(cosx°+sinx°) The Attempt...

I can't solve it myself, please help. http://i754.photobucket.com/albums/xx187/asdhkbbq2/MA1.jpg Thanks all for who help.

Homework Statement There are three complex numbers a, b and c. Show that these propositions are equals. 1. ABC (triangle from the three points in complex plane) is equilateral (T1). 2. j or j2 is the solution for az2 + bz + c = 0. 3. a2 + b2 + c2 = ab + bc + ca Homework Equations...

Homework Statement Harder: given that √(−15 − 8i) = ±(1 − 4i) obtain the two solutions of the equation z² + (−3 + 2i)z + 5 − i = 0Homework Equations I can easily prove √(−15 − 8i) = ±(1 − 4i) but that's not important The Attempt at a Solution I would of thought that a compex solution would...

Homework Statement I have came up with an example to illustrate my question. There is a rod, which can turn around p1. p1p2 = (-1+j) m p1p3 = (-3 + 3j) m p1p4 = (1 - j ) m F1 = (1+3j) N F3 = (-1 - 2j ) N F4 = unknown, orthogonal to the rod compute F2_n, orthogonal component of F2 to the...

Homework Statement I was working on a problem dealing with complex numbers. I had to add two phasors together to get the combined phasor. I converted both numbers to rectangular form, added them and converted the result back to polar form. My magnitude was correct, but my phase was not...

Can anyone prove to me why each subfield of the field of complex numbers contains every rational numers?

Homework Statement f(z)=\sqrt{(z.^3+8)} How many branches (solutions) and branch points does the funtion f(z) have? Homework Equations The first part of the question was working out the roots of z^3+8=0 which I found to be -2, 1+i\sqrt{3} and 1-i\sqrt{3} The Attempt at a Solution...

Homework Statement Find real numbers p and q such that the following equation is true: \frac{p}{q+5i}=4e^{\frac{-i\pi}{4}} Homework Equations Euler's formula The Attempt at a Solution Ok so I converted the right side to rectangular form using Euler's formula and solved for p...

I can't come up with the code to solve for a and b in terms of x and y. x + I y = Sqrt[a + I b] In[84]:= Clear["Global`*"] Solve[x + I y == Sqrt[a + I b], {a, b}] During evaluation of In[84]:= Solve::svars: Equations may not give solutions for all "solve" variables. >> Out[85]=...

When there are, say, two complex numbers that are equal. What can we say about their equality? Can we say that the real part of one is equal to the real part of the other? Similarly, can we say that the complex part of one is equal to the complex part of the other? Is this what it means when...

We have already shown 1+ w+ w^2 =0 If w is the complex number exp(2*Pi*i/3) , find the power series for; exp(z) +exp(w*z) + exp (z*w^2) We have already shown 1+ w+ w^2 =0

So I am trying to compute all possible Jordan forms over the complex numbers given a minimal polynomial. My question is this: If the roots of the minimal polynomial are both real, should I proceed as if all of the possible forms are over real numbers?

[RESOLVED] Quick complex numbers question in QM (probability amplitues) Im a little confused here. I am reading in my textbook about probability amplitudes in Stern Gerlach measurements, and it says this: We find the resulting probabilities for deflection of...