Complex numbers Definition and 730 Threads

I'm working my way through Shaum's Outline on linear algebra and in it they define a complex number as an ordered pair of real numbers (a,b). So given a real number a, its complex counterpart would be (a,0). Operations of addition and multiplication of real numbers work under the...

I have read from my algebra book that the product of two complex numbers is still a complex number: (a+bi)(c+di)= (ac-db)+(bc +ad)i I was thinking that since complex numbers can be used to represent vectors, the product of two vectors should still be a vector. But I have also read from my...

Homework Statement Let z1 = a (cos (pi/4) + i sin (pi/4) ) and z2 = b (cos (pi/3) + i sin (pi/3)) Express (z1/z2)^3 in the form z = x + yi. ]2. Homework Equations [/b] The Attempt at a Solution a(cos (pi/4) + i sin (pi/4)) b (cos (pi/3) + i sin (pi/3)) I then multiplied...

Homework Statement Given that arg((z1z2)/2i) = \pi, find a value of k. Homework Equations arg z2=2\pi=0 arg z1=\pi/6 The Attempt at a Solution ((\pi/6)^k x (2\pi))/ (2i) = \pi I'm not sure what to do with the imaginary number i..

Homework Statement What does the following subring of the complex numbers look like: {a(x)/b(x) | b(x) ϵ C[x], b(x) is not a member of (x)} ? Homework Equations The Attempt at a Solution

Kind of stuck (embarrassingly) on determining what poles of the function: 1/(z^6 + 1) lie above the y-axis (I'm solving a contour integral using the residues theorem). What's the easiest way to do this? Normally I'd write z as e^(theta + 2kpi)/6 , where theta is the angle that -1 forms...

Homework Statement Calculate sin(z) = i Homework Equations Sin, cos, sinh, cosh exponential formulas? The Attempt at a Solution I tried expanding the sin out with exponentials, I think I could take e^iz and substitute that for z, so I'd have: (z+ z^-1)/2i = i Maybe I could...

Homework Statement sin-1{z-1/i} where z is non real, can be angle of a triangle if 1)Re(z) = 1, Im(z) = 2 2)Re(z) = –1, 0<Im(z)≤1 3)Re(z) + Im(z) = 0 4)none of these The Attempt at a Solution -1≤z-1/i≤1, but inequalities don't hold for complex numbers. How to solve this one?

Homework Statement If n is an integer which leaves remainder one when divided by three, then (1+√3i)n + (1-√3i)n equals a) -2n+1 b) 2n+1 c) -(-2)n d) -2nThe Attempt at a Solution Multiply and divide each of the two expression inside bracket by 2. 2n(cosπ/3 + isinπ/3)n + 2n(cosπ/3-isinπ/3)n On...

Homework Statement [PLAIN]http://img689.imageshack.us/img689/6294/08052011193408.jpg Homework Equations The Attempt at a Solution Hi, I would really appreciate some help on part a, I'm simply getting no where. This is classed as a geometry question, but I haven't covered...

I was playing around with complex numbers in Matlab this evening and noticed this interesting pattern: Given: a = (e^{x})^{i \pi/2} When x is incremented an integer power (0,1,2,3), the a is rotated {\pi/2} radians in the complex plane. It started out at 0 radians with x = 0 and then...

Homework Statement So, i have this equation, and it is asked of me to find the number of complex numbers that satisfy the equation. (z=x+iy)Homework Equations z-\overline{z}+|z-i|=4-2iThe Attempt at a Solution I tried replacing the numbers and i got something like this...

Hello guys! I have a question related to complex numbers. How would i calculate, for example (\frac{\sqrt{3}+i}{2})^{2010} without using the De Moivre's theorem?

Attempt solution attached Manual says the answers is -1 + i

If c=\sqrt{a^{2}+b^{2}} would i be correct in saying c=|a+ib| ?

I can't find a proof for the multiplicative inverse of complex numbers... can anybody please tell me the proof (i already know what the formula is)

Homework Statement I am told to try and solve <x - ty, x - ty> where t = <x,y>/<y,y> However, I am stuck at that equation and unable to manipulate it to get rid of the * Homework Equations The Attempt at a Solution <x - ty, x - ty> = <x,x> - <x,ty> - <ty,x> + <ty,ty>...

Hey there, I have a question to answer and I'm unsure what exactly it is asking for, could anyone shine some light as to what this means? Is it two RC circuits in parallel?? "A circuit consisting of a 280 Ohm resistor in series with a 0.3 uF capacitor is connected to a supply operating at a...

Complex Numbers, I just don't understand what they are supposed to represent. I understand how they are used, to some degree in solving equations etc. but the definitions leave me clueless. The idea that they "extend the idea of the one-dimensional number line to the two-dimensional complex...

I am not sure whether there is any difference between differentiating complex and real numbers... I am just trying to differentiate: e^(2+3i)x = (2+3i)e^(2+3i)x Is this correct? I have a feeling its not this simple.

Hi fellows, Homework Statement Prove that: \sqrt{\frac{7}{2}}\leq|z+1|+|1-z+z²|\leq\sqrt{\frac{7}{6}} for all complex numbers with |z|=1. Homework Equations The Attempt at a Solution I've tried something like this: Starting by the following property...

hello, can anyone help me with this question! http://img855.imageshack.us/i/img0401m.jpg/

Homework Statement express z^7 + 1 as a product of four non-trivial factors and given that z is a complex number Homework Equations The Attempt at a Solution

Homework Statement Solve the equation in the complex numbers set (this is as best as i could translate since English is not my native language :D) \left|z\right|-z=1+2i Homework Equations |z|=sqrt{x^2+y^2} z=x+iy The Attempt at a Solution Well i started by supposing y=1 and...

Complex numbers - "parallel lines meet at infinity"? What does it mean? We started learning about complex numbers last week. One of the first things my teacher said was that "We learned that parallel lines never meet. But as it turns out, they meet at infinity." I'm willing to accept it...

Hi All, I have been trying to solve a complex number equation for character impedance with MATLAB but it continues to tell me that "indexing must appear last on index addressing". I am new to MATLAB so I think that my code is wrong. What I am using is complex sqrt (R+jwL)(G+jwC)...

q. find all the solution of equation z^3= j Attempt okie now we know we have its argument as pie/2 but my friend did this and he placed the argument of it as THETA + 2*pie*k/ n i want to ask first why the argument change ? second , i thought in demoivers theorem we...

Hello. I'm currently following a course in Complex Analysis, but I'm often afraid of manipulating certain expressions. It is well known that certain "intuitively obvious" actions which are true for real numbers are not true for complex numbers, a simple one being \sqrt{-1}\sqrt{-1} \neq \sqrt...

Homework Statement express in terms of n and r: \sum { ^{n} C _{r} } \times cos(rx) from r=0 up to n Homework Equations well i know de moivre's theorem also, in class we have been finding ways to represents a series of cosines added up as the sum of a geometric series so i tried it on this...

Hi guys. What is really the need for complex numbers? Is there any physical meaning associated with it?

show there is a some constant c, independent of n, s.t. if {Z_j} are complex numbers and sum of |Z_j| from 1 to n >= 1, then there's a subcollection {Z_j_k} of {Z_j} s.t. sum of |Z_j_k| >= c. Any hint on how to start this?

Homework Statement I'm given that the sum from k = 0 to N-1 of e^(j*2*pi*k/N) + 0. Then there's some code. tt = 0:1:1000; xx = 0*tt; for kk=5:11 xx = xx + 99*cos(0.006*pi*tt + 0.25*pi*kk); end plot(tt,xx), title(’SECTION of a SINUSOID’), xlabel(’TIME (sec)’) The plot made from the vector xx...

Homework Statement Hi, I have to solve this exercise: "Given that (a+bi)^2 = 3+4i obrain a pair of simultaneous equations involving a and b. Hende find the two square roots of 3+4i. Hence find the two suqre roots of 3+4i." I don't really know to do do it. 2. The attempt at a solution...

Homework Statement Simplify the expression e^(i6theta)[ (1+e^(-i10theta))/(1+e^i2theta)] Answer should be in terms of cosines but i don't know how to start this problem? :S Also, does e^(-iwt) = - coswt -jsinwt? K so I am thinking about Eulers formula, and I get an expression with Sines...

Im am very curios as to what role complex numbers might have in nature?

Dear all, I'm fighting against complicated expressions which contain complex numbers that, after some calculations and simplifications, appear in a form like: 1. + 0.I. I think that this weighs down the next calculations. The question is: how can I put zero the imaginary part? or, in general...

In Non-Standard analysis, the "real" numbers are extended by adding infinitesimal elements and their reciprocals, infinite elements. These numbers are referred to as hyperreals and are logically sound and analytically rigorous. When one considers the "Standard Part" function st(x), one can...

hi, I am looking for any example (complete calculus) using Complex Numbers to solve real problems. is the right approach: 1- starting with real variables 2- adding the imagine part 3- calculation... 4- extracting only the real value from the result. ? thx.

Well it's all in the title: i don't understand why we define vectors and complex numbers differently, with different properties (eg vector dot product and cross product and complex multiplication). After all, all a complex number is is a 2-uplet of real numbers, but that's exactly the same as a...

Homework Statement the problem that i attached bellow is related to how you can obtain a transfer function from its squared magnituded. my question is not on the problem it self as its just a solved example from my book. what i find difficult to understand as you can see from my...

Homework Statement Problem 1. Create a program to display a complex number (or a list of complex numbers) as an arrow (or arrows) on the complex plane. i know RandomComplex[] will give me a random complex number, and i know RandomComplex[{1 - I, 1 + I}, 5, WorkingPrecision ->...

Homework Statement Find the five values of (1+i√3)^(3/5) This question was from my recent end of year exam, I hadn't come across a question like it in my revision, does it mean find the five roots of (1+i√3)^(3/5) ?:confused:

Hi all, I'm trying to solve this question , can anyone help?? Suppose that D is an open connected set , fn ->f uniformly on compact subsets of D. If f is nonconstant and z in D , then there exists N and a sequence zn-> z such that fn ( zn ) = f(z) for all n > N. hint: assume that...

Hi all, I'm trying to solve 4.15 from the attached file, can anyone help? I tried to use residue thm , i.e the integral of f over the curve gamma-r equals winding number of z0 over gamma-r and residue of z0 of f. I can't see how b-a relates to the winding number of z0. Can anyone help please?

Homework Statement The complex numbers, z and w satisfy the inequalities |z-3-2i|<=2 and |w-7-5i|<=1 Find the least possible value of |z-w| Homework Equations No clue at all. The Attempt at a Solution Since its |z-w| i figured that the least possible value will only be...

Homework Statement Really new to complex numbers so please forgive my ignorance. Prove that if the complex number z is an nth root of the real number x then the complex conjugate z (z with horizontal line on top) is also an nth root of x.Homework Equations i2=-1The Attempt at a Solution nsqr(z)...

Homework Statement If cosx - isin2x and sinx - icos2x are conjugates of each other, then what is the value of x? The Attempt at a Solution Since the given complex numbers are conjugates of each other, their modulus must be same. i.e. cos2x + sin22x = sin2x + cos22x cos2x = cos4x On solving...

This is supposed to be a proof of trigonometric multiplication of complex numbers: What happened at the =...= point? I understand everything up to that.

I got this out of An Imaginary Tale: The Story of Sqrt(-1). In section 1.5 of the book, the author explains that Bombelli took x3 = 15x + 4 and found the real solutions: 4, -2±sqrt(3). But if you plug the equation into the Cardan forumla you get imaginaries...

i have 1 question.. the question is: Given any 2 distinct real numbers a and b, exactly one a<b or b<a must be true. The real numbers are said to be ordered. Show that there is no ordering of the complex numbers. my problems is not understand that orders~~anybody help me?