Complex numbers Definition and 724 Threads

Hello all, I am trying to find the algebraic representation of the following numbers: \[rcis(90^{\circ}+\theta )\] and \[rcis(90^{\circ}-\theta )\] The answers in the book are: \[-y+ix\] and \[y+ix\] respectively. I don't get it... In the first case, if I take 90 degrees (working with...

Hello all, I wish to plot and following complex numbers on a plane, and to find out which shape will be created. I find it hard to figure out the first one, I believe that the others will follow more easily (the forth is also tricky). \[z_{1}=\frac{2}{i-1}\] \[z_{2}=-\bar{z_{1}}\]...

Hiya all, I need your assistance with the following problem: A) Show that the equation \[z^{2}+i\bar{z}=(-2)\] has only two imaginary solutions. B) If Z1 and Z2 are the solutions, draw a rectangle which has the following vertices: Z1+3 , Z2+3 , Z1+i , Z2+i I do not know how to even...

Hello everyone, I have a complex number problem that i would greatly appreciate some help with. Thanks in advance to anyone offering their time to make a contribution. Q) Write the following in polar form: I have attempted the question (please see my working below) and have been advised that i...

It's not a homework question. I just thought up a method of finding answers to problems where a number is raised to a complex number and I need to know if I am right. If we have to find e^(i), can we do it by; first squaring it to get, e^(-1) which is 1/e and then taking its square root to get...

Homework Statement exp(z)=-4+3i, find z in x+iy form Homework Equations See attached image. The Attempt at a Solution See attached image. exp(z)=exp(x+iy)=exp(x)*exp(iy)=exp(x)*[cos(y)+isin(y)] ... y=inv(tan(-3/4)=-.6432 ... mag(-4+3i)=5, x= ln (5)..exp(ln(5))=5 ...

Hi everyone. I was looking at complex numbers, eulers formula and the unit circle in the complex plane. Unfortunately I can't figure out what the unit circle is used for. As far as I have understood: All complex numbers with an absolut value of 1 are lying on the circle. But what about...

Homework Statement I've used z* to mean z conjugate. Given the equation z + 2iz* = 8 + 7i, express z in the form a + ib. From SQA Advanced Higher Mathematics 2005 Exam Paper Homework Equations n/a The Attempt at a Solution I substituted a+ib and its conjugate in for z and z*, which, after...

Homework Statement This is a CIE A'level maths P3 question out of an exam from 2013 in October/November. As there is no markscheme ( I at least can't find one), I would be grateful if someone could look at my solution to the problem and correct me if I made a mistake. The problem is 8.(b)...

Can someone please explain what's going on at 47:40 Thanks in advance.

Homework Statement well this is not exactly a homework, i had an argument whith my teacher about my grade in a test, because i put a complex number in the form of R,theta and she claims that the form was costheta+isentheta, and i know that but i need to prove in a book that...

Is the fact that QM uses complex numbers should be considered as a math artefact (as it is the case when complex numbers are used for alternate current circuit analysis), or, alternatively, it has some deep and important relation to the nature (or at least to the nature of the quantum theory)...

Can we order Complex Numbers ? I searched a bit most places says it can but not like the real numbers. I am confused a bit.And I am not sure abouth the truth of those sources. Thanks

Homework Statement http://prntscr.com/eqhh2p http://prntscr.com/eqhhcg Homework EquationsThe Attempt at a Solution I don't even know what these are, it is not outlined in my textbook. I'm assuming I am is image? But how do you calculate image even? As far as I'm concerned I am has to do wtih...

Could you give me a hint how to attack this problem? Find a complex number z = a+i*b such that f(t)=Re e^(z*t) where f(t)=cos(2*pi*t) I have begun as follows: e^((a+i*b)*t)=e^(a*t)*(cos(b)+i*sin(b)) Re e^(z*t)= e^(a*t)*cos(b) What to do now?

There a simple math example that I am confused ##(\sqrt {-4})^2## Theres two ways to think 1-##\sqrt {-4}=2i## so ##(2i)^2=4i^2## which its ##-4## 2-##\sqrt {-4}##.##\sqrt {-4}##=##\sqrt {-4.-4}=\sqrt{16} =4## I think second one is wrong but I couldn't prove how, but I think its cause ##\sqrt...

Homework Statement If $$C = 1+cos\theta+...+cos(n-1)\theta,$$ $$S = sin\theta+...+sin(n-1)\theta,$$prove that $$C=\frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}} cos\frac{(n-1)\theta}{2} \enspace and \enspace S = \frac{sin\frac{n\theta}{2}}{sin\frac{\theta}{2}}sin\frac{(n-1)\theta}{2}$$...

Homework Statement If Z1+Z2+Z3=0 and Z1*Z2 + Z2*Z3 + Z3*Z1=0 and Z1, Z2, Z3 are all complex, what is the value of (|z1|+|z2|+|z3|)/(|z1*z2|+|z2*z3|+|z3*z1|) Homework EquationsThe Attempt at a Solution I tried to multiply the equations by the product of all conjugates and reach some...

Homework Statement So we have been doing complex numbers for about 2 weeks and there is this one equation I just can't solve. It's about showing the set of solutions in graphical form (on "coordinate" system with the imaginary and the real axis). So here is the equation: Homework Equations...

I have to find a primitive function below using the Euler formulas for ##\sin x## and ## \cos x## The problem $$ \int e^{2x} \sin 3x \ dx $$ Relevant equations ## \cos x = \frac{e^{ix}+e^{-ix}}{2} \\ \sin x = \frac{e^{ix}-e^{-ix}}{2i} \\ \\ \int e^{ix} \ dx = \frac{e^{ix}}{i} ## The attempt...

Homework Statement I am having trouble solving systems of equations when they contain complex numbers. The context is circuit theory and phasors. For example, I am given this And the goal is to find I2 and Voc, which you can see the answers for. I just don't know how to manipulate the numbers...

I was taught a scalar is a quantity that consists of a number (positive or negative) and it might include a measuring unit, e.g. 6, 5 kg, -900 J, etc. I was wondering if complex numbers like 3 + 7j (where j is the square root of minus 1) can be considered scalar quantities too, or is it that...

Homework Statement Sketch the loci, find centre point and the radius of the circle. args((z-3i)/((z+4))=π/6[/B] Homework Equations args(x/y)=args(x)-args(y) Circle theorem - inclined angle theoremThe Attempt at a Solution I sketched the circle with major arc. Radius= using Pythagorus I got...

Homework Statement Prove that each subfield of the field of complex numbers contains every rational number. ' From Hoffman and Kunze's Linear Algebra Chapter 1 Section 2 Homework EquationsThe Attempt at a Solution Suppose there was a subfield of the complex numbers that did not contain every...

I have been trying to show that $$\lim_{U\rightarrow\infty}\int_C \frac{ze^{ikz}}{z^2+a^2}dz = 0 $$ Where $$R>2a$$ and $$k>0$$ And C is the curve, defined by $$C = {x+iU | -R\le x\le R}$$ I have tried by using the fact that $$|\int_C \frac{ze^{ikz}}{z^2+a^2}dz| \le\int_C...

I'm having trouble figuring out to get the answers from the 2 equations. The phasors and complex numbers confuse me. Do I need to change the phasor form? How do I go about doing this thanks! (Not homework question I am trying to figure this for my exam!)

Homework Statement Express the complex number (−3 +4i)3 in the form a + bi Homework Equations z = r(cos(θ) + isin(θ)) The Attempt at a Solution z = -3 + 4i z3 = r3(cos(3θ) + isin(3θ)) r = sqrt ((-3)2 + 42) = 5 θ = arcsin(4/5) = 0.9273 ∴ z3 = 53(cos(3⋅0.9273) + isin(3⋅0.9273)) a = -117 b...

I've just had my first batch of lectures on complex numbers (a very new idea to me). Algebraic operations and the idea behind conjugates are straightforward enough, as these seem to boil down to vectors. My problem is sketching. I have trouble defining the real and imaginary parts, and I don't...

Homework Statement Consider 3 nonzero complex numbers $$z_1,z_2,z_3$$ each satisfying $$z^2=i \bar{z}$$. We are supposed to find $$z_1+z_2+z_3, z_1z_2z_3, z_1z_2+z_2z_3+z_3z_1$$. The answers- 0, purely imaginary , purely real respectively. Homework EquationsThe Attempt at a Solution I have...

Homework Statement ask to find all the values in z to the equation to be true[/B]Homework Equations [/B]The Attempt at a Solution this is my atemp of solution i don't know what else do, because the problem ask for z values well i must add that i am thinking about x=0 and y= pi/2 will solve...

Homework Statement Write this complex number in the form "a+bi" a) cos(-pi/3) + i*sin(-pi/3) b) 2√2(cos(-5pi/6)+i*sin(-5pi/6)) Homework Equations my only problem is that I am getting + instead of - on the cosinus side.(real number) The Attempt at a Solution a) pi/3 in the unit circle is 1/2...

Homework Statement Homework Equations Theta = arctan (y/x) The Attempt at a Solution Hopefully this is the right section to post in, but I am a bit confused with complex numbers. I am working on the problems above and I just wanted to make sure I am doing each part correctly. I think A...

Hi all, I have spent a couple of hours on this perplexing question. Solve the simultaneous equations: z = w + 3i + 2 and z2 - iw + 5 - 2i = 0 giving z and w in the form (x + yi) where x and y are real. I tried various methods, all to no avail. I have substituted z into z2 , I got the wrong...

The problem The following equation ##z^4-2z^3+12z^2-14z+35=0## has a root with the real component = 1. What are the other solutions? The attempt This means that solutions are ##z = 1 \pm bi##and the factors are ##(z-(1-bi))(z-(1+bi)) ## and thus ## (z-(1-bi))(z-(1+bi)) =...

If I have 2 complex numbers, A and B, what is the correct way to evaluate this expression: ## E = AB - B^*A^*## I was under the impression that when taking the product of complex numbers, you always conjugate one factor, but in this instance, it is quite important which one is conjugated, no...

We should note that we can write any complex number as $\displaystyle \begin{align*} z = r\,\mathrm{e}^{\mathrm{i}\,\theta} \end{align*}$ where $\displaystyle \begin{align*} r = \left| z \right| \end{align*}$ and $\displaystyle \begin{align*} \theta = \textrm{arg}\,\left( z \right) + 2\,\pi\,n ...

First let's write this number in its polar form. $\displaystyle \begin{align*} \left| z \right| &= \sqrt{\left( -2 \right) ^2 + 2^2} \\ &= \sqrt{4 + 4} \\ &= \sqrt{8} \\ &= 2\,\sqrt{2} \end{align*}$ and as the number is in Quadrant 2 $\displaystyle \begin{align*} \textrm{arg}\,\left( z...

I'm reading "Time Series Analysis and Its Applications with R examples", 3rd edition, by Shumway and Stoffer, and I don't really understand a proof. This is not for homework, just my own edification. It goes like this: Σt=1n cos2(2πtj/n) = ¼ ∑t=1n (e2πitj/n - e2πitj/n)2 = ¼∑t=1ne4πtj/n + 1 + 1...

Homework Statement [/B] Suppose q(z) = z^3 − z^2 + rz + s, is a complex polynomial with 1 + i and i as zeros. Find r and s and the third complex zero. The Attempt at a Solution [/B] (z-(1+i)(z-i) = Z^2-z-1-2iz+i (Z^2-z-1-2iz+i)(z+d) = Z^3+z^2(d-1-zi)-z(d+1+2di-i)-d(1-i) Z^2 term...

Homework Statement Showing all necessary working solve the equation ##iz^2+2z-3i=0## giving your answer in the form ##x+iy## where x and y are real and exact,Homework EquationsThe Attempt at a Solution ##iz^2+2z-3i=0, z^2+(2/i)z-3=0##,using quadratic formula →##(-2/i± √8)/2 , z= √2+1/i## and...

Homework Statement How would I go about solving 1/z=(-4+4i) The answer that I keep on getting is wrong The Attempt at a Solution [/B] What I did z=1/(-4+4i)x(-4-4i)/(-4-4i) z=(-4-4i)/(16+16i-16i-16i^2) z=(-4-4i)/32 z=-1/8-i/8 This is the answer that I got however it says that it is...

1. Give a formula for the values on m such that z^m=z z=cos(7pi/6)+i*sin(7pi/6) 2. If i use de movires i get 3. m*7pi/6=7pi/6 + k*2pi But then i get the value that k=12/7, Which is the wrong formula. The correct answer is 1+12k for k=0,1,2...

Homework Statement in a given activity: solve for z in C the equation: z^3=1 Homework Equations prove that the roots are 1, i, and i^2 The Attempt at a Solution using z^3-1=0 <=> Z^3-1^3 == a^3-b^3=(a-b)(a^2+2ab+b^2) it's clear the solution are 1 and i^2=-1 but i didn't find "i" as a solution...

Homework Statement Let ##z_1,z_2,z_3## be three complex numbers that lie on the unit circle in the complex plane, and ##z_1+z_2+z_3=0##. Show that the numbers are located at the vertices of an equilateral triangle. Homework EquationsThe Attempt at a Solution As far as I understand, I need to...

Homework Statement Reflection of the line ##\bar{a}z + a\bar{z} = 0## in the real axis is Homework EquationsThe Attempt at a Solution I know that a line in the complex plane is represented as ##\bar{a}z + a\bar{z} + b= 0## and that its slope ##μ = \dfrac{-a}{\bar{a}}##. I'm not sure how to do...

Or basically anything that isn't a positive integer. So I can prove quite easily by induction that for any integer n>0, De Moivre's Theorem (below) holds. If ##\DeclareMathOperator\cis{cis} z = r\cis\theta, z^n= r^n\cis(n\theta)## My proof below: However I struggle to do this with...

This may be a simple thing but due to some reason I am not able to understand. I am not able to understand an example from Brown-Churchill book. I have provided the screenshot in the attached picture. Request help.

Homework Statement I am not able to understand an example from Brown-Churchill book. I have provided the screenshot in the attached picture. Request help. Homework Equations No The Attempt at a Solution No

Homework Statement a) The complex number ## 1-i ## is denoted by ##u##. On an argand diagram, sketch the loci representing the complex numbers ## z## satisfying the equations ## |z-u|= |z| and |z-i|=2 ## b) Find the argument of the complex numbers represented by the points of intersection of...

Hello, I am enrolled in calculus 2. Just having started a section in our textbook about integration by partial fractions, I eagerly began trying to use this integration technique wherever I could. After messing around for multiple days, I ran into this problem: ∫ 1/(x^2+1)dx I immediately...