Complex numbers Definition and 724 Threads

since ##|z|≤3## →##z=0+0i##, therefore we shall have centre##(0,0)## and radius ##3##, find my sketch below,

Anyone know what the simplest possible self-contained numeral system for complex numbers would be, analogous to signed ternary for integers? My guess would be quarter-imaginary base (https://en.wikipedia.org/wiki/Quater-imaginary_base.)

I have this problem in my book: Show that ##\mathbb{C}## can be obtained as 2 × 2 matrices with coefficients in ##\mathbb{R}## using an arbitrary 2 × 2 matrix ##J## with a characteristic polynomial that does not contain real zeros. In the picture below is the given solution for this: I...

Using the boundary conditions where psi is 0, I found that k = n*pi/a, since sin(x) is zero when k*a = 0. I set up my normalization integral as follows: A^2 * integral from 0 to a of (((exp(ikx) - exp(-ikx))*(exp(-ikx) - exp(ikx)) dx) = 1 After simplifying, and accounting for the fact that...

Solve ##Z^2\bar{Z}=8i## i am confused on how to proceed i have tried to substitute ##z=a+ib## solve the conjugate and the square, then separate the real from the imaginary and put all in a system, but becomes too complicated

Summary:: Hello, my question asks if the complex exponential equation 4e^(-j) is equal to 4 ∠-180°. I tried to use polar/rectangular conversions: a+bj=c∠θ with c=(√a^2 +b^2) and θ=tan^(-1)[b/a] 4e^(-j)=4 ∠-180° c=4, 4=(√a^2 +b^2) solving for a : a=(√16-b^2) θ=tan^(-1)[b/a]= -1 b/(√16-b^2)=...

i have to find such domain z=x+iy , y,x∈ℝ , |y-x|<=2, |x|<=2 i'm confused with |y-x|<=2, how should i proceed ? with abs of x i am ok.

Let $z_1=18+83i,\,z_2=18+39i$ and $z_3=78+99i$, where $i=\sqrt{-1}$. Let $z$ be the unique complex number with the properties that $\dfrac{z_3-z_1}{z_2-z_1}\cdot \dfrac{z-z_2}{z-z_3}$ is a real number and the imaginary part of $z$ is the greatest possible. Find the real part of $z$.

Hello everyone, I've been struggling quite a bit with this problem, since I'm not sure how to approach it correctly. The inequality form reminds me of the equation of a circle (x^2 + y^2 = r^2), but I have no idea how to be sure about it. Would it help just to simplify the inequality in terms...

Given the domain of the integral for the Fourier transform is over the real numbers, how does the Fourier transform transform functions whose independent variable is complex? For example, given \begin{equation} \begin{split} \hat{f}(k_{\mathbb{C}}) &= \int_{\mathbb{R}} f(z_{\mathbb{C}})...

Hello! :smile: I am locked in an exercise. I must find (and graph) the complex numbers that verify the equation: ##z^2=\bar z^2 ## If ##z=x+iy## then: ##(x+iy)^2=(x-iy)^2 ## and operating and simplifying, ##4.x.yi=0 ## and here I don't know how to continue... can you help me with ideas? thanks!

Is it possible two find complex numbers ##C_n##, so that both equations are satisfied \sum^{\infty}_{n=1}nC_n=0 and \sum^{\infty}_{n=1}|C_n|^2=1 ?

reducing it to various forms: for example, the one in the title, or 2*pi*k(ln m) = a(ln(n/m)), and so forth. My gut feeling is that it is true (that no such foursome exists), but manipulations have not got me anywhere. Anyone push me in the right direction? I am probably overlooking something...

I know this topic was raised many times at numerous forums and I read some of these discussions. However, I did not manage to find an answer for the following principal question. I gather one deals with the same set in both cases equipped it with two different structures (it is obvious if one...

Solution to the problem tells us that ##S_5 + i S_6## is the sum of the terms of a geometric sequence and thus the solutions should be : $$S_5 = \frac{\sin( (n+1) x)}{\cos^n(x) \sin(x)},\,\,\,\, S_6 = \frac{\cos^{n+1}(x) - \cos((n+1)x)}{\cos^n(x) \sin(x)} , x \notin \frac{\pi}{2} \mathbb{Z}$$...

All i was able to think was that i have to find a point (x,y) such that sum of its distances from points (0,0),(1,0),(0,1) and (3,4) is minimum.I tried by assuming the point to be centre of circle passing through any of the above 3 points,But it didn't helped me.

Summary: Trouble with infinity and complex numbers, just curious. I'm not too familiar with set theory ... but <-∞, ∞> contains just real numbers? Does something similar to <-∞, ∞> exist in Complex numbers? My question, is it "wrong"?

Summary: Which properties of ##\mathbb{C}## are actually necessary? The following is speculative as well as a honestly meant question about the way QM is modeled. I don't want to create a new theory, just understand the necessities of the old one. Physicists use complex numbers for QM. But...

--Continued-- 7) Let ##\sum_{k=0}^9 x^k = 0## Find smallest positive argument. Same thing as previous question, but I guess I can expand to ##z+z_{2}+z_{3}+...+z_{9}=0## ##z=re^{iθ}## ##re^{iθ}+re^{2iθ}+re^{3iθ}+...## What do I do to proceed on? Cheers

Hello all! Thanks for helping me out so far :) Really appreciate it. I don't seem to understand some of the questions presented to me, so if anyone has an idea on how to start the questions, please do render your assistance :) 4) Take ##3+7i## is a solution of ##3x^2+Ax+B=0## Since ##3+7i## is...

Hello, here with some complex number questions which I need some assistance in checking :) 1) z=3+5i1+3iz=3+5i1+3i Find Re(z) and Im(z) My answer is 9595 and −25−25 respectively. Checked by Wolfram 2) Find principal argument of the complex numberz=−5+3iz=−5+3i and express it in radians up to 2...

I'm kind of stuck over here at one part, but something is telling me that it might be wrong too :( Do assist, thanks.

[Note from mentor: This was split off from another thread, which you can go to by clicking the arrow in the quote below] Actually they are not. See https://www.amazon.com/dp/3319658662/?tag=pfamazon01-20 Sec. 5.1.

Hi, I'm working on an assignment for circuit theory, and I'm wondering if someone could let me know if I'm heading in the right direction? 1) I have a voltage value of 120 /_0 (polar form), from this can I assume that Arctan (a/b) =0, so voltage =120 in phase? Therefore, V =120+J0, where V...

Hello I thought is would be fun to try a problem in which I had a complex number elevated to a complex power. To do this, I first tried to manipulate the general equation ## z^{w} ## (where ##z ## and ##w## are complex numbers) to look a bit more approachable. My work is as follows: ##z^{w}##...

Homework Statement Write ##5-3i## in the polar form ##re^\left(i\theta\right)##. Homework Equations $$ |z|=\sqrt {a^2+b^2} $$ The Attempt at a Solution First I've found the absolute value of ##z##: $$ |z|=\sqrt {5^2+3^2}=\sqrt {34} $$. Next, I've found $$ \sin(\theta) = \frac {-3} {\sqrt...

If we have solution of an equation as x=1, it may be expressing, depending on context, 1 apple, 1 excess certain thing, etc. And, if we have solution of an equation as x=-1, it may be expressing, depending on context, 1 deficient apple, 1 deficient certain thing, etc. Is there any experience...

$\textsf{ If $z$ and $u$ are complex numbers, show that}$ $$\displaystyle\bar{z}u=\bar{z}\bar{u} \textit{ and } \displaystyle \left(\frac{z}{u} \right)=\frac{\bar{z}}{\bar{u}}$$ok couldn't find good example on what this is and I'm not good at 2 page proof systemsso much help is mahalo

1. The complex number are not ordered. Which else number are not ordered? 2. Are the infinitesimally numbers are ordered numbers? It there a difference between infinitesimally number to another infinitesimally number?

Homework Statement Find ##z## in ##z^{1+i}=4##. Is my solution correct Homework Equations ##\log(z_1 z_2)=\log(z_1)+\log(z_2)## such that ##z_1, z_2\in \{z\in\Bbb{C} : (z=x+iy) \land (x\in\Bbb{R}) \land -\infty \lt y \lt +\infty\}## ##re^{i\theta}=r(\cos\theta + i\sin\theta)## The Attempt at a...

Homework Statement If ##\lim_{n \rightarrow \infty} x_n = L## then ##\lim_{n\rightarrow\infty}cx_n = cL## where ##x_n## is a sequence in ##\mathbb{C}## and ##L, c \epsilon \mathbb{C}##. Homework Equations ##\lim_{n\rightarrow\infty} cx_n = cL## iff for all ##\varepsilon > 0##, there exists...

Homework Statement $$\frac{1}{z}+\frac{1}{2-z}=1$$ Homework Equations Quadratic-formula and algebra The Attempt at a Solution Been struggling with this one.. I keep getting the wrong answer, but that isn't the worst part, I can live with a wrong answer as long as the math behind it is...

Homework Statement So the problem I have is this silly little equation.. $$\frac {z - 7}{z + 3} = i $$Homework Equations This is the thing, I don't think you need anything more advanced than basic algebra to solve this problem. The Attempt at a Solution And I've tried solving it doing the...

1. Homework Statement . Figure 1 shows a 50 Ω load being fed from two voltage sources via their associated reactances. Determine the current i flowing in the load by: (a) Thevenin's theorem (b) Superposition (c) Transforming the two voltage sources and their associated reactances into current...

Hi particular solution only. As an example of what I am talking about, this method works for this DE: $$ 4y' + 2y = 10\cos(x) \\ \\ 10 \cos(x) = \Re( 10 e^{j(x)} ) = \Re(e^{j(x)} \cdot e^{j(0)} ) \rightarrow \text{complex number that captures the amplitude and phase of 10 cos x is} \\ 10...

Homework Statement Refer given image. Homework Equations Expansion of determinant. w^2+w+1=0 where w is cube root of 1. The Attempt at a Solution Expanding the determinant I got cw^2+bw+a-c=0. Well after that I have no idea how to proceed.

Hi. If you have seen the above image which shows a parabola then you can also see that there is a colored portion of the parabola that have solution in "another dimension" - the "another dimension" can give me new numbers to form a solution of a function like f(x) = x2 + 1. 1. Is this "another...

DSP Guide .com has the highly rated textbook for digital signal processing. Chapter 30 pg 561 on Complex Numbers http://www.dspguide.com/ch30.htm (chapters are free to download) Hes talking about representing sinusoids with a complex number. Author states "Multiplying complex numbers A and...

I have a problem to put the complex number in mode (1000/3, ∠36.87), apparently the division simbol gives some syntax error

I just need to know. Why exactly what's the complex number i=√–1 put in the wave function for matter. Couldn't it have just been exp(kx–wt)?

Homework Statement if ## x + iy## = ## \frac a {b+ cos ∅ + i sin ∅} ## then show that ##(b^2-1)(x^2+y^2)+a^2 = 2abx##Homework EquationsThe Attempt at a Solution i let ## ... ##x + iy = ## \ a(b+cos ∅ - i sin ∅)##/ ##(b + cos ∅)^2 + sin^2∅ ##...got stuck here... alternatively i let ## b +...

Quick question. While going over complex numbers in my book, I think I came across a typo and I wanted to be sure I had the right information. In the paragraph going over dividing complex numbers, my book has: ##|\frac{z_1}{z_2}|=|\frac{z_1}{z_2}|## That's obviously true. Should that be...

Say you have an un-damped harmonic oscillator (keep it simple) with a sine or cosine for the forcing function. We can exploit Euler's equation and solve for both possibilities (sine or cosine) at the same time. Then, once done, if the forcing function was cosine, we choose the real part as the...

Homework Statement Use NVA to solve for ##i##. Enter your answer in polar form with the angle in degrees. Homework EquationsThe Attempt at a Solution My nodes are as follows: ##V_1## on the left middle junction, ##V_2## is the junction in the very center, and ##V_3## is the junction on the...

I am searching for a shortcut in the calculation of a proof. The question is as follows: 2.12 Prove that: $$|z_1|+|z_2| = |\frac{z_1+z_2}{2}-u|+|\frac{z_1+z_2}{2}+u|$$ where $z_1,z_2$ are two complex numbers and $u=\sqrt{z_1z_2}$. I thought of showing that the squares of both sides of the...

My residue is wrong. What is the solutions and the steps to achieve it ?

Hi I was hoping some of you would give me a clue on how to solve this complex number task. z = (1 +(√3 /2) + i/2)^24 → x=(1 +(√3 /2), y= 1/2 I think there must be some nice looking way to solve it. My way was to calculate |z| which was equal to [√(3+2√3)]/2 → cosθ = x/|z|, sinθ= y/|z| After...

Hello everyone. Iam reading about complex numbers at the moment ad Iam quite confused. I know how to use them but Iam not getting a real understanding of what they actually are :-( What exactly is the imaginary part of a complex number? I read that it could in example be phase... Thanks in...

Homework Statement Homework EquationsThe Attempt at a Solution I attempted to use the formula zj = xj + iyj to substitute both z's. Further simplification gave me (x1 + x2)cosθ + (y2 - y1)sinθ or, Re(z2 + z1)cosθ + Im(z2 - z1)sinθ. Is this a valid answer? Or are there any other identities...

Hello all, Three consecutive elements of a geometric series are: m-3i, 8+i, n+17i where n and m are real numbers. I need to find n and m. I have tried using the conjugate in order to find (8+i)/(m-3i) and (n+17i)/(8+i), and was hopeful that at the end I will be able to compare the real and...