Complex numbers Definition and 724 Threads

Homework Statement Homework EquationsThe Attempt at a Solution I tried to attempt the question but I am not sure how to start it, at least for part (i). My biggest question, I think, is how does the multiplication of a random complex number to a Fourier-Transformed signal (V(f)) have an...

Homework Statement Describe the set of points determined by the given condition in the complex plane: |z - 1 + i| = 1 Homework Equations |z| = sqrt(x2 + y2) z = x + iy The Attempt at a Solution Tried to put absolute values on every thing by the Triangle inequality |z| - |1| + |i| = |1|...

I came across this strange relationship when deriving the degree-4 equation for a torus. First thing that comes to mind is the 'Freshman's Dream'. Apparently, it was pure coincidence that they are equal. But, I don't believe in coincidences when it comes to a math expression. There is something...

How relevant is complex analysis to physics? I really want to take differential equations but I would have to change my schedule around way more than I want to. So, would anyone advise a physics major to to take complex analysis? Should I just change my schedule around so I can take differential...

hello can you tell me please why we introduced complex numbers? what was the problem that we couldn't express with rest of algebra and we introduced complex numbers? I am basically interested in why we introduced complex number to describe and analyze AC circuits, like voltage, current and...

Hi All, I'm working out a program to emulate a quantum computer (definitely in a nascent stage), and I'm struggling with a piece of the math. I looked at the math sections in these forums, but thought this might be more appropriate to post it. I'll try to conceptually outline the problem, and...

Suppose we have two equation x1=Aeiωt + Be-iωt and x2=A*e-iωt + B*eiωt . Where A and B are complex number and A* B* are their conjugate correspondingly. Now if we want to make x1 and x2 exactly equivalent all the time, one way to do it is to have A=B* and B=A* so that x1 and x2 are equivalent...

Homework Statement If arg(\frac{z-ω}{z-ω^2}) = 0, \ then\ prove \ that\ Re(z) = -1/2 Homework Equations ω and ω^2 are non-real cube roots of unity. The Attempt at a Solution arg(z-ω) = arg(z-ω^2) So, z-ω = k(z-w^2) Beyond that, I'm not sure how to proceed. Using the rotation formula may also...

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: Superposition Theorem Homework Equations [/B]The Attempt at a Solution :[/B] see attached files as I can not write in itex and...

Is there in a nutshell an explanation or even a single reason why complex numbers have so many fascinating consequences and give rise to so much deep stuff like analytic functions (with all its stunning properties), Riemann surfaces, analytic continuations, modular forms, zeta function, its...

Within the context of real numbers, the square root function is well-defined; that is, the function ##f## defined by: ##f(x) = \sqrt{x}## Refers to the principal root of any real number x. Is it true that this is not the case when dealing with complex numbers? Does ##\sqrt{z}##, where ##z ∈ ℂ##...

Mod note: This thread was moved from a technical math section, so doesn't include the homework template. I know this has been asked before, but none of the other posts have helped me. I cannot for the life of me figure out how to solve a system of equations with complex numbers. Here is a very...

Homework Statement Find the Geometric image of; 1. ## | z - 2 | - | z + 2| < 2; ## 2. ## 0 < Re(iz) < 1 ## Homework EquationsThe Attempt at a Solution In both cases i really am struggling to begin these questions, complex numbers are not my best field. There are problems before this one...

Homework Statement let z' = (a,b), find z in C such that z^2 = z' Homework EquationsThe Attempt at a Solution let z = (x,y) then z^2 = (x^2-y^2, 2xy) since z^2 = z', we have, (x^2-y^2, 2xy) = (a,b) comparing real and imaginary components we have; x^2-y^2 = a, 2xy = b. Now, this...

Homework Statement The Attempt at a Solution -1 < (z-w) /(1-z*w) < 1 [/B] Hi can give clue. I am clueless

Homework Statement 1) 2w+iz = 3; 2) (3-i)w - z = 1 +3i 2wi - z = 3i; 3w - iw - z = 1 + 3i Substract (2) from (1): 2wi - z - (3w-iw-z) = 3i - (1+3i) 2wi -3w +iw = -1 3iw - 3w = -1 3w(i-1) = -1 3w = -1/(i-1) = -0.5i - 0.5 w = -i/6 - 1/6 But the answer is i/6 +...

Homework Statement goal: solve for t; all else are constants $$cos(\omega t)=1-e^{-(\frac{t}{RC})}$$Homework Equations noneThe Attempt at a Solution i turned the cos to complex notation & rearranged $$e^{i\omega t}+e^{-(\frac{t}{RC})}=1$$ $$ln(e^{i\omega t}+e^{-(\frac{t}{RC})})=0$$ and i...

Homework Statement Write down number 1+i and 1+i\sqrt{3} in trigonometry form.[/B]Homework Equations For complex number z=x+iy \rho=|z|=\sqrt{x^2+y^2} \varphi=arctg\frac{y}{x} And [/B]The Attempt at a Solution Ok. For z=1+i \rho=\sqrt{1+1}=\sqrt{2}...

Homework Statement Solve the following complex equation for z: zi = sqrt(3) - i Homework EquationsThe Attempt at a Solution Do I have to equate the real and imaginary parts ?, this is what I tried zi = (x+iy)i = exp(i*log(x+iy))

Homework Statement This problem is very easy, but I'm not sure how best to "prove" it. This part of the question just states: Prove that (1/z)* = 1/(z*) where z* is the complex conjugate of z. Homework Equations The Attempt at a Solution So the complex conjugate of z = x + iy is defined is...

This is just a follow on from this thread, https://www.physicsforums.com/threads/complex-numbers-and-vector-multiplication.509944/ Basically I've noted that, in 2d at least that the complex multiplication of A and B is equal to (A dot conj(B)) + (A cross B)i Would that then mean his initial...

Hi all, $$f(x) = 3x^2+2x+10$$ I recognized that this a quadratic and used the quadratic formula. I came up with $$-1/3+-\sqrt{29}/3$$. But the answer has a $$i$$ for imaginary. When I was under the \sqrt{116}, I broke that down, but didn't realize there would be an $$i$$ Can someone explain...

Homework Statement I'd like to separate this function to U(x) + i*V(y) form. It's a homework problem that is asking if it is an analytic function. Searching thru trig substitutions, but looking ahead I don't see much luck... Any suggestions or help is greatly appreciated. Homework...

Homework Statement "Put each of the following into the form Acos(ωt+θ)..." (a.) 4ejt+4e-jt Homework Equations Euler's Identity: ejθ = cos(θ)+jsin(θ) Phasor Analysis(?): Mcos(ωt+θ) ←→ Mejθ j = ej π/2 Trignometric Identities The Attempt at a Solution I attempted to use phasor analysis to...

Do we use imaginary numbers just in the intermediary steps of a predictive theory? For example, in QM, in order to make predictions in the real world, you square the wave function. The wave function might have have all the information, but in order to predict something you must operate on it to...

Hello! I am very unsure of how to solve this question. The question states z^2=a+bi, where a and b belong to real numbers. Find all possible solutions for z. I think that the solution includes the De Moivre's formula, however I am very confused by how to do this or what the formula means...

Hi! Can you tell me how the theta changes into theta minus pi/2? Can you show me, please?

Evaluate the following expressions, expressing answers in rectangular form. 1. $\cos(1+j)$ 2.$\sinh(4-j3)$ can you help me on how to solve these problems. thanks in advance!

We know that i^3 is -i . But I am getting confused, because I thought that i can be written as √(-1) and i^3 = √(-1) × √(-1) × √(-1) = √(-1 × -1 × -1) = √( (-1)^2 × -1) = √(1× -1) = √(-1) = i ( and not -i ). Please help.:rolleyes: Sorry I couldn't use superscript because I was using my phone.

Please forgive me as I may have to edit this post to get the equations to show properly. I am doing some work with AC circuits and part of one of my phasor equations has this in it: \frac {2i} {1+cos(θ) + i sin(θ)} - i , where i is the imaginary number \sqrt{-1}. However, knowing the...

is there a way to solve this without performing the tedious expansion of $(1+j)^8$? here's the problem $\text{Im}[(1+j)^8(x+jy)]$

I just wanted to check something. If I have a complex number of the form a = C * \exp(i \phi) where C is some non-complex scalar constant. Then the phase of this complex number is simply \phi. Is that correct?

The problem H(e^{j0.2\pi}) = {\frac{1 - 1.25e^{-j0.2\pi}}{1 - 0.8e^{-j0.2\pi}}} Solves to H(e^{j0.2\pi}) = 1.25e^{j0.210\pi} Attempts I'm really not sure how to get that answer, but I've tried a number of different approaches Multiplying by complex conjugate Multiplying by...

Dear All, Hi! I am about to begin a Diploma in Aeronautical Engineering and would like to know if anyone could help me understand if in my future career of being an Aeronautical Engineer I would at any time be required to use Complex numbers to solve problems. If yes can you suggest examples...

Hello, please I need help. I have no idea how to start this. Can someone guide me? This is not homework, I'm just studying on my own and I really don't know how to begin this.

Homework Statement 8i = ( 2x + i ) (2y + i ) + 1 The final answers is [x =0, 4] [y=4, 0] Homework Equations The Attempt at a Solution The final answer in the book is stated as above but if I follow the solution I will get the real parts which would...

This problem is from Boas Mathermatical Methods 3ed. Section 16, problem 1. Show that if the line through the origin and the point z is rotated 90° about the origin, it becomes the line through the origin and the point iz. Use this idea in the following problem: Let z = ae^iωt be the...

Are the less than (<) and greater than(>) relations applicable among complex numbers? By complex numbers I don't mean their modulus, I mean just the raw complex numbers.

If a is a complex number, and a^2-a+1=0, then a^2011=? I tried using De Moivre's theorem, Taking a=cosθ+isinθ, but didn't get anywhere, got stuck at cos2θ+isin2θ-cosθ-isinθ+1=0. What do I do?

I'm not sure whether this should go in this forum or another. feel free to move it if needed Homework Statement Suppose that z_0 \in \mathbb{C}. A polynomial P(z) is said to be dvisible by z-z_0 if there is another polynomial Q(z) such that P(z)=(z-z_0)Q(z). Show that for...

Hello, Homework Statement The complex numbers z_{1} = \frac{a}{1 + i} and z_{2} = \frac{b}{1+2i} where a and b are real, are such that z_{1} + z_{2} = 1. Find a and b. Homework Equations The Attempt at a Solution This looked like a time for partial fractions to me, so I went...

Problem: Let $\dfrac{1}{a_1-2i},\dfrac{1}{a_2-2i},\dfrac{1}{a_3-2i},\dfrac{1}{a_4-2i},\dfrac{1}{a_5-2i}, \dfrac{1}{a_6-2i},\dfrac{1}{a_7-2i},\dfrac{1}{a_8-2i}$ be the vertices of regular octagon. Find the area of octagon (where $a_j \in R$ for $j=1,2,3,4,5,6,7,8$ and $i=\sqrt{-1}$). Attempt...

Homework Statement Suppose that u and v are real numbers for which u + iv has modulus 3. Express the imaginary part of (u + iv)^−3 in terms of a polynomial in v.Homework Equations The Attempt at a Solution |u+iv|=3 then sort(u^2+i^2) = 3 then u = 3 and v=0 or u=0 and v=3(0+3i)^-3 i swear i am...

Homework Statement problem in a pic attached Homework Equations The Attempt at a Solution i solved i and ii a , when it came to b , i just said that every one of the 3 roots will be squared having 2 roots 1 + and 1 - but then i read the marking schemes ( also attached) , and i got...

I want to solve y''+y'+y=(sin(x))^2 and try to use y=Ae^{ix} but then when I square it I get A^2 e^{2ix} I found y' and y'' and solved for A and it didn't work I guess I could use the formula for reducing powers but I would like to try and get around that.

Homework Statement Let f(z) = z3-8 and g(z) = f(z-1). This information applies to questions 1-5. 1. Express g(z) in the form g(z) = z3+az2 +bz + c 2. Hence, solve g(z) = 0. Plot solutions on an Argand diagram. Homework Equations Factorisation i2=-1 The Attempt at a Solution I have done...

Homework Statement Hello Assume that we have n complex numbers u: u_1,u_2,...,u_n, and n complex numbers v:v_1,v_2,...v_n I would like to prove that: |\Sigma_{i=1}^nRe(u_i\bar{v_i})| \le |\Sigma_{i=1}^nu_i\bar{v_i}| I guess this can be written simpler: |\Sigma_{i=1}^nRe(z_i)| \le...

I just finished reading the "Reality Bits" in a recent copy of NewScientist. It discusses attempts to purge mathematics of the need for complex numbers. Started me thinking(danger, danger) of not how to get rid of the square root of negative one, but, more easily, simply find out where it enters...

Homework Statement Solve the equations: 3(z-2) = 2j(2z+1) and (i-2)z-z*=3i+1 where z* is the complex conjugate of z. (I am assuming z and z* are the unknowns. i and j are basically the same since they're defined as i2 = j2 = -1?) Homework Equations Rules for solving...

Hi, We know that if we have two complex numbers in polar format (i.e., magnitude and exponential), that for two complex vectors z1 = A*exp(iB) z2 = C*exp(iD) If z1 and z2 are equal, then A = C and B = D. However, this is assuming these values are all real. What if they are complex...