Complex number Definition and 436 Threads

Hi, I had a question I was working on a while back, and whilst I got the correct answer for it, I was told that there was a second solution to it that I missed. Here is the question. ] I worked my answer out to be sqrt(2)(cos(75)+i(sin(75))), however, it appears there is a second solution...

Homework Statement ##z## is a complex number such that ##z = \frac{a+bi}{a-bi}##, where ##a## and ##b## are real numbers. Prove that ##\frac{z^2+1}{2z} = \frac{a^2-b^2}{a^2+b^2}##. Homework EquationsThe Attempt at a Solution I calculated \begin{equation*} \begin{split} z = \frac{a+bi}{a-bi}...

Homework Statement For the expression $$r = \frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha L)-i\delta\sinh(\alpha L)} \tag{1}$$ Where ##\alpha=\sqrt{\kappa^{2}-\delta^{2}}##, I want to show that: $$\left|r\right|^{2} = \left|\frac{i\kappa\sinh(\alpha L)}{\alpha\cosh(\alpha...

Show geometrically that if |z|=1 then, $Im[z/(z+1)^2]=0$ I am unsure how to begin this problem. I have sketched out |z|=1 but can't work out how to sketch the Imaginary part of the question.

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...

So say a wave is described by Acos(Φ), completely real. Then the to use Euler's Eq, we we say the wave is AeiΦ, which is expanded to Acos(Φ) + iAsin(Φ). We tell ourselves that we just ignore the imaginary part and only keep the real part. And if intensity is |AeiΦ|2, which is (Acos(Φ) +...

if a is a complex number then sqrt(a^2)=? Is it is similar to Real Number? Help me please

For this, f and g are real functions, and k is a real constant. I have ##\psi = fe^{ikx}+ge^{ikx}## and I want to find ##\left|\psi \right|^2##. I went about this two different ways, and got two different answers, meaning I must be doing something wrong. Method 1: ##\psi =(f+g)e^{ikx}##...

Hi all, I was trying the understand theory behind Fourier and Laplace Transform (especially in the context of control theory) by reading the book "Complex Variables and the Laplace Transform for Engineers" written by "Wilbur R. LePage". In section 6-10 of the book the author touches on the...

$\tiny{s10.03.25}$ $\textsf{Write complex number in rectangular form}$ \begin{align*}\displaystyle z&=4\left[\cos\frac{7\pi}{4} + i\sin \frac{7\pi}{4} \right]\\ \end{align*} $\textit{ok from the unit circle: $\displaystyle\cos{\left(\frac{7\pi}{4}\right)}=\frac{\sqrt{2}}{2}$}\\$ $\textit{and...

How is it possible to ignore the addition sign and imaginary number without contradicting fundamental Mathematics? I find it difficult to understand.

Hello all, Given a complex number: \[z=r(cos\theta +isin\theta )\] I wish to find the polar representation of: \[-z,-z\bar{}\] I know that the answer should be: \[rcis(180+\theta )\] and \[rcis(180-\theta )\] but I don't know how to get there. I suspect a trigonometric identity, but I...

Homework Statement Given that a complex number z and its conjugate z¯ satisfy the equation z¯z¯ + zi = -i +1. Find the values of z. Homework EquationsThe Attempt at a Solution

Why does every subfield of Complex number have a copy of rational numbers ? Here's my proof, Let ##F## is a subield of ##\Bbb C##. I can assume that ##0, 1 \in F##. Lets say a number ##p \in F##, then ##1/p \ p \ne 0## and ##-p## must be in ##F##. Now since ##F## is subfield of ##\Bbb C##...

Homework Statement We are given Z, and are asked to find the magnitude of the expression. See attached picture(s) Homework Equations See attached pictures(s) The Attempt at a Solution When I solved it on the exam, I did it the long way using De Moivre's theorem. I ended up making a few sign...

I'm working on a school project and my goal is to recognize objects. I started with taking pictures, applying various filters and doing boundary tracing. Fourier descriptors are to high for me, so I started approximating polygons from my List of Points. Now I have to match those polygons, which...

Homework Statement $$z^2 + z|z| + |z|^2=0$$ The locus of ##z## represents- a) Circle b) Ellipse c) Pair of Straight Lines d) None of these Homework Equations ##z\bar{z} = |z|^2## The Attempt at a Solution Let ##z = r(cosx + isinx)## Using this in the given equation ##r^2(cos2x + isin2x) +...

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...

So the question is show that $$S=\left\{ \begin{pmatrix} a & b\\ -b & a \end{pmatrix} :a,b \in \Bbb{R} ,\text{ not both zero}\right\}$$ is isomorphic to $\Bbb{C}^*$, which is a non-zero complex number considered as a group under multiplication So I've shown that it is a group homomorphism by...

As we know that √-5×√-5=5 i.e multiplication with it self My question is that according to this √-1×√-1=1.but it does not hold good in case of i(complex number). I.e i^2 =-1. Why?

Homework Statement Homework Equations r=sqrt(a^2+b^2) θ=arg(z) tan(θ)=b/a The Attempt at a Solution for a)[/B] finding the polar form: r=sqrt(-3^2+(-4)^2)=sqrt(7) θ=arg(z) tan(θ)=-4/-3 = 53.13 ° 300-53.13=306.87° -3-j4=sqrt(7)*(cos(306.87+j306.87) I don't know if my answer is correct...

This is a question from a competitive entrance exam ...I just want to check whether my approach is correct as i don't have the answer keys . here is the question : How many complex numbers z are there such that |z+ 1| = |z+i| and |z| = 5? (A) 0 (B) 1 (C) 2 (D) 3 My approach : let z = x+iy...

Homework Statement Find the modulus and argument of z=((1+2i)^2 * (4-3i)^3) / ((3+4i)^4 * (2-i)^3 Homework Equations mod(z)=sqrt(a^2+b^2) The Attempt at a Solution In order to find the modulus, I have to use the formula below. But I'm struggling with finding out how to put the equation in...

Homework Statement \frac{z-1}{z+1}=i I found the cartesian form, z = i, but how do I turn it into polar form?The Attempt at a Solution |z|=\sqrt{0^2+1^2}=1 \theta=arctan\frac{b}{a}=arctan\frac{1}{0} Is the solution then that is not possible to convert it to polar form?

Homework Statement Write the given complex number in polar form first using an argument where theta is not equal to Arg(z) z=-7i The Attempt at a Solution 7isin(\frac{-\pi}{2}+2\pi n) The weird part about this problem it asks me to not use the argument, The argument is the smallest angle...

I have learned that if I multiply a vector, say 3i + 4j, by a scalar that is a real number, say 2, the effect of the operation is to expand the size of the magnitude of the original vector, by 2 in this case, and the result would be 6i + 8j. What would be the effect on a vector, like 3i + 4j...

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...

I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...

Hello. Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z. ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π <...

so i am starting with the equation x3 = √(3) - i first : change to a vector magnitude = √[ (√(3))2 + 12] = 2 and angle = tan-1( 1/√(3) ) = 30 degrees (in fourth quadrant) so i have a vector of 2 ∠ - 30 so i plot the vector on the graph and consider that : 1. the fundamental theorum of...

I felt upon a mistake I made but cannot understand. I consider the following rotation transformation inspired from special relativity : $$\left(\begin{array}{c} x'\\ict'\end{array}\right)=\left (\begin {array} {cc} cos(\theta) & -sin(\theta) \\ sin(\theta) & cos(\theta) \end...

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 am under the impression that the following cannot be stated, a < b, if the a term is a complex number and the b term is either a natural number or a complex number, or any other type of number for that matter. Firstly am I correct? Secondly, if I am, does there exist a theorem of some sort...

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...

Complex numbers If z=rcis(theta) FIND: |iz^2| I am confused about how I incorporate the i into the absolute value. I can't remember what it means. Please help and show exactly how I complete the workings. I can easily find the absolute value of z^2 I just really don't understand how to put the...

Homework Statement If modulus of z=x+ iy(a complex number) is 1 I.e |z|=1 then find the argument of z/(1+z)^2 Homework Equations argument of z = tan inverse (y/x) where z=x+iy modulus of z is |z|=root(x^2+y^2) The Attempt at a Solution z/(1+2z+z^2) = x+iy / 1+2(x+iy)+( x+iy)2 ...

Question 1: (a) Show that the complex number i is a root of the equation x^4 - 5x^3 + 7x^2 - 5x + 6 = 0 (b) Find the other roots of this equation Work: Well, I thought about factoring the equation into (x^2 + ...) (x^2+...) but I couldn't do it. Is there a method for that? Anyways the reason I...

$\displaystyle \begin{align*} z^3 + 1 &= 0 \\ z^3 &= -1 \\ z^3 &= \mathrm{e}^{ \left( 2\,n + 1 \right) \,\pi\,\mathrm{i} } \textrm{ where } n \in \mathbf{Z} \\ z &= \left[ \mathrm{e}^{\left( 2\,n + 1 \right) \, \pi \,\mathrm{i}} \right] ^{\frac{1}{3}} \\ &= \mathrm{e}^{ \frac{\left( 2\,n + 1...

Homework Statement p(x) = x^3 − x^2 + ax + b is a real polynomial with 1 + i as a zero, find a and b and find all of the real zeros of p(x).The Attempt at a Solution [/B] 1-i is also a zero as it is the conjugate of 1+i so (x-(1+i))(x-(1-i))=x^2-2x+2 let X^3-x^2+ax+b=x^2-2x+2(ax+d)...

Homework Statement Verify that i2=-1 using (a+bi)(c+di) = (ac-bd)(ad+bc)i Homework Equations (a+bi)(c+di) = (ac-bd)(ad+bc)i The Attempt at a Solution I tried choosing coefficients so that it would be (i)(i) = (0 - 1)+(0+0)i = -1 so then I get i^2 = -1 But I was told that this was wrong and...

Given A(2√3,1) in R^2 , rotate OA by 30° in clockwise direction and stretch the resulting vector by a factor of 6 to OB. Determine the coordinates of B in surd form using complex number technique. i try to rewrite in Euler's form and I found the modulus was √13 but the argument could not be...

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...

Prove that $\arg[(a+bi)(c+di)]=\arg(a+bi)+\arg(c+di)$.

Homework Statement Finding "polar" and "rectangular" representation of a complex number? Make a table with three columns. Each row will contain three representations of a complex number z: the “rectangular” expression z = a + bi (with a and b real); the “polar” expression |z|, Arg(z); and a...

I find this interesting. You can approximate pi/4 with the Gregory and Leibniz series pi /4 = 1/1 - 1/3 + 1/5 - 1/7 + 1/9 ... (1) btw it takes a lot of terms to get a reasonable approximation for pi. The formuli is pi / 4 = [ ( -1 ) ^ ( k + 1 ) ] / ( 2 * k -1)...

Homework Statement a) Solve equation z + 2i z(with a line above it i.e. complex conjugate) = -9 +2i I want it in the form x + iy and I am solving for z. b) The equation |z-9+9i| = |z-6+3i| describes the straight line in the complex plane that is the perpendicular bisector of the line segment...

What's the quickest way to calculate the argument of $\displaystyle \pi e^{-\frac{3i\pi}{2}}$?

One problem I sometimes encounter is with complex numbers. When a formula including functions of complex variables runs in Matlab, I obtain the corresponding result but if I write that formula in different forms (for example when I arrange the long formula in simpler form) I obtain another...

Homework Statement In the argand plane z lies on the line segment joining # z_1 = -3 + 5i # and # z_2 = -5 - 3i # . Find the most suitable answer from the following options . A) -3∏/4 B) ∏/4 C) 5∏/6 D) ∏/6 2. MY ATTEMPT AT THE SOLUTION We get two points ( -3 , 5 ) & ( -5 , -3 ) => The...

Homework Statement Arg z≤ -π /4 Homework EquationsThe Attempt at a Solution I'm confused whether the answer to that would be more than -45° or less. Should the approach to arguments be the same as in negative numbers?