What is Polar form: Definition and 103 Discussions
In mathematics, a complex number is a number that can be expressed in the form a + bi, where a and b are real numbers, and i is a symbol called the imaginary unit, and satisfying the equation i2 = −1. Because no "real" number satisfies this equation, i was called an imaginary number by René Descartes. For the complex number a + bi, a is called the real part and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols
C
{\displaystyle \mathbb {C} }
or C. Despite the historical nomenclature "imaginary", complex numbers are regarded in the mathematical sciences as just as "real" as the real numbers and are fundamental in many aspects of the scientific description of the natural world.Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation
(
x
+
1
)
2
=
−
9
{\displaystyle (x+1)^{2}=-9}
has no real solution, since the square of a real number cannot be negative, but has the two nonreal complex solutions −1 + 3i and −1 − 3i.
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule i2 = −1 combined with the associative, commutative and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field that has the real numbers as a subfield. The complex numbers form also a real vector space of dimension two, with {1, i} as a standard basis.
This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely expressing in terms of complex numbers some geometric properties and constructions. For example, the real numbers form the real line which is identified to the horizontal axis of the complex plane. The complex numbers of absolute value one form the unit circle. The addition of a complex number is a translation in the complex plane, and the multiplication by a complex number is a similarity centered at the origin. The complex conjugation is the reflection symmetry with respect to the real axis. The complex absolute value is a Euclidean norm.
In summary, the complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.
In the following%3A%20https://pubs.rsc.org/en/content/articlehtml/2013/sm/c3sm00140g?casa_token=3O_jwMdswQQAAAAA%3AaSRtvg3XUHSnUwFKEDo01etmudxmMm8lcU4dIUSkJ52Hzitv2c_RSQJYsoHE1Bm2ubZ3sdt6mq5S-w'] paper, the surface velocity for a moving, spherical particle is given as (eq 1)...
The equation I'm trying to graph on desmos is this with A & B as numbers, but I'm unsure how as it is a vector.
r = (A cosθ sinθ cscθ - B sinθ cscθ) i + (A cosθ sinθ cscθ + B sinθ cscθ) j
Hello,
Today I started to think about why graphs, of the same equation, look different on the Cartesian plane vs. the polar grid. I have this visualization where every point on the cartesian plane gets mapped to a point on the polar grid through a transformation of the grids themselves...
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)=...
Express -3-3i in polar form.
I know that r=3√2.
And I understand that now we take tan^-1(b/a) which I did. tan^-1(-3/-3) = π/4. So I put my answer as z = 3√2 [cos(π/4) + isin(π/4)].
However the answer manual told me this was incorrect I am unsure of where I went wrong...
I don't fully understand how to work out the impedance from the given equation (5j-5)x(11j-11)/(5j-5)+(11j-11). Any help would be greatly appreciated. Thanks.
The answer needs to be in rectangular and polar form.
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...
Convert the equation to polar form
8x=8y
I thought it would be
8*r*cos(theta)=8*r*sin(theta)
Said it was incorrect
then I thought I needed to divide by 8 to remove it, giving me:
r*cos(theta)=r*sin(theta)
But that was also incorrect and now I am stuck
Hi,
I am going around in circles, excuse the pun, with phasors, complex exponentials, I&Q and polar form...
1. A cos (ωt+Φ) = Acos(Φ) cos(ωt) - Asin(Φ)sin(ωt)
Right hand side is polar form ... left hand side is in cartesian (rectangular) form via a trignometric identity?
2. But then...
Homework Statement
It's not a homework problem itself, but rather a general method that I imagine is similar to homework. For a given elementary complex function in the form of the product, sum or quotient of polynomials, there are conventional methods for converting them to polar form. The...
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...
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...
Homework Statement
you are given the standard form z = 3 - 3i
Homework EquationsThe Attempt at a Solution
so to convert this to polar form, i know that ##r = 3√2## but how do i find theta here? There are so many mixed answers it seems online that I can't tell... i know that ##(3,-3)## is in...
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...
Hi all,
I was doing some math and I stumbled upon a very interesting thing. When I do ln(-1), I get πi, and when I turn that into polar coordinates on the calculator, it gives me πeiπ/2 . Why is that? I'm very curious to know, because they are so intertwined!
Thank you
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...
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
can someone explain about the formula of the circled part?
Why dA will become r(dr)(dθ)?
Homework EquationsThe Attempt at a Solution
A = pi(r^2)
dA will become 2(pi)(r)(dr) ?
why did 2(pi) didnt appear in the equation ?
Homework Statement
I have the following complex numbers : -3,18 +4,19i
I must put it in polar form.
Homework Equations
r=(a^2+b^2)^(1/2)
cos x = a/r
sin x = b/r
The Attempt at a Solution
I was able to find with cos x = a/r that the x = 127,20
But when I do it with sin x = b/r I obtain like...
Homework Statement
A lamina has constant density \rho and takes the shape of a disk with center the origin and radius R. Use Newton's Law of Gravitation to show that the magnitude of the force of attraction that the lamina exerts on a body of mass m located at the point (0,0,d) on the positive...
Hello
Excuse me, but how do I sketch the phasor of a voltage that it's V=5cos(10t+30degrees) and how the V=5sin(10t+30degrees) ?
I know that these can be converted as the R<angle polar form, with R being the Vmax, ie the 5, and the angle the phase.
But what doesn't it matter if I have cos or...
Greetings. I have been teaching myself Calculus. To do this I ordered a used Larson's 8th Edition Calculus and a used TI-81 graphing calculator. When I got to Chapter 10, I ran into a problem: the chapter introduces equations in polar form and when I whipped out my TI-81, I had no idea how to...
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...
another question:
convert $|\frac{1-i}{3}|$ to polar form
i am getting $\frac{\sqrt{2}}{3} e^{\frac{i\pi}{4}}$
but the solutions say:
$e^{\frac{-i\pi}{4}}$
i did
$ x = r\cos(\theta)$ and $y=r\sin(\theta)$
so
$\frac{1}{3} = {\frac{\sqrt{2}}{3}}\cos(\theta)$
$\frac{1}{3} = \cos(\theta)$
And...
I started of with attempting to convert the numerator first
$ | 1 + i | = \sqrt{1^2+i^2}$
$= \sqrt{1-1} = 0$ ? this is wrong obviously, i don't see why its $\sqrt{2}$
for the second part
$ |\sqrt{3} - i|= \sqrt{3+1} = 2$
$ x = r \cos\theta$ $ y = r\sin\theta$
$x = 2\cos\theta$ $...
Homework Statement
Hello Guys, I am reading Hobson's General Relativity and I have come across an exercise problem, part of which frustrates me:
3.20 (P. 91)
In the 2-space with line element
ds^2=\frac{dr^{2}+r^{2}d\theta^{2}}{r^{2}-a^{2}}-\frac{r^{2}dr^{2}}{{(r^{2}-a^{2})}^{2}}
and...
Homework Statement A 90Ω resistor, a 32 mH inductor, and a 5μF capacitor are connected in series across the terminals of a sinusoidal voltage source Vs = 750cos(5000t + 30)V.
Calculate the phasor current.
[Broken]Homework Equations
phasor current i = V/Z
V in polar form = (Magnitude)(cos a...
Homework Statement
Solve the BVP:
r^{2}u_{rr} + ru_{r} + u_{ψψ} = 0
0 ≤ r ≤ 1, 0 < ψ < 2π
u(1,ψ) = 0.5(π - ψ)
Homework Equations
The Attempt at a Solution
I've derived the general solution of u(r,ψ) = C + r^{n}Ʃ_{n}a_{n}cos nψ + b_{n}sin nψ, where a,b, C are...
Homework Statement
4{cos(13∏/6)+isin(13∏/6)}
= 4((√3/2)+(i/2))
= 2√3+2i
Homework Equations
The Attempt at a Solution
This is an example from my textbook. The part which I do not understand is how to convert the cos and sin of radians into those fractions. Any help is greatly appreciated.
Homework Statement
Find the polar form for zw by first putting z and w into polar form.
z=2√3-2i, w= -1+i
Homework Equations
Tan-1(-√3/3)= 5∏/6
The Attempt at a Solution
r= √[(2√3)2+(-2)2]=4
tanθ= -2/(2√3)=-1/√3=-√3/3=> acording to above... tan-1(-√3/3)= 5∏/6
so, in polar form z should be...
Homework Statement
How do I write 1-2i in polar form?
Homework Equations
The Attempt at a Solution
I know r=√5, and when using x=rcosθ, I get angle of 63.43 or 296.57. However, when I take the sin inverse of-2/√5 I get -63.43. I am really confused.
Homework Statement
\int_0^2 \int_0^\sqrt{2x-x^2} xy,dy,dx
I know the answer, but how does the 2 in the outer integral become pi/2?? I'm fine with everything else, I just can't get this...
Homework Statement
An impedance 8 + j7 Ω is connected in parallel with another impedance of 5 + j6 Ω. this circuit is then connected in series with another impedance, comprising a resistance of 5 Ω in series with a capacitive reactance of 7 Ω. The complete circuit is then connected to 150...
Homework Statement
Write z = 1 + √3i in polar form
Homework Equations
z = r (cos\varphi + sin\varphii)
The Attempt at a Solution
Found the modulus by
|z| = √4 = 2
Now I am stuck on this part of finding the argument:
Tan-1 (√3)
now I am not sure how to go from that to...
Homework Statement
Write the equation
x^2 + y^2 = 1 + sin^2(xy)
in polar form assuming
x = rcos(\phi)
y = rsin(\phi)
0<r, 0<= \phi < 2pi
solve for r as a function of \phi
The Attempt at a Solution
(rcos(\phi))^2 + (rsin(\phi))^2 = 1 + sin^2(r^2cos(\phi)sin(\phi))...
Not homework as such, just need some clarification.
When finding \alpha do you have to take the signs into account when finding tan^{-1} x/a. Does it matter if a or x are negative?
Next question is about quadrants
1: \theta = \alpha
2: \theta = \pi - \alpha
3: \theta = -\pi -...
I couldn't find any book discussing all of this.
===================================================
U+jV=f(x+jy) W=f(z)
Ux=Vy
Uy= -Vx
jWx=Wy <--Cauchy-Riemann equation
Uxx+Uyy=0
Vxx+Vyy=0 <--harmonic condition...
Homework Statement
Consider the complex number z=(i^201+i^8)/(i^3(1+i)^2).
(a) Show that z can be expressed in the Cartesian form 1/2+(1/2)i.
(b) Find the modulus of 4z − 2z*. (z* meaning z-bar/complex conjugate of z)
(c) Write 2z in polar form.
(d) Write 8z^3 in polar exponential form...
Homework Statement
express the arg(z) and polar form of
(1/\sqrt{2}) - (i/\sqrt{2})
Homework Equations
The Attempt at a Solution
Ok so I did \sqrt{(1/\sqrt{2})^{2}+(1/\sqrt{2})^{2}} = 1
so tan^{-1}(1) = \pi/4 so arg(z)=5\pi/4
but they had the answer as -3\pi/4
Am I...
Homework Statement
Given that z_{1}z_{2} ≠ 0, use the polar form to prove that
Re(z_{1}\bar{z}_{2}) = norm (z_{1}) * norm (z_{2}) \Leftrightarrow θ_{1} - θ_{2} = 2n∏, where n is an integer, θ_{1} = arg(z_{1}), and θ_{2} = arg(z_{2}). Also, \bar{z}_{2} is the conjugate of z_{2}. Homework...
Homework Statement
I'm trying to find the best solution for solving a problem in which I must form an operation with three vectors in polar form, ending with a sum in rectangular form. The operation is as follows:
(5 \angle 0°) + (20 \angle -90°) - (6 \angle180°) =
Homework Equations...
Homework Statement
Alright, here's the question, A stream function for a plane, irrotational, polar-coordinate flow is ψ=9r^2sin^θ. Find out the velocity potential in Cartesian Co-ordinate!
Homework Equations
The Attempt at a Solution
Well, I can easily find out the velocity...