# 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.

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1. ### A Converting this vector into polar form

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)...
2. ### Help graphing Vectors in polar form

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
3. ### I Cartesian to Polar form.... Is it just a transformation of the plane?

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...
4. ### Complex numbers: convert the exponential to polar form

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)=...
5. ### MHB Hankel transform on the polar form of the Laplacian.

Why first terms equal to zero ? ? ?
6. ### Finding Polar Form Expressions: -3-3i & 2√3-2i

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...
7. ### MHB Finding the impedance in rectangular and polar form

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.

24. ### Finding polar form of complex number

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...
25. ### Gravity of a disk acting on a mass on the z axis

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...
26. ### Understanding Phasors: How to Sketch a Voltage Phasor in Polar Form

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...
27. ### Calculators Graphing in polar form on the TI-81

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...
28. ### Complex Numbers converting from Polar form to Acos(wt + x)

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...
29. ### MHB Converting from Cartesian to polar form

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...
30. ### MHB Converting to polar form

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 $|\sqrt{3} - i|= \sqrt{3+1} = 2$ $x = r \cos\theta$ $y = r\sin\theta$ $x = 2\cos\theta$ \$...
31. ### Line element under coordinate transformation to get polar form

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...
32. ### Find phasor current (impedance, etc.), finding polar form

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...
33. M

### Laplace Equation Polar Form

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...
34. ### Write the polar form of a complex number in the form of a+ib

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.
35. ### Surds in polar form of imaginary number

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...
36. ### Write 1-2i in Polar Form - Solve Confusion

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.
37. ### Problem with limits of integration - converting double integral to polar form

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...
38. ### Calculating Impedance and Power in AC Circuits: A Step-by-Step Guide

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...
39. ### Find the locus of a pt in polar form

the question is showed below i know that x=rcos θ and y= rsinθ and x^2 + y^2 = R^2 but i just dun know how to find the locus is polar form any clue ?
40. ### Writing in polar form a complex number

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...
41. ### Cube Roots of 1 in Polar Form: Stephen's Question

Hi all, There is a question that asks? Determine the cube roots of 1 in polar form? Does that mean I can use De Moirve Formula? Stephen
42. ### Writing x^2 + y^2 = 1 + sin^2(xy) in polar form

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))...
43. ### Complex numbers polar form

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 -...
44. ### Multiple integrals in polar form

Homework Statement do you see how the integral of r is .5? I don't get how that follows?
45. ### Cauchy-Riemann equation polar form

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...
46. ### Help with finding the modulus, polar form and polar exponential form

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...
47. ### Polar form and arg(z) problem

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...
48. ### Complex Analysis: Using polar form to show arg(z1) - arg(z2) = 2n*pi

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...
49. ### Form an op w/ three vectors in polar form.

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...
50. ### How to convert velocity potential from polar form to Cartesian coordinate form

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...