Complex Numbers in Polar Form and Operations
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Questions and Answers

What does multiplication by i mean geometrically?

Rotating a complex number by 90 degrees counterclockwise in the complex plane.

Write (8eiπ/3)/(2eiπ/2) in Cartesian form.

2 + 2√3i

Write 2eiπ/4 + 3eiπ/3 in Cartesian form.

(√2 + 3/2) + (√2 - 3√3/2)i

Write 1 + √3i in polar form.

<p>2(cos(π/3) + i sin(π/3))</p> Signup and view all the answers

Find all solutions of the equation (z - i)3 = 1 + √3i. Express the solutions in Cartesian form. Your answer may involve trigonometric expressions.

<p>z = 1+i , z = -√3/2 + (1/2)i , z = √3/2 - (1/2)i</p> Signup and view all the answers

Match the following programming languages with their primary usage:

<p>Python = General-purpose programming JavaScript = Client-side scripting for web applications SQL = Database queries CSS = Styling web pages</p> Signup and view all the answers

Write 5eiπ/4 in Cartesian form.

<p>-5√2/2 - 5√2/2i</p> Signup and view all the answers

Write (cosθ + i sinθ)7 in the form cos β + i sin β.

<p>cos(7θ) + i sin(7θ)</p> Signup and view all the answers

Write (cos(π/4) + i sin(π/4))-2 in the form cos β + i sin β.

<p>cos(π/2) + i sin(π/2)</p> Signup and view all the answers

Flashcards

Complex number in Cartesian form

A complex number expressed as a + bi, where a and b are real numbers and i is the imaginary unit.

Complex number in polar form

A complex number expressed as re^(iθ), where r is the modulus and θ is the argument (or angle) of the complex number.

Modulus of a complex number

The distance of a complex number from the origin in the complex plane.

Argument of a complex number

The angle between the positive real axis and the line connecting the origin to the complex number in the complex plane.

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De Moivre's Theorem

A theorem that relates the powers and roots of complex numbers in polar form.

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nth roots of a complex number

Solutions to z^n = a, where a is a complex number and n is an integer.

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Complex plane

A two-dimensional coordinate system where the horizontal axis represents the real part and the vertical axis represents the imaginary part of a complex number.

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Multiplication by i geometrically

A rotation of 90 degrees counterclockwise in the complex plane.

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Cartesian form of a complex #

Standard algebraic form a + bi where a and b are real numbers and i is the imaginary unit.

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Polar form of a complex #

A way to express a complex number using modulus and argument (r, θ).

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Trigonometric Form

Expressing complex numbers in the form re^{iθ}.

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z̄

Complex conjugate of z, obtained by changing the sign of the imaginary part.

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|z|

Modulus of z, the distance from the origin z to the complex point z in the complex plane.

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e^(iθ)

A way to concisely express the function cos(θ) + i sin(θ).

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Complex Conjugate

Changing the sign of the imaginary part of a complex number.

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z^3

A complex number z raised to the power of 3.

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e^(iπ)

The complex value derived by Euler's equation, equals to -1 + 0i.

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Argument of a complex number

The angle of a complex number in the complex plane relative to the positive real axis.

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Study Notes

Complex Numbers in Polar Form

  • Convert complex numbers from rectangular (Cartesian) to polar form (r cis θ)
  • Polar form: r(cos θ + i sin θ) or reiθ
  • r is the modulus (distance from origin), θ is the argument (angle from the positive real axis)
  • Examples provided for conversion, including specific values like -1+i.

Complex Number Operations

  • Multiplication: Multiply moduli and add arguments.
  • Division: Divide moduli and subtract arguments.
  • Powers: Raise the modulus to the power and multiply the argument by the power.
  • Roots: Take the nth root of the modulus and divide the argument by n.

Complex Plane

  • Plotting complex numbers: x-axis is the real part, y-axis is the imaginary part.
  • Visualizing operations like multiplication by i on the complex plane.

Equations in the Complex Plane

  • Equations involving complex numbers often describe geometric shapes in the complex plane
  • Specific equations shown describing lines and circles/

De Moivre's Theorem

  • Used to find powers and roots of complex numbers in polar form.

Trigonometric Identities

  • Using De Moivre's theorem to express cos(3θ) and sin(3θ) in terms of cos θ and sin θ.
  • These identities involving powers of trig functions of an angle.

Third Roots of a Complex Number

  • Find the third roots of complex numbers
  • Expressing solutions both in polar and rectangular/Cartesian form.
  • Graphing these roots on the complex plane.

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Related Documents

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Description

This quiz explores the conversion of complex numbers from rectangular to polar form and covers operations such as multiplication, division, and roots in the complex plane. Additionally, it examines the geometric representations of complex equations and their impact on visualizing complex numbers. Enhance your understanding of complex analysis and its applications.

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