Complex Numbers in Polar Form and Operations

What does multiplication by i mean geometrically?

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

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

Write 1 + √3i in polar form.

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

Match the following programming languages with their primary usage:

Write 5eiπ/4 in Cartesian form.

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

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

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 re^iθ
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.

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.