Polar Representation of Complex Numbers: A Complete Tutorial

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Published on September 25, 2024 11:30:36 AM ● Video Link: https://www.youtube.com/watch?v=rLoUIyfwtoQ

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The polar representation of complex numbers expresses a complex number in terms of its magnitude (or modulus) and angle (or argument) relative to the origin in the complex plane. Instead of the standard form z=a+biz=a+bi (where aa and bb are the real and imaginary parts), the polar form is written as:
z=r(cos⁡θ+isin⁡θ)
z=r(cosθ+isinθ)

or more compactly using Euler's Formula as:
z=reiθ
z=reiθ

where r=∣z∣r=∣z∣ is the magnitude, and θθ is the argument or angle.

Polar representation simplifies many operations with complex numbers, such as multiplication, division, and finding powers and roots, making it essential in fields like electrical engineering, signal processing, and physics.
Key Sub-Parts of the Topic:

Magnitude (Modulus) of a Complex Number:
The magnitude rr of a complex number z=a+biz=a+bi is given by:
r=∣z∣=a2+b2
r=∣z∣=a2+b2

It represents the distance of the complex number from the origin in the complex plane.

Argument (Angle) of a Complex Number:
The argument θθ is the angle formed between the positive real axis and the line joining the complex number to the origin, measured in radians. It is calculated as:
θ=tan⁡−1(ba)
θ=tan−1(ab​)

where aa is the real part and bb is the imaginary part of the complex number.

Conversion Between Cartesian and Polar Form:
To convert from Cartesian to polar form, use the formulas for rr (modulus) and θθ (argument).
To convert from polar form z=reiθz=reiθ to Cartesian form z=a+biz=a+bi, use the relationships:
a=rcos⁡θandb=rsin⁡θ
a=rcosθandb=rsinθ

Powers and Roots:
Polar form simplifies taking powers using De Moivre's Theorem and finding roots of complex numbers.

Applications of Polar Representation:
Electrical Engineering: Polar form is used to represent alternating currents and voltages in AC circuit analysis.
Signal Processing: The representation is key for analyzing waves and oscillations.
Physics: Polar coordinates are often used in solving physical problems involving rotations or wave functions.


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