Convert The Following Complex Number Into Its Polar Representation

Converting Complex Numbers into Polar Representation: A Comprehensive Guide

Converting a complex number from its rectangular (or Cartesian) form to its polar form is a fundamental operation in complex analysis with numerous applications in fields like electrical engineering, signal processing, and physics. This comprehensive guide will walk you through the process, explaining the underlying concepts and providing numerous examples to solidify your understanding.

Understanding Complex Numbers and Their Representations

A complex number, denoted by 'z', is a number that can be expressed in the form z = a + bi, where:

• a is the real part (Re(z))
• b is the imaginary part (Im(z))
• i is the imaginary unit, defined as √(-1)

This is the rectangular or Cartesian representation. We can visualize this on a complex plane, where the real part is plotted on the x-axis and the imaginary part on the y-axis. The complex number then represents a point in this plane.

Introducing Polar Representation

The polar representation offers an alternative way to represent a complex number. Instead of using the real and imaginary components, it uses:

• r: The magnitude (or modulus) of the complex number, representing the distance from the origin (0,0) to the point representing the complex number on the complex plane.
• θ: The argument (or phase) of the complex number, representing the angle between the positive real axis and the line connecting the origin to the point representing the complex number. This angle is measured counter-clockwise.

The polar form is expressed as: z = r(cos θ + i sin θ)

This can be further simplified using Euler's formula: e^(iθ) = cos θ + i sin θ

Therefore, the polar form can also be written as: z = re^(iθ)

The Conversion Process: From Rectangular to Polar

The conversion involves calculating the magnitude (r) and the argument (θ) from the real (a) and imaginary (b) parts of the complex number.

Calculating the Magnitude (r)

The magnitude r is calculated using the Pythagorean theorem:

r = √(a² + b²)

This represents the distance from the origin to the point (a, b) on the complex plane.

Calculating the Argument (θ)

The argument θ is calculated using trigonometry:

θ = arctan(b/a)

However, this formula only provides the principal value of θ, which lies in the range of -π/2 to π/2. To obtain the correct argument, you need to consider the quadrant in which the point (a, b) lies on the complex plane:

• First Quadrant (a > 0, b > 0): θ = arctan(b/a)
• Second Quadrant (a < 0, b > 0): θ = arctan(b/a) + π
• Third Quadrant (a < 0, b < 0): θ = arctan(b/a) - π
• Fourth Quadrant (a > 0, b < 0): θ = arctan(b/a)

Alternatively, you can use the atan2(b, a) function, which is available in most programming languages and calculators. This function automatically handles the quadrant and provides the correct argument in the range of -π to π.

Worked Examples

Let's convert some complex numbers from rectangular to polar form:

Example 1: z = 3 + 4i

1. Calculate the magnitude (r): r = √(3² + 4²) = √(9 + 16) = √25 = 5

2. Calculate the argument (θ): θ = arctan(4/3) ≈ 0.93 radians (Since both a and b are positive, this is in the first quadrant)

Therefore, the polar representation is: z = 5(cos(0.93) + i sin(0.93)) or z = 5e^(i0.93)

Example 2: z = -2 + 2i

1. Calculate the magnitude (r): r = √((-2)² + 2²) = √(4 + 4) = √8 = 2√2

2. Calculate the argument (θ): Since a is negative and b is positive, we are in the second quadrant. Using atan2: θ = atan2(2, -2) = ¾π radians or 135 degrees.

Therefore, the polar representation is: z = 2√2(cos(¾π) + i sin(¾π)) or z = 2√2e^(i¾π)

Example 3: z = -1 - i

1. Calculate the magnitude (r): r = √((-1)² + (-1)²) = √(1 + 1) = √2

2. Calculate the argument (θ): Since both a and b are negative, we're in the third quadrant. θ = arctan(-1/-1) = π/4 However, this is not correct. Using atan2: θ = atan2(-1, -1) = -3π/4 radians or -135 degrees.

Therefore, the polar representation is: z = √2(cos(-3π/4) + i sin(-3π/4)) or z = √2e^(-i3π/4)

Example 4: z = 5 (a purely real number)

1. Calculate the magnitude (r): r = √(5² + 0²) = 5

2. Calculate the argument (θ): θ = arctan(0/5) = 0 radians

Therefore, the polar representation is: z = 5(cos(0) + i sin(0)) or z = 5e^(i0)

Example 5: z = 2i (a purely imaginary number)

1. Calculate the magnitude (r): r = √(0² + 2²) = 2

2. Calculate the argument (θ): θ = arctan(2/0) which is undefined. However, using our understanding of the complex plane, we know the point (0,2) lies on the positive imaginary axis, thus the angle is π/2 radians or 90 degrees.

Therefore, the polar representation is: z = 2(cos(π/2) + i sin(π/2)) or z = 2e^(iπ/2)

Applications of Polar Representation

The polar form of complex numbers is particularly useful in several applications:

• Signal Processing: Representing sinusoidal signals, analyzing their amplitude and phase.
• Electrical Engineering: Analyzing AC circuits, where the magnitude represents the amplitude and the argument represents the phase shift.
• Quantum Mechanics: Describing quantum states using complex amplitudes.
• Fractals: Generating and understanding various fractal patterns, like the Mandelbrot set.
• Rotation in Two Dimensions: Multiplying complex numbers in polar form is equivalent to rotating and scaling a vector on the complex plane. This property finds numerous applications in computer graphics and robotics.

Conclusion

Converting complex numbers from rectangular to polar form is a crucial step in many mathematical and scientific applications. Understanding the underlying principles of magnitude and argument calculation, along with the considerations for different quadrants, ensures accurate conversions. By mastering this fundamental concept, you gain a powerful tool for solving complex problems across diverse fields. Remember to utilize the atan2 function for robust and accurate argument calculation. The examples provided illustrate the process clearly, and applying these methods will enhance your proficiency in working with complex numbers.