Convert The Following Complex Number Into Its Polar Representation

Converting Complex Numbers to 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 widespread applications in various fields, including electrical engineering, physics, and signal processing. This comprehensive guide will delve into the intricacies of this conversion, providing a detailed explanation, numerous examples, and practical tips to solidify your understanding.

Understanding Rectangular and Polar Representations

Before diving into the conversion process, let's refresh our understanding of the two representations:

Rectangular Form: A complex number z is expressed as z = x + iy, where x is the real part and y is the imaginary part. This representation is straightforward and easily visualized on a Cartesian plane, where x represents the horizontal coordinate and y represents the vertical coordinate.

Polar Form: The polar form expresses the complex number using its magnitude (or modulus) r and argument (or angle) θ. It's written as z = r(cos θ + i sin θ), or more compactly using Euler's formula as z = re^(iθ). Here, r represents the distance from the origin to the point representing z in the complex plane, and θ is the angle the line connecting the origin and z makes with the positive real axis.

The Conversion Process: From Rectangular to Polar

The conversion involves calculating the magnitude r and the argument θ from the real and imaginary parts x and y.

1. Calculating the Magnitude (Modulus):

The magnitude r is calculated using the Pythagorean theorem:

r = √(x² + y²)

This formula represents the distance from the origin (0, 0) to the point (x, y) in the complex plane. It's always a non-negative real number.

2. Calculating the Argument (Angle):

The argument θ is the angle between the positive real axis and the line connecting the origin to the point (x, y). It's calculated using the inverse tangent function (arctan or tan⁻¹):

θ = arctan(y/x)

However, this formula alone isn't sufficient because the arctan function only provides a value in the range (-π/2, π/2). To accurately determine θ in all four quadrants of the complex plane, we need to consider the signs of x and y:

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

Alternatively, you can use the atan2(y, x) function which is readily available in most programming languages and calculators. This function automatically handles the correct quadrant, eliminating the need for manual adjustments based on the signs of x and y.

3. Expressing the Polar Form:

Once you've calculated r and θ, the polar form of the complex number is:

z = r(cos θ + i sin θ) = re^(iθ)

Examples: Converting Complex Numbers to Polar Form

Let's illustrate the conversion process with several examples:

Example 1: z = 3 + 4i

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

2. Calculate the argument: θ = arctan(4/3) ≈ 0.93 radians or ≈ 53.13 degrees (First Quadrant)

3. Polar Form: z = 5(cos(0.93) + i sin(0.93)) = 5e^(i0.93)

Example 2: z = -2 + 2i

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

2. Calculate the argument: Since x < 0 and y > 0, we're in the second quadrant. θ = arctan(2/-2) + π = -π/4 + π = 3π/4 radians or 135 degrees.

3. Polar Form: z = 2√2(cos(3π/4) + i sin(3π/4)) = 2√2e^(i3π/4)

Example 3: z = -1 - i

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

2. Calculate the argument: Since x < 0 and y < 0, we're in the third quadrant. θ = arctan((-1)/(-1)) - π = π/4 - π = -3π/4 radians or -135 degrees.

3. Polar Form: z = √2(cos(-3π/4) + i sin(-3π/4)) = √2e^(-i3π/4)

Example 4: z = 1 - i

1. Calculate the magnitude: r = √(1² + (-1)²) = √2

2. Calculate the argument: Since x > 0 and y < 0, we are in the fourth quadrant. θ = arctan((-1)/1) = -π/4 radians or -45 degrees.

3. Polar Form: z = √2(cos(-π/4) + i sin(-π/4)) = √2e^(-iπ/4)

Handling Special Cases: Zero and Purely Real/Imaginary Numbers

• z = 0: The magnitude is 0, and the argument is undefined.

• Purely Real Numbers (y = 0): The argument is 0 if x > 0 and π if x < 0.

• Purely Imaginary Numbers (x = 0): The argument is π/2 if y > 0 and -π/2 if y < 0.

Applications of Polar Representation

The polar form of a complex number is particularly useful in various contexts:

• Multiplication and Division: Multiplying complex numbers in polar form simply involves multiplying their magnitudes and adding their arguments. Division involves dividing their magnitudes and subtracting their arguments. This simplifies calculations significantly compared to the rectangular form.

• Powers and Roots: DeMoivre's theorem provides a straightforward method for calculating powers and roots of complex numbers in polar form.

• Signal Processing: Polar representation is crucial in representing sinusoidal signals and analyzing their phase and amplitude.

• Electrical Engineering: In AC circuit analysis, the polar form is used to represent phasors, simplifying calculations involving impedance, voltage, and current.

• Physics: Polar coordinates are essential in many physics problems involving rotation, oscillations, and wave phenomena.

Conclusion

Converting a complex number from rectangular to polar representation is a critical skill in complex analysis and its applications. Understanding the process of calculating the magnitude and argument, along with the nuances of handling different quadrants, is essential. By mastering this conversion, you'll gain a powerful tool for simplifying complex calculations and enhancing your understanding of various mathematical and scientific concepts. Remember to utilize the atan2 function whenever possible to ensure accurate argument calculation, regardless of the quadrant. Practice with diverse examples to solidify your understanding and become proficient in this fundamental aspect of complex number manipulation.