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 significant applications in various fields like electrical engineering, physics, and signal processing. This guide provides a thorough explanation of the process, exploring the underlying concepts and offering practical examples to solidify your understanding.

Understanding Complex Numbers and Their Representations

A complex number, denoted as 'z', is a number that can be expressed in the form a + bi, where 'a' and 'b' are real numbers, and 'i' is the imaginary unit (√-1). 'a' is the real part (Re(z)) and 'b' is the imaginary part (Im(z)). This is the rectangular representation.

The polar representation offers an alternative perspective. It expresses the complex number using its magnitude (or modulus) and argument (or phase).

The Polar Form: Magnitude and Argument

The magnitude (r) of a complex number z = a + bi is the distance from the origin (0, 0) to the point (a, b) in the complex plane. It's calculated using the Pythagorean theorem:

r = |z| = √(a² + b²)

The argument (θ) is the angle between the positive real axis and the line connecting the origin to the point (a, b). It's measured counter-clockwise from the positive real axis. The argument can be found using trigonometry:

θ = arctan(b/a)

However, the arctan function only provides the principal argument, which lies within the range (-π/2, π/2). To determine the correct argument for all quadrants, consider the signs of 'a' and 'b':

• 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) + 2π (or -2π)

The Polar Representation: r(cos θ + i sin θ)

Once we have the magnitude (r) and argument (θ), we can express the complex number in its polar form:

z = r(cos θ + i sin θ)

This form is also often written using Euler's formula:

z = re^(iθ)

Step-by-Step Conversion: A Practical Approach

Let's illustrate the conversion process with a detailed example. Consider the complex number z = -3 + 4i.

Step 1: Find the Magnitude (r)

r = √((-3)² + 4²) = √(9 + 16) = √25 = 5

Step 2: Find the Argument (θ)

Since a = -3 (negative) and b = 4 (positive), the point lies in the second quadrant.

θ = arctan(4/-3) + π ≈ -0.927 + π ≈ 2.214 radians (or approximately 126.87°)

Step 3: Express in Polar Form

Using the calculated magnitude and argument, the polar representation of z = -3 + 4i is:

z = 5(cos 2.214 + i sin 2.214) or z = 5e^(i2.214)

Handling Different Quadrants: Avoiding Common Pitfalls

The key to accurate conversion lies in correctly determining the argument. The arctan function alone is insufficient; careful consideration of the quadrant is crucial. Let's look at another example to highlight this:

z = -2 - 2i

Step 1: Find the Magnitude (r)

r = √((-2)² + (-2)²) = √(4 + 4) = √8 = 2√2

Step 2: Find the Argument (θ)

Both a and b are negative, placing the point in the third quadrant.

θ = arctan((-2)/(-2)) - π = arctan(1) - π = π/4 - π = -3π/4 radians (or -135°)

Step 3: Express in Polar Form

z = 2√2(cos(-3π/4) + i sin(-3π/4)) or z = 2√2e^(-i3π/4)

Advanced Considerations: Multiple Arguments

It's important to note that the argument θ is not unique. Adding or subtracting multiples of 2π (or 360°) to the argument yields an equivalent polar representation. For instance, in the previous example, we could also represent the argument as -3π/4 + 2π = 5π/4. The choice of argument often depends on the specific application or context.

Applications of Polar Representation

The polar representation offers significant advantages in certain contexts:

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

• Powers and Roots: De Moivre's Theorem provides a straightforward method for raising a complex number to a power or finding its roots when the number is in polar form.

• Signal Processing: In signal processing, the magnitude represents the amplitude of a signal, and the argument represents its phase. This is crucial for analyzing and manipulating signals.

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

Conclusion

Converting a complex number to its polar representation involves calculating its magnitude and argument. This conversion simplifies various operations in complex analysis and is crucial in numerous applications across different scientific and engineering fields. Mastering this conversion process is essential for anyone working with complex numbers. By understanding the underlying principles and carefully considering the quadrant of the complex number, you can confidently navigate the conversion and unlock the benefits of the polar representation. Remember to always check your work and consider the implications of using different representations of the argument, especially in practical applications.