Problem 52 - Algebra and Trigonometry

Question

Write the complex number in polar form with argument \(\theta\) between 0 and 2\(\pi\) $$ -\pi i $$

Step-by-Step Solution

Answer

The polar form is \(\pi(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2})\).

Step 1: Identify the Real and Imaginary Parts

The complex number \(-\pi i\) has a real part of 0 and an imaginary part of \(-\pi\). Thus, the complex number can be written as \(0 + (-\pi)i\).

Step 2: Calculate the Magnitude

The magnitude (or modulus) of the complex number is given by \(|z| = \sqrt{a^2 + b^2}\) where \(a\) is the real part and \(b\) is the imaginary part. Here, \(a=0\) and \(b=-\pi\). So, \(|z| = \sqrt{0^2 + (-\pi)^2} = \sqrt{\pi^2} = \pi\).

Step 3: Determine the Argument

The argument \(\theta\) of a complex number is the angle it makes with the positive x-axis. Since our complex number is \(-\pi i\), it lies on the negative imaginary axis. Therefore, \(\theta = \frac{3\pi}{2}\) as it corresponds to a downward direction from the origin on the complex plane.

Step 4: Write in Polar Form

The polar form of a complex number is \(z = r(\cos\theta + i\sin\theta)\), where \(r\) is the magnitude and \(\theta\) is the argument. Here, \(r = \pi\) and \(\theta = \frac{3\pi}{2}\). Thus, the polar form is \(z = \pi(\cos\frac{3\pi}{2} + i\sin\frac{3\pi}{2})\).

Key Concepts

Polar Form

Complex numbers can be elegantly represented in polar form, which provides a different perspective compared to their standard rectangular representation. The polar form expresses a complex number as a point in the plane, defined by its distance from the origin and the angle it makes with the positive x-axis. Opening up this concept a bit more:

• The polar form of a complex number is given by \(z = r(\cos\theta + i\sin\theta)\).
• Here, \(r\) is the magnitude of the complex number, and \(\theta\) is the argument.

To illustrate, consider the complex number \(-\pi i\). By converting it to polar form, we find it sits \(\pi\) units away from the origin, on the negative imaginary axis, which shifts the perspective from purely real and imaginary components to a more cohesive geometric angle and length view.

Magnitude of Complex Numbers

Determining the magnitude of a complex number is a crucial step when converting between different forms. The magnitude, sometimes called the modulus, indicates how far the number is from the origin on the complex plane. To calculate it:

• Use the formula \(r = |z| = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts respectively.
• For the complex number \(-\pi i\), \(a = 0\) and \(b = -\pi\), hence \(|z| = \sqrt{0^2 + (-\pi)^2} = \pi\).

This tells us that \(-\pi i\) is \(\pi\) units away from the origin, along the imaginary axis. The magnitude plays a pivotal role in expressing complex numbers in polar form, providing the 'radius' of the circle that marks the number's location.

Argument of Complex Numbers

The argument of a complex number represents the angle the number makes with the positive real axis, an essential aspect when transitioning to polar form. It is defined typically within the interval \([0, 2\pi)\):

• For the complex number \(-\pi i\), this number lies entirely on the negative imaginary axis.
• This placement directly corresponds to an angle of \(\frac{3\pi}{2}\), as it can be visualized as a point reached by rotating \(\frac{3\pi}{2}\) radians counterclockwise from the positive x-axis.

Understanding where the complex number is located in relation to these axes aids in determining the correct angle, ensuring accurate conversion to and from polar form. This understanding completes our journey in visualizing and confidently working with complex numbers geometrically.