Problem 52 - Precalculus Mathematics for Calculus

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 \( \pi(0 - i) \).

Step 1: Identify the complex number

We start by identifying the given complex number, which is given as \(-\pi i\). This can be written in the form \(0 - \pi i\), where the real part is 0 and the imaginary part is \(-\pi\).

Step 2: Calculate the Absolute Value (Modulus)

The modulus \( r \) of a complex number \( a + bi \) is given by the formula:\[ r = \sqrt{a^2 + b^2}\]For our number, \( a = 0 \) and \( b = -\pi \), thus:\[ r = \sqrt{0^2 + (-\pi)^2} = \sqrt{\pi^2} = \pi \]So the modulus is \( \pi \).

Step 3: Determine the Argument \(\theta\)

The argument \( \theta \) is the angle the line representing the complex number forms with the positive real axis. For a purely imaginary negative number, the argument is always \( \frac{3\pi}{2} \) since it is located on the negative imaginary axis.Therefore, \( \theta = \frac{3\pi}{2} \).

Step 4: Write the Polar Form

The polar form of a complex number is given by:\[r(\cos(\theta) + i\sin(\theta))\]Substitute \( r = \pi \) and \( \theta = \frac{3\pi}{2} \):\[\pi \left( \cos\left(\frac{3\pi}{2}\right) + i\sin\left(\frac{3\pi}{2}\right) \right)\]Thus, the polar form is \( \pi(0 - i) = -\pi i \).

Key Concepts

Complex Numbers

Complex numbers are numbers that consist of two parts: a real part and an imaginary part. They are usually expressed in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. The imaginary unit \(i\) is defined as the square root of \(-1\), i.e., \(i^2 = -1\). This may seem a bit abstract, but complex numbers are very powerful in mathematics and engineering.

• Real numbers can be visualized on a number line, but complex numbers require a plane to be visualized. This plane is called the complex plane.
• The horizontal axis (x-axis) of this plane represents real numbers.
• The vertical axis (y-axis) represents imaginary numbers.

In this context, the complex number \(0 - \pi i\) is purely imaginary, meaning it lies entirely on the imaginary axis.

Modulus

The modulus of a complex number is a measure of its "size" or "length". It's similar to the concept of absolute value for real numbers. For a complex number \(a + bi\), the modulus is denoted as \(|a + bi|\) or sometimes simply \(r\), and calculated using:\[ r = \sqrt{a^2 + b^2} \]This formula represents the distance of the complex number from the origin \((0,0)\) in the complex plane.

• For \(-\pi i\), we have: \(a = 0\) and \(b = -\pi\).
• Thus, the modulus is \(\sqrt{0^2 + (-\pi)^2} = \pi\).

The modulus helps us convert a complex number into polar form by defining its radius in a plane, which is crucial for graphical representation and further computations.

Argument of Complex Number

The argument of a complex number is the angle \(\theta\) that a line connecting the origin to the complex number makes with the positive real axis, usually measured counterclockwise.

• The argument provides directional information in the complex plane.
• It is typically given in radians and can vary from 0 to \(2\pi\).

For the complex number \(-\pi i\), situated entirely on the negative imaginary axis, the standard argument is \(\frac{3\pi}{2}\). Knowing the argument allows us to determine the position of the complex number in its polar representation, a crucial step for functions involving rotation or phase shifts.

Imaginary Numbers

Imaginary numbers are a core component of complex numbers. They arise when we extend real numbers to include solutions for equations like \(x^2 = -1\), something not possible with only real numbers. This is how the imaginary unit \(i\) is defined, representing \(\sqrt{-1}\).

• The term "imaginary" doesn't mean these numbers aren't "real" in a mathematical sense, rather they extend the real number line into a plane.
• Imaginary numbers can be represented on the vertical axis of the complex plane, where each imaginary number corresponds to a real multiple of \(i\).

In the given problem, \(-\pi i\) is a purely imaginary number, falling directly on this imaginary axis, showcasing the power and usability of complex numbers beyond purely real calculations.