Problem 81 - Precalculus

Question

Use a calculator to express the complex number in polar form. $$a-2 a i, \text { where } a>0$$

Step-by-Step Solution

Answer

Polar form: \( a\sqrt{5}(\cos(\arctan(-2)) + i\sin(\arctan(-2))) \).

Step 1: Identify Real and Imaginary Components

The complex number is given as \( a - 2ai \). Here, the real part is \( a \) and the imaginary part is \( -2a \).

Step 2: Calculate Magnitude

To find the magnitude (or modulus) of the complex number, use the formula \( |z| = \sqrt{a^2 + (-2a)^2} \). Simplify it to \( |z| = \sqrt{a^2 + 4a^2} = \sqrt{5a^2} = a\sqrt{5} \).

Step 3: Determine the Argument

The argument \( \theta \) of the complex number can be found using \( \tan \theta = \frac{\text{Imaginary part}}{\text{Real part}} = \frac{-2a}{a} = -2 \). Thus, \( \theta = \arctan(-2) \).

Step 4: Convert to Polar Form

The polar form of a complex number is given by \( z = r(\cos \theta + i\sin \theta) \), where \( r \) is the magnitude \( a\sqrt{5} \) and \( \theta \) is the argument \( \arctan(-2) \). Therefore, the polar form is \( a\sqrt{5}(\cos(\arctan(-2)) + i\sin(\arctan(-2))) \).

Key Concepts

Polar Form

Complex numbers can be represented in different forms. One of these forms is the polar form. It represents a complex number using the magnitude and the angle. This is often useful in various mathematical applications. For a given complex number, you can write it as:

• Magnitude (also called the modulus)
• Argument (the angle in the polar coordinate system)

The polar form is expressed as:

• \[ z = r(\cos \theta + i\sin \theta) \]

where:

• \( r \) is the modulus or magnitude of the complex number
• \( \theta \) is the argument or angle of the complex number

Using the polar form makes it easier to multiply and divide complex numbers. This is because multiplying involves adding angles (arguments) and multiplying magnitudes, which simplifies the calculation process.

Modulus of a Complex Number

The modulus of a complex number gives us an idea of how far the number is from the origin in the complex plane. In simple terms, it measures the "size" of the complex number, irrespective of its direction. For a complex number of the form \( z = a + bi \), the modulus is calculated using:

• \[ |z| = \sqrt{a^2 + b^2} \]

The modulus is always a non-negative number. It gives you the distance from the origin \((0,0)\), to the complex number \((a,b)\) on the Argand plane. For the exercise given, the modulus of \( a - 2ai \) is found by calculating:

• Real part: \( a \)
• Imaginary part: \( -2a \)

Using the formula, the modulus becomes:

• \[ |z| = \sqrt{a^2 + (-2a)^2} = a\sqrt{5} \]

This shows the magnitude of the complex number based on its real and imaginary parts.

Argument of a Complex Number

The argument of a complex number indicates the angle that the line representing the complex number makes with the positive real axis in the Argand plane. It's an essential component in the polar form representation. To find the argument \( \theta \) of a complex number \( z = a + bi \), use the formula:

• \[ \tan \theta = \frac{b}{a} \]

In some cases, especially when the complex number has a negative real or imaginary part, it helps to first determine the correct quadrant for \( \theta \). The angle can then be adjusted accordingly.
For the given number \( a - 2ai \), the calculation of the argument involves:

• Real part: \( a \)
• Imaginary part: \( -2a \)

The argument is:

• \[ \theta = \arctan\left(\frac{-2a}{a}\right) = \arctan(-2) \]

This result provides the angle necessary to express the complex number in its complete polar form. Properly accounting for the sign of the real and imaginary parts ensures the correct direction for this angle on the Argand plane.