Problem 12 - Precalculus Mathematics for Calculus

Question

Graph the complex number and find its modulus. $$-1-\frac{\sqrt{3}}{3} i$$

Step-by-Step Solution

Answer

The graph is at (-1, -√3/3) and the modulus is 2√3/3.

Step 1: Understand the Complex Number

The given complex number is \(-1-\frac{\sqrt{3}}{3} i\). This can be written in the form \(a + bi\) where \(a = -1\) and \(b = -\frac{\sqrt{3}}{3}\). Note that \(a\) is the real part and \(b\) is the imaginary part of the complex number.

Step 2: Graph the Complex Number

To graph the complex number \(-1-\frac{\sqrt{3}}{3} i\), plot the point \((-1, -\frac{\sqrt{3}}{3})\) on the complex plane. The horizontal axis represents the real part \(-1\) and the vertical axis represents the imaginary part \(-\frac{\sqrt{3}}{3}\).

Step 3: Calculate the Modulus

The modulus of a complex number \(a + bi\) is given by the formula \(\sqrt{a^2 + b^2}\). Substitute \(a = -1\) and \(b = -\frac{\sqrt{3}}{3}\):\[\sqrt{(-1)^2 + \left(-\frac{\sqrt{3}}{3}\right)^2}\].Calculate the squares:\[= \sqrt{1 + \left(\frac{3}{9}\right)} = \sqrt{1 + \frac{1}{3}} = \sqrt{\frac{3}{3} + \frac{1}{3}} = \sqrt{\frac{4}{3}}\].Simplify to find the modulus:\[= \frac{2}{\sqrt{3}}\].Multiply the numerator and the denominator by \(\sqrt{3}\) to rationalize:\[= \frac{2\sqrt{3}}{3}\].

Key Concepts

Graphing Complex Numbers

Graphing complex numbers is a visual way to understand their structure. Complex numbers have a real part and an imaginary part. In the exercise, the complex number \(-1-\frac{\sqrt{3}}{3} i\) can be seen as \(a + bi\) form where \(a = -1\) and \(b = -\frac{\sqrt{3}}{3}\).
To graph this number, we use the complex plane, where the horizontal axis is reserved for real numbers and the vertical axis handles the imaginary numbers.

• Identify the real part: \(-1\). Move one unit to the left on the real axis.
• Identify the imaginary part: \(-\frac{\sqrt{3}}{3}\). From the resulting position on the real axis, move downwards, since the imaginary part is negative.
• Plot this point: \((-1, -\frac{\sqrt{3}}{3})\).

Now, these coordinates represent our complex number on the plane. Each complex number has a unique position, helping us visualize their addition, subtraction, or multiplication behaviors.

Modulus of Complex Numbers

The modulus of a complex number is like its distance from the origin in the complex plane. It's the length of the vector represented by the complex number.
For a complex number \(a + bi\), the modulus is given by the formula \(\sqrt{a^2 + b^2}\).
In our exercise, the complex number is \(-1-\frac{\sqrt{3}}{3} i\). Let's break it down:

• Identify \(a = -1\) and \(b = -\frac{\sqrt{3}}{3}\).
• Calculate each square: \((-1)^2 = 1\) and \((-\frac{\sqrt{3}}{3})^2 = \frac{1}{3}\).
• Find the sum of the squares: \(1 + \frac{1}{3} = \frac{4}{3}\).
• Take the square root of this sum: \(\sqrt{\frac{4}{3}}\) = \(\frac{2}{\sqrt{3}}\).

This modulus gives us the exact length of the line from the origin to our plotted point. It's crucial for understanding magnitude and direction in physics and engineering.

Rationalizing Denominators

Rationalizing denominators is a crucial step in simplifying expressions. When we encounter expressions with radicals in the denominator, simplifying them aids in clearer presentation and further calculations.
In our exercise, the modulus was initially found as \(\frac{2}{\sqrt{3}}\). It's not standard practice to leave the square root in the denominator.

• Multiply both the numerator and the denominator by \(\sqrt{3}\) to remove the radical: \(\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}\).
• This simplifies to: \(\frac{2\sqrt{3}}{3}\).

Rationalizing transforms our expression into a more standard form, paving the way for consistent operations and comparisons. Common in algebra, it's vital for handling complex expressions effectively.