Problem 79 - Precalculus Student Solutions Manual 5th

Question

Find each quotient. Write the answer in standard form \(a+b i .\) $$\frac{2}{3 i}$$

Step-by-Step Solution

Answer

-\frac{2}{3}i

Step 1: Understand the Problem

Given expression is \(\frac{2}{3i}\). The goal is to rewrite it in standard form \(a + bi\).

Step 2: Multiply Numerator and Denominator by the Conjugate of the Denominator

Since the denominator is \(3i\), multiply both the numerator and denominator by \(-3i\) (the conjugate of \(3i\)) to simplify the expression: \[\frac{2}{3i} \cdot \frac{-i}{-i} = \frac{2 \times -i}{3i \times -i}\].

Step 3: Simplify the Expression

Calculate the denominator: \(3i \times -i = -3i^2\). Since \(i^2 = -1\), this becomes \(-3 \times (-1) = 3\). Calculate the numerator: \(2 \times -i = -2i\). Now the expression is: \[\frac{-2i}{3}\].

Step 4: Write in Standard Form

Divide the numerator by the denominator: \(-2i \div 3 = -\frac{2}{3}i\). So the expression in standard form is \(0 - \frac{2}{3}i\) or simply \(-\frac{2}{3}i\).

Key Concepts

Standard Form

The standard form of a complex number is written as \(a + bi\). This format includes a real part \(a\) and an imaginary part \(bi\). Writing complex numbers in this way:

• Helps to clearly distinguish the real and imaginary components
• Makes it easier to perform operations such as addition, subtraction, multiplication, or division

By breaking down the complex number into these parts, we can handle computations more effectively and consistently. In our exercise, the goal is to convert the fraction \(\frac{2}{3i}\) into standard form, resulting in \(0 - \frac{2}{3}i\). This way, we see that the real part is \(0\) and the imaginary part is \( - \frac{2}{3}i\).

Imaginary Unit

The imaginary unit is denoted by \(i\) and is defined as \(\sqrt{-1}\). In other words, \(i^2 = -1\). Understanding the imaginary unit is crucial because:

• It allows for the extension of the real number system to include solutions to equations like \(x^2 + 1 = 0\).
• It provides the foundation for constructing complex numbers in the form \(a + bi\).

In the given exercise, the term \(3i\) in the denominator necessitated multiplying by the conjugate \(-i\). This step simplifies the denominator by utilizing the property \(i^2 = -1\), leading to a real number and allowing the expression to be written in the standard form.

Conjugate Multiplication

Multiplying by the conjugate is a technique used to simplify the division of complex numbers. The conjugate of a complex number \(a + bi\) is \(a - bi\). When you multiply a complex number by its conjugate, the result is always a real number:

• \((a + bi) \times (a - bi) = a^2 - (bi)^2 = a^2 + b^2 \)\.
• The imaginary parts cancel each other out.

In our solution, the denominator was \(3i\). To eliminate the imaginary unit in the denominator, we multiplied both the numerator and the denominator by \(-i\). This step resulted in:

• The denominator becoming a real number, specifically:
  \(3i \times -i = -3i^2 = -3(-1) = 3\)
• The numerator transforming into a simpler form.

This process led us to the final result in standard form: \(0 - \frac{2}{3}i\).