Problem 43 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ \frac{4+6 i}{3 i} $$

Step-by-Step Solution

Answer

The expression evaluates to \( 2 - \frac{4}{3}i \).

Step 1: Simplify the Complex Fraction

We begin by simplifying the expression \( \frac{4+6i}{3i} \). To do this, we will separate the fraction into two parts: \( \frac{4}{3i} \) and \( \frac{6i}{3i} \). This gives us: \[ \frac{4+6i}{3i} = \frac{4}{3i} + \frac{6i}{3i}. \]

Step 2: Simplify \( \frac{6i}{3i} \)

Next, we'll simplify the expression \( \frac{6i}{3i} \). Notice that \( i \) cancels out in the numerator and denominator, simplifying the expression to 2. Thus, \( \frac{6i}{3i} = 2 \).

Step 3: Rationalize \( \frac{4}{3i} \)

To simplify \( \frac{4}{3i} \), we need to rationalize the denominator. Multiply the numerator and denominator by \( i \): \[ \frac{4}{3i} \times \frac{i}{i} = \frac{4i}{3i^2}. \] Recall that \( i^2 = -1 \), so it becomes \( \frac{4i}{-3} = -\frac{4}{3}i \).

Step 4: Combine Real and Imaginary Parts

Now, we combine the results from Step 2 and Step 3 to express the answer in standard form \( a + bi \). From Step 3, we have the imaginary part \( -\frac{4}{3}i \), and from Step 2, the real part is 2. Therefore, \[ \frac{4+6i}{3i} = 2 - \frac{4}{3}i. \]

Key Concepts

Simplifying Complex Fractions

When working with complex fractions, like \( \frac{4+6i}{3i} \), the goal is to make the expression simpler and easier to work with. Complex fractions often have both real and imaginary numbers in the numerator and the denominator. In this problem, we have a fraction in which the denominator is purely imaginary. We simplify the fraction by splitting it into two smaller fractions: \( \frac{4}{3i} \) and \( \frac{6i}{3i} \).

This step is key because it allows us to treat each part separately. By doing so, we can make the process of simplifying less complicated and more systematic. After splitting the fractions, we can further simplify each part in subsequent steps.

Rationalizing the Denominator

Rationalizing the denominator is a vital step when dealing with complex numbers, especially when the denominator has an imaginary part. In our problem, we need to simplify \( \frac{4}{3i} \). Since we want to avoid having \( i \) in the denominator, we use a technique called rationalizing.

To rationalize \( \frac{4}{3i} \), multiply both the numerator and the denominator by \( i \). This results in \( \frac{4i}{3i^2} \). We know that \( i^2 = -1 \), which changes the expression to \( \frac{4i}{-3} = -\frac{4}{3}i \).

By rationalizing, we've effectively eliminated the imaginary unit from the denominator, turning it into a real number, which is easier to handle.

Imaginary Unit i

The imaginary unit, represented as \( i \), plays a crucial role in understanding complex numbers. By definition, \( i \) is the square root of -1. It's a fundamental building block that allows us to work with numbers involving the square roots of negative values, which traditional real numbers can't handle.

Here are some important properties of \( i \):

• \( i^2 = -1 \)
• When squared, it results in a negative value, shifting the real number operation into the complex number realm.
• Exponent rules: \( i^3 = i^2 \cdot i = -i \) and \( i^4 = 1 \). This cyclical nature helps in simplifying expressions involving powers of \( i \).

Understanding these properties helps in operations like rationalizing denominators and simplifying complex number expressions.

Combining Real and Imaginary Parts

After simplifying and rationalizing the expression, the next step is to combine the real and imaginary parts into a single expression. Our problem results in two simplified components: the real number 2 and the imaginary number \(-\frac{4}{3}i\).

Combining these gives us the final answer in the standard form \( a + bi \). The real part is 2, and the imaginary part is \(-\frac{4}{3}i\), so the expression becomes \( 2 - \frac{4}{3}i \).

Here's why writing complex numbers in this form is beneficial:

• Consistency: It allows you to compare and perform operations on complex numbers more easily.
• Clarity: Distinguishes between the "real" effects and "imaginary" effects in calculations.

By consistently combining real and imaginary parts, you ensure clarity and precision in mathematical communication.