Problem 28 - College Algebra Essentials

Question

In Exercises \(21-28,\) divide and express the result in standard form. $$ \frac{3-4 i}{4+3 i} $$

Step-by-Step Solution

Answer

\(\frac{3-4i}{4+3i} = -i\)

Step 1: Identify the conjugate of the denominator

The complex conjugate of \(4+3i\) is \(4-3i\). Remember, to form the conjugate of a complex number, the imaginary part changes sign.

Step 2: Multiply numerator and denominator by the conjugate

Multiply the numerator \((3-4i)\) and the denominator \((4+3i)\) by the conjugate of the denominator \((4-3i)\). This gives: \[\frac{(3-4i)(4-3i)}{(4+3i)(4-3i)}.\]

Step 3: Expand and simplify

Expanding the numerator and denominator results to: \[\frac{12-9i-16i+12i^2}{16-9i^2}.\] Now, because \(i^2=-1\), simplify the numerator and denominator to get: \[\frac{12-25i-12}{16+9} = \frac{-25i}{25}.\] Divide -25i by 25 to find the result.

Step 4: Final result

After division, the final result is \(-i\). So, \(\frac{3-4i}{4+3i} = -i\) in standard form.

Key Concepts

Conjugate of a Complex Number

Complex numbers often take the form \(a + bi\), where \(a\) represents the real part and \(b\) represents the imaginary part. The conjugate of a complex number is a simple yet potent concept. It involves changing the sign of the imaginary part.
For a complex number like \(4 + 3i\), its conjugate is \(4 - 3i\). It is vital for various operations, especially in division, as you will see later. By multiplying a complex number by its conjugate, you eliminate the imaginary part, leading you to handle real numbers alone.
In practical terms, using the conjugate helps remove any imaginary components from the denominator when dividing complex numbers. This maneuver simplifies fractions involving complex numbers, making computations more manageable.

Division of Complex Numbers

Dividing complex numbers can seem daunting. However, it becomes straightforward once you grasp the process. The key is using the conjugate. Here's how it's done:

• First, identify the conjugate of the denominator. For instance, the complex number \(4 + 3i\) has a conjugate \(4 - 3i\).
• Next, multiply both the numerator and the denominator by this conjugate. This step resembles rationalizing a denominator in fractions.
• Through this multiplication, the imaginary unit \(i\) gets eliminated from the denominator, simplifying the expression significantly.

For example, dividing \(\frac{3-4i}{4+3i}\) becomes easier by multiplying the numerator and denominator by \(4-3i\). This computation leads to a real denominator, enabling the expression to convert into a simpler form.
After simplifying, it results in a purely imaginary number, showing how order and methodical processes can transform a possibly complex division into an accessible solution.

Standard Form of Complex Numbers

When solving or expressing complex numbers, putting them into a standard form \(a + bi\) is essential. This standard form neatly separates the real part \(a\) from the imaginary part \(bi\).
In exercises involving complex numbers, after performing operations like division, it is necessary to return to this form. It not only provides a clearer presentation but also facilitates any further operations or interpretations needed in your calculations.
In the example \(\frac{3-4i}{4+3i} = -i\), through the calculation, you'll realize the answer appears as \(0 - 1i\). This can be rewritten simply as \(-i\). This demonstrates that every complex number can be succinctly expressed in this standard structure. Keeping results in standard form enables ease of understanding and furthers mathematical communication.