Problem 32 - College Algebra

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ \frac{2-3 i}{4+3 i} $$

Step-by-Step Solution

Answer

The simplified complex number is \(-\frac{1}{25} - \frac{18}{25}i\).

Step 1: Write down the expression

The given expression is \( \frac{2-3i}{4+3i} \). We are asked to perform the division and simplify the result into a complex number of the form \( a + bi \).

Step 2: Form the conjugate of the denominator

The conjugate of a complex number \( a + bi \) is \( a - bi \). Therefore, the conjugate of \( 4 + 3i \) is \( 4 - 3i \).

Step 3: Multiply numerator and denominator by the conjugate

To eliminate the imaginary part in the denominator, multiply both the numerator and the denominator by the conjugate of the denominator. This gives:\[\frac{(2-3i)(4-3i)}{(4+3i)(4-3i)}\]

Step 4: Expand the products

First, expand the numerator:\((2 - 3i)(4 - 3i) = 2 \cdot 4 + 2 \cdot (-3i) - 3i \cdot 4 + (-3i) \cdot (-3i)\)\(= 8 - 6i - 12i + 9i^2\)Since \(i^2 = -1\), we have \(8 - 6i - 12i - 9 = -1 - 18i\).Expand the denominator:\((4+3i)(4-3i) = 4^2 - (3i)^2 = 16 - 9(-1) = 25\).

Step 5: Simplify the expression

The expression now is:\[\frac{-1 - 18i}{25}\]Divide each term in the numerator by the denominator:\[-\frac{1}{25} - \frac{18}{25}i\]

Step 6: Write the simplified complex number

Thus, the simplified complex number is:\[-\frac{1}{25} - \frac{18}{25}i\].

Key Concepts

Complex Numbers

Complex numbers are a blend of real and imaginary numbers into a single number of the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) represents the imaginary unit. They provide a way to work with numbers that are not real, such as the square root of negative numbers.
For example, in the complex number \(2 - 3i\) from our expression, 2 is the real part, while \(-3i\) is the imaginary part. The beauty of complex numbers lies in their ability to simplify expressions involving square roots of negative numbers, enabling mathematical operations that involve both real and imaginary components to be performed seamlessly.

Conjugate Multiplication

Conjugate multiplication is an essential process when working with the division of complex numbers. The conjugate of a complex number \(a + bi\) is \(a - bi\). By multiplying a complex number by its conjugate, you eliminate the imaginary component, resulting in a real number.
For example, to simplify \(\frac{2-3i}{4+3i}\), we form the conjugate of the denominator, \(4 - 3i\), and multiply both the numerator and denominator by it:\[\frac{(2-3i)(4-3i)}{(4+3i)(4-3i)}\].
This operation turns the complex denominator into a straightforward real number, making it easier to divide and simplify.

Imaginary Unit

The imaginary unit \(i\) is defined as the square root of \(-1\) and it’s central to understanding complex numbers.
In practice, \(i^2 = -1\), and this property is key when simplifying complex expressions. During multiplication, terms involving \(i^2\) can be converted to real numbers by substituting \(-1\) for \(i^2\).
For instance, in the expansion step of the expression \((2 - 3i)(4 - 3i)\), you encounter \(9i^2\), which simplifies to \(-9\) because \(9i^2 = 9 \times -1 = -9\). This simplification is crucial to further reducing the expression into a pure real and imaginary part format.

Division of Complex Numbers

Dividing complex numbers can seem tricky, but it becomes straightforward with the right technique. The essential strategy is to multiply both the numerator and the denominator by the conjugate of the denominator. This helps clear out any imaginary number from the denominator, converting it into a real number.
In our task, we start with \(\frac{2-3i}{4+3i}\). By using conjugate multiplication — where \(4 + 3i\) is multiplied by its conjugate \(4 - 3i\) — the denominator simplifies to \(25\), a real number. The equation then becomes \(\frac{-1-18i}{25}\). By dividing terms separately, simplicity is achieved, resulting in \(-\frac{1}{25} - \frac{18}{25}i\).
With practice, dividing complex numbers becomes an effortless routine, always leading to a neat result in the form of \(a + bi\).