Problem 26 - College Algebra

Question

In Exercises \(21-28,\) divide and express the result in standard form. $$\frac{-6 i}{3+2 i}$$

Step-by-Step Solution

Answer

The division of the given complex numbers simplifies to \( \frac{-18}{13}i + \frac{12}{13}\)

Step 1: Identify the conjugate

The conjugate of the denominator \(3+2i\) is \(3-2i\). Using the conjugate, we can change the denominator to a real number.

Step 2: Multiply numerator and denominator by the conjugate

Multiply both the numerator and denominator by \(3-2i\). This yields: \(\frac{-6i(3-2i)}{(3+2i)(3-2i)}\).

Step 3: Perform multiplication

Multiply the elements to yield: \(\frac{-18i+12}{9+4}\). Simplify to get: \(\frac{-18i+12}{13}\).

Step 4: Divide each term by the real number

Divide each term in the numerator by the denominator to yield: \( \frac{-18}{13}i + \frac{12}{13}\).

Key Concepts

Complex Conjugate

The complex conjugate is an essential concept when dealing with division involving complex numbers. A complex conjugate takes a complex number and changes the sign of its imaginary part. For a complex number expressed as \( a + bi \), its conjugate is \( a - bi \).
Using the complex conjugate allows us to transform the denominator into a real number. This technique is crucial in simplifying expressions where complex numbers appear in the denominator.

• For example, for the complex number \( 3 + 2i \), the conjugate is \( 3 - 2i \).

By multiplying both the numerator and the denominator by the conjugate, we effectively eliminate the imaginary part from the denominator. This step transforms the denominator from \( 3 + 2i \) to \( (3 + 2i)(3 - 2i) = 9 + 4 = 13 \), which is a real number. This process is a vital step in expressing the division of complex numbers in standard form.

Standard Form

The standard form of a complex number is the format where it is written as \( a + bi \), where \( a \) and \( b \) are real numbers.
This format clearly separates the real and imaginary components, making it easier to perform arithmetic operations and compare complex numbers.
When dividing complex numbers, the goal is to express the quotient in this standard form.

• Following the multiplication of the numerator and denominator by the conjugate, as in the example \( \frac{-18i+12}{13} \), the next step is to divide each term in the numerator by the real denominator (13 in this case).

By performing this division, we achieve the standard form of the complex number, breaking it into two clear parts: the real part \( \frac{12}{13} \) and the imaginary part \( \frac{-18}{13}i \).
Ultimately, this ensures that the result of the division is neatly expressed for further analysis or computation.

Imaginary Unit

The imaginary unit, symbolized by \( i \), is a fundamental component in complex numbers. It is defined such that \( i^2 = -1 \).
This property allows for the representation of numbers that are not real within the broader complex number system.

• Imaginary numbers take the form \( bi \), where \( b \) is a real number.

When working with complex numbers, the imaginary unit allows us to separate and handle the imaginary parts of expressions.
In our example of dividing \( \frac{-6i}{3+2i} \), the property of \( i^2 = -1 \) is particularly useful in simplifying multiplication and conversion to standard form.
As the complex conjugate process involves subtracting the imaginary components, understanding \( i \) is crucial for grouping and simplifying terms effectively.