Problem 26 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$\frac{-2+6 i}{3 i}$$

Step-by-Step Solution

Answer

\(a = -2\), \(b = -\frac{2}{3}\)

Step 1: Identify the Complex Number

The given complex number is \(-2 + 6i\) divided by \(3i\). We need to rewrite this expression in the form \(a + bi\).

Step 2: Multiply by the Conjugate of the Denominator

To simplify, multiply both the numerator and the denominator by the complex conjugate of the denominator. The conjugate of \(3i\) is \(-3i\). Thus, we have:\[\frac{-2 + 6i}{3i} \cdot \frac{-3i}{-3i}\]

Step 3: Simplify the Denominator

The denominator becomes real after multiplying by its conjugate:\[3i \cdot (-3i) = 9i^2 = 9(-1) = -9\]

Step 4: Apply Multiplication to the Numerator

Distribute and simplify the numerator:\[(-2 + 6i)(-3i) = -2(-3i) + 6i(-3i) = 6i - 18i^2\]

Step 5: Simplify Using \(i^2 = -1\)

Replace \(i^2\) with \(-1\):\[6i - 18(-1) = 6i + 18\]

Step 6: Divide Numerator by Simplified Denominator

Now combine and divide each term by \(-9\):\[\frac{18 + 6i}{-9} = \frac{18}{-9} + \frac{6i}{-9} = -2 - \frac{2}{3}i\]

Step 7: Present the Final Answer

The expression \(-2 + 6i\) divided by \(3i\) is equivalent to \(-2 - \frac{2}{3}i\). Thus, \(a = -2\) and \(b = -\frac{2}{3}\).

Key Concepts

Imaginary Unit

To fully understand complex numbers, we must first grasp the concept of the imaginary unit, denoted as

• The imaginary unit is symbolized as \( i \), and is defined by the property that \( i^2 = -1 \).
• This definition is crucial because it forms the basis of complex numbers, allowing us to work with square roots of negative numbers.

Complex numbers are expressed in the form \( a + bi \), where \( a \) and \( b \) are real numbers. Here, \( a \) is the real part, while \( bi \) is the imaginary part. The ability to handle complex numbers, especially considering their imaginary components, enhances mathematical operations and solutions, especially in fields such as engineering and physics. When working with these numbers, keeping the equation \( i^2 = -1 \) in mind is essential, as it helps convert powers of \( i \) into straightforward numerical values.

Complex Conjugate

The complex conjugate is a fundamental aspect of simplifying and managing complex number expressions, especially during division.

• For any complex number \( a + bi \), its conjugate is \( a - bi \).
• The complex conjugate essentially "flips" the sign of the imaginary part, leaving the real part intact.

The importance of the complex conjugate arises largely in division. By multiplying the numerator and the denominator of a complex fraction by the conjugate of the denominator, we eliminate any imaginary component in the denominator.
For the example in the exercise, where the denominator is \( 3i \), its conjugate is \( -3i \). Multiplying both the numerator and the denominator by \( -3i \) helps transform the denominator into a real number, simplifying the division process.

Simplification

Simplification in the context of complex numbers often requires a few steps. The primary goal is to express complex fractions in the standard form \( a + bi \).

• Begin by rewriting the expression using the conjugate to make the denominator real.
• This involves multiplying both the numerator and the denominator of the fraction by the conjugate of the denominator.
• Simplify the mathematical expressions by using the property \( i^2 = -1 \).

An important note during simplification involves recognizing when terms involving \( i^2 \) appear. Since \( i^2 = -1 \), this substitution turns imaginary terms into real numbers.
This action is depicted in the exercise when \( 18i^2 \) turns into \( 18(-1) = -18 \). Such conversions are crucial for reducing expressions to their simplest forms, where all terms are easy to calculate and interpret as real and imaginary parts.

Division of Complex Numbers

Dividing complex numbers can initially seem tricky, but by using the conjugate, it becomes manageable.

• As shown in the exercise, the approach involves multiplying both the numerator and the denominator by the complex conjugate of the denominator.
• This process removes the imaginary part from the denominator, turning it into a real number, which simplifies the division.
• In the exercise solution, dividing by the simplified denominator \(-9\) is straightforward.

After eliminating the imaginary part from the denominator, the remaining real number allows for standard division.
Thereafter, divide each term of the resulting expression by this real denominator, providing a clean result in the form \( a + bi \) that is easy to interpret and use.