Problem 63 - A Graphical Approach to College Algebra

Question

Multiply as indicated. Write each product in standard form. $$(2+3 i)(2-3 i)$$

Step-by-Step Solution

Answer

The product is -5.

Step 1: Recognize the Pattern

The expression \((2+3i)(2-3i)\) is a product of two conjugates. In general, \((a+bi)(a-bi)\) equals \(a^2+b^2i^2\).

Step 2: Multiply the Conjugates

Using the formula for multiplying conjugates: \((2)^2 + (3i)^2 = 4 + 9(-1)\).

Step 3: Simplify the Expression

Calculate to obtain \(4 - 9 = -5\).

Step 4: Write in Standard Form

The standard form of a complex number is \(a + bi\), where \(b = 0\) in this case. Thus, the final result is \(-5\).

Key Concepts

Conjugates of Complex Numbers

Conjugates are pairs of complex numbers that have the same real part but opposite imaginary parts. Understanding conjugates is key when working with complex numbers, especially in multiplication and division.

• Definition: If you have a complex number, for example, \( a + bi \), its conjugate is \( a - bi \).
• Importance: The product of a complex number and its conjugate always results in a real number. This is because the imaginary parts cancel each other out.
• Example: Consider \( (2 + 3i) \) and its conjugate \( (2 - 3i) \). Their product is \( (2)^2 + (3i)^2 = 4 + 9(-1) = 4 - 9 = -5 \), which is a real number.

Applying conjugates simplifies complex number calculations and is especially useful in rationalizing denominators.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part.

• Simplified Understanding: Think of it as a way to write complex numbers that highlights both their real and imaginary components.
• Why It Matters: Writing complex numbers in standard form makes them easier to work with in operations like addition, subtraction, and multiplication.
• Example: After performing the steps of multiplication or addition, ensure the result is in standard form. For the expression \( (2 + 3i)(2 - 3i) \), after simplifying, you get \( -5 + 0i \) or simply \( -5 \), which is still considered a valid standard form with no imaginary component.

Using standard form is crucial for recognizing the distinct contributions of the real and imaginary parts.

The Imaginary Unit (i)

The imaginary unit, denoted as \( i \), is a fundamental component in complex numbers. It's defined by the property that \( i^2 = -1 \).

• Definition: \( i \) is used to represent the square root of -1, solving equations that don't have real solutions.
• Role in Complex Numbers: A complex number generally looks like \( a + bi \), where \( b \) is multiplied by \( i \), indicating the imaginary part.
• Example in Use: In the problem expression \( (2 + 3i)(2 - 3i) \), the term \( (3i)^2 \) evaluates to 9\( i^2 \). Given that \( i^2 = -1 \), this becomes \( 9(-1) = -9 \).

Knowing \( i \) and its properties is essential for understanding complex numbers as it allows extending real-number concepts into new dimensions of mathematical exploration.