Problem 63 - A Graphical Approach to Precalculus with Limits

Question

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

Step-by-Step Solution

Answer

The product is 13.

Step 1: Identify the Formula

The expression given is \((2 + 3i)(2 - 3i)\). This is a product of two complex conjugates, which can be simplified using the difference of squares formula: \((a + bi)(a - bi) = a^2 + b^2\).

Step 2: Apply the Formula

Substitute \(a = 2\) and \(b = 3\) into the formula: \[a^2 + b^2 = (2)^2 + (3)^2\].

Step 3: Calculate Each Term

Calculate \(a^2\) as \((2)^2 = 4\) and \(b^2\) as \((3)^2 = 9\).

Step 4: Add the Results

Combine the results: \(4 + 9 = 13\).

Step 5: Express in Standard Form

The product is \((2 + 3i)(2 - 3i) = 13\). This is already in standard form for a complex number, where there is no imaginary part.

Key Concepts

Complex Conjugates

Complex conjugates are a pair of complex numbers that have identical real parts but opposite imaginary parts. Essentially, if you have a complex number expressed as \(a + bi\), its complex conjugate will be \(a - bi\). Here, \(i\) represents the imaginary unit, which satisfies \(i^2 = -1\).

This concept is useful in many mathematical operations, particularly in simplifying expressions involving complex numbers. When you multiply a complex number by its conjugate, the resulting product has a unique property: it is a real number. This is because the imaginary parts cancel each other out. For example, in the given expression \((2+3i)(2-3i)\), the complex conjugate of \(2+3i\) is \(2-3i\).

• The real parts (2s in this case) remain the same.
• The imaginary parts (3i and -3i) are negatives of each other.

This conjugate property simplifies certain calculations and leads us directly to the use of the difference of squares.

Difference of Squares

The difference of squares is a powerful algebraic identity often used to simplify expressions involving complex numbers. It states that for any expressions \(a\) and \(b\), \((a + b)(a - b) = a^2 - b^2\). In the realm of complex numbers, however, when dealing with complex conjugates \(a+bi\) and \(a-bi\), this formula adapts slightly to become \(a^2 + b^2\).

The reason this formula works is due to the cancellation of the imaginary parts during multiplication. Let's explore this further using our example \((2+3i)(2-3i)\):

• First, identify \(a = 2\) and \(b = 3\).
• Apply the formula: \(a^2 + b^2 = 2^2 + 3^2\).
• On calculating, it simplifies to \(4 + 9 = 13\).

This gives us a real number, aligning with the earlier mentioned property of complex conjugates. Thus, the difference of squares is a powerful tool when multiplying complex conjugates.

Standard Form

The standard form for complex numbers is \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. The goal of most exercises involving complex numbers, especially multiplication, is to bring the result back into this standard form.

In our completed calculation, we arrived at 13 after multiplying the conjugates in \((2 + 3i)(2 - 3i)\). As there is no imaginary component in this result, the number is already in standard form. This is because the imaginary parts have cancelled each other out, leaving us with a purely real number.

• In cases where a result is still complex, you typically separate the real and imaginary parts.
• If from a conjugate multiplication, however, the result usually simplifies to a real number, as demonstrated.

Thus, understanding the standard form helps in visually interpreting the result of complex processes and calculations.