Problem 14 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$(4+9 i)(4-9 i)$$

Step-by-Step Solution

Answer

The expression simplifies to 97.

Step 1: Identify the Form

The given expression is in the form \( (a+bi)(a-bi) \), where \( a = 4 \) and \( b = 9 \). This matches the formula for the difference of squares of complex numbers.

Step 2: Apply the Difference of Squares Formula

Use the identity \((a+bi)(a-bi) = a^2 - (bi)^2\). Here, \( a = 4 \) and \( b = 9 \).

Step 3: Calculate \(a^2\)

Find the square of the real part: \( a^2 = 4^2 = 16 \).

Step 4: Calculate \((bi)^2\)

Compute the square of the imaginary part: \((bi)^2 = (9i)^2 = 81i^2\). Recall that \( i^2 = -1 \), so \( 81i^2 = 81(-1) = -81 \).

Step 5: Combine the Results

Subtract the result of the imaginary square from the real square: \( 16 - (-81) = 16 + 81 = 97. \)The expression in the form \( a + bi \) is \( 97 + 0i \), which simplifies to \( 97 \) since the imaginary part is zero.

Key Concepts

Difference of Squares

The difference of squares is a mathematical technique used to simplify expressions by utilizing the identity \( (a+b)(a-b) = a^2 - b^2 \). In the context of complex numbers, this identity also applies when dealing with expressions like \((a+bi)(a-bi)\), where the portions \(bi\) and \(-bi\) represent the imaginary components.

• The formula transforms into \(a^2 - (bi)^2\).
• This can further simplify complex expressions into a form that reveals their real and imaginary parts clearly.

In our original exercise, we applied this formula to calculate \( (4+9i)(4-9i) \) as \( 4^2 - (9i)^2 \). By doing so, we simplified the equation to its most reduced form in terms of real and imaginary components.

Imaginary Unit

The imaginary unit, denoted by \(i\), is a fundamental concept in complex numbers. It is defined by the property \(i^2 = -1\). This unique characteristic allows us to work with numbers that aren't purely real.

• It helps in converting negative squares, a concept that doesn't exist in the realm of real numbers, into meaningful quantities.
• When squaring an imaginary number, \((bi)^2\), it transforms through \(b^2i^2 = b^2(-1)\), ultimately turning a negative imaginary squared value into a real one.

In our step-by-step solution, the term \((9i)^2\) becomes \(-81\) by using the property of \(i\). Hence, understanding the behavior of \(i\) is essential in solving and simplifying complex number expressions.

Real and Imaginary Parts

In complex numbers, every number is expressed in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Identifying and separating these parts help in simplifying and working with complex expressions.

• The real part represents the position of the number along the horizontal axis in the complex plane.
• The imaginary part influences the vertical axis, showcasing the number's imaginary magnitude.

When solving the exercise, rewriting the expression as \(97 + 0i\) highlights that our solved expression has a real part of 97 and an imaginary part of 0. This effectively means the final expression is a real number. Being able to distinguish these parts allows for greater maneuverability in calculations involving complex numbers.