Problem 23 - Precalculus: Functions and Graphs

Question

Write the expression in the form \(a+b i,\) where \(a\) and \(b\) are real numbers. $$\frac{-4+6 i}{2+7 i}$$

Step-by-Step Solution

Answer

The expression is \(\frac{34}{53} + \frac{40}{53}i\).

Step 1: Identify the Complex Conjugate

To convert a complex fraction into the form \(a + bi\), multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 + 7i\) is \(2 - 7i\).

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by \(2 - 7i\):\[\frac{-4 + 6i}{2 + 7i} \times \frac{2 - 7i}{2 - 7i} = \frac{(-4 + 6i)(2 - 7i)}{(2 + 7i)(2 - 7i)}\]

Step 3: Expand the Numerator

Use the distributive property (FOIL method) to expand the numerator:\[(-4 + 6i)(2 - 7i) = (-4 \cdot 2) + (-4 \cdot -7i) + (6i \cdot 2) + (6i \cdot -7i)\]Calculating gives:\[-8 + 28i + 12i - 42i^2 = -8 + 40i + 42\]

Step 4: Simplify using \(i^2 = -1\)

Since \(i^2 = -1\), replace \(-42i^2\) with \(+42\). Thus, the numerator simplifies to:\[-8 + 40i + 42 = 34 + 40i\]

Step 5: Calculate the Denominator

To simplify the denominator \((2 + 7i)(2 - 7i)\), use the formula \((a + bi)(a - bi) = a^2 + b^2\):\[(2 + 7i)(2 - 7i) = 2^2 + 7^2 = 4 + 49 = 53\]

Step 6: Write in Standard Form

Combine the results from the expanded numerator and simplified denominator to finish simplifying the expression:\[\frac{34 + 40i}{53} = \frac{34}{53} + \frac{40}{53}i \rightarrow a + bi\]

Step 7: Final Expression

The expression \(\frac{-4 + 6i}{2 + 7i}\) in the form \(a + bi\) is:\[\frac{34}{53} + \frac{40}{53}i\]

Key Concepts

Complex Conjugate

In complex numbers, every complex number has a complex conjugate. A complex conjugate is created by changing the sign between the real and imaginary parts of a complex number. For example, the conjugate of the complex number \(2 + 7i\) is \(2 - 7i\).

• To find the conjugate of any complex number, if the number is in the form \(a + bi\), then its conjugate is \(a - bi\).

Multiplying by the conjugate is a common technique to remove the imaginary part from the denominator of a complex fraction. It simplifies the expression by making the denominator a real number, which is quite helpful in expressing complex fractions in standard form \(a + bi\).

Using the complex conjugate in division ensures you are working with real numbers in the denominator, enabling you to clearly see the real and imaginary components of the result.

Distributive Property

The distributive property is an important rule in algebra that allows you to multiply a sum by a number. It's expressed generally as \(a(b + c) = ab + ac\).

When dealing with complex numbers, this property plays a crucial role in multiplying expressions. For example, when multiplying \((-4 + 6i)(2 - 7i)\), each term in the first expression must be multiplied by each term in the second expression. Thus, applying the distributive property makes it clear:

• \((-4 \cdot 2) + (-4 \cdot -7i) + (6i \cdot 2) + (6i \cdot -7i)\)

This breaks the multiplication down into manageable parts, ensuring that every component is accounted for. The distributive property facilitates the process of expanding and simplifying complex polynomials.

FOIL Method

The FOIL method is a specific application of the distributive property, used primarily when multiplying two binomials, like \((a + b)(c + d)\). FOIL stands for First, Outside, Inside, Last, referring to the order in which you multiply the terms:

• **First**: Multiply the first terms from each binomial.
• **Outside**: Multiply the outermost terms.
• **Inside**: Multiply the inner terms.
• **Last**: Multiply the last terms from each binomial.

With complex numbers, using FOIL helps to simplify expressions by calculating each pair separately. For example, with \((-4 + 6i)(2 - 7i)\), you compute:

• **First**: \(-4 \cdot 2 = -8\)
• **Outside**: \(-4 \cdot -7i = 28i\)
• **Inside**: \(6i \cdot 2 = 12i\)
• **Last**: \(6i \cdot -7i = -42i^2\)

This method ensures all terms are correctly multiplied and helps in collecting like terms, easing the simplification in problems involving complex numbers.

Imaginary Unit

The imaginary unit \(i\) is a fundamental concept in complex numbers, defined as \(i = \sqrt{-1}\). This means \(i^2 = -1\), making the arithmetic with complex numbers unique.

• The imaginary unit allows for the extension of the real number system to include solutions to equations that don't exist within the real numbers alone, such as \(x^2 + 1 = 0\).

When multiplying complex numbers, expressions involving \(i^2\) often appear. Remembering that \(i^2 = -1\) helps you convert quadratic terms involving \(i\) into real numbers. For instance, in the expression \(-42i^2\), substituting \(i^2\) with \(-1\) changes the term to \(+42\), which simplifies the computation.Understanding the imaginary unit and its properties is essential when dealing with complex algebraic expressions to ensure precise mathematical manipulations.