Problem 23 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 1-34: 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 simplifies to \( \frac{34}{53} + \frac{40}{53}i \).

Step 1: Identify the Expression

The given expression is \( \frac{-4+6i}{2+7i} \). We need to simplify it to the form \( a + bi \), where \( a \) and \( b \) are real numbers.

Step 2: Use the Conjugate

To eliminate the imaginary number in the denominator, multiply the numerator and the denominator by the conjugate of the denominator, which is \( 2 - 7i \).

Step 3: Multiply the Expressions

Multiply the numerator and denominator:\[ (-4 + 6i)(2 - 7i) \] and \[ (2 + 7i)(2 - 7i) \].

Step 4: Simplify the Numerator

Using the distributive property (FOIL method), expand the numerator:\(((-4)(2) + (-4)(-7i) + (6i)(2) + (6i)(-7i)) = -8 + 28i + 12i - 42i^2\).Recall that \( i^2 = -1 \). Thus, \(-42i^2 = 42\).Therefore, the simplified numerator is \( 34 + 40i \).

Step 5: Simplify the Denominator

Expand and simplify the denominator using the difference of squares:\((2 + 7i)(2 - 7i) = 4 - (7i)^2 = 4 + 49 = 53\).

Step 6: Divide Terms

The fraction now becomes:\( \frac{34 + 40i}{53} \).Divide both the real and imaginary parts by 53:\( \frac{34}{53} + \frac{40i}{53} \).

Step 7: Write in the Form of a + bi

The expression is now written as:\(\frac{34}{53} + \frac{40}{53}i\).Identify \( a = \frac{34}{53} \) and \( b = \frac{40}{53} \).

Key Concepts

Imaginary Numbers

An imaginary number is a fascinating concept in mathematics that helps us deal effectively with negative square roots. They are generally represented using the symbol \( i \), where \( i \) is the imaginary unit defined as \( \sqrt{-1} \). This concept allows us to extend the real number system to include numbers that can be used when solving equations like \( x^2 + 1 = 0 \).
For example, multiplying imaginary numbers results in interesting outcomes. Remember when \( i \) is squared, it becomes \( i^2 = -1 \). Adding real numbers to these creates complex numbers, exemplified by expressions like \( a + bi \), where \( a \) and \( b \) are real.

• Real numbers like 3 or -5 are familiar entities along the number line.
• Imaginary numbers fall outside the real line and are essential in engineering and physics.

When dealing with expressions like \( \frac{-4+6i}{2+7i} \), recognizing \( i \) as the core of the imaginary number is crucial in manipulation and simplification.

Conjugate

The conjugate of a complex number helps us simplify expressions, especially useful in divisions involving imaginary numbers. The conjugate of a complex number \( a + bi \) is denoted \( a - bi \). Essentially, you keep the real part the same but reverse the sign of the imaginary portion.
The importance of conjugates comes to light when you aim to eliminate the imaginary component from a denominator. By multiplying both numerator and denominator by the conjugate, you transform the denominator into a real number. For example, multiplying \( 2+7i \) by its conjugate \( 2-7i \) gives:

• Results in \( (2+7i)(2-7i) = 4 - 49i^2 = 4 + 49 = 53 \)
• Eliminates the imaginary, making division straightforward.

It’s a handy trick when simplifying any complex expression into its \( a + bi \) form.

FOIL Method

The FOIL method is a straightforward technique used to expand expressions involving binomials, especially applicable in multiplying complex numbers. The name FOIL stands for First, Outer, Inner, Last, reminding us of the sequence:

• First terms in each binomial are multiplied.
• Outer terms (one from each binomial).
• Inner terms.
• Last terms in each binomial.

In the context of complex numbers, this method helps to expand expressions like \( (a+bi)(c+di) \). For example, \((-4 + 6i)(2 - 7i)\) deploys this method, yielding:

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

Following these steps, the result simplifies to \(34 + 40i\), making further calculations possible.

Real Numbers

Real numbers embody the familiar world of everyday numbers, including those we use in desks, markets, and classrooms. They are the building blocks of both simple and complex mathematical expressions. This set includes:

• Positive numbers like 5, 3.14 (not just whole numbers).
• Negative numbers such as \(-\frac{2}{3}\), \(-1\), representing opposites.
• Zero, serving as the neutral element of addition.

In a complex number, such as \(a+bi\), \(a\) is the real number portion.
Understanding real numbers is vital since they form half of the story in any complex number scenario. When dividing expressions like \( \frac{-4+6i}{2+7i} \), reaching a real number in the denominator enhances manageability. After simplification, the expression \( \frac{34}{53} + \frac{40}{53}i \) reveals its real components, \(a = \frac{34}{53}\). It guides us toward writing numbers in the pure \(a+bi\) format, connecting back to simpler arithmetic concepts.