Problem 102 - A Graphical Approach to Precalculus with Limits

Question

Find the complex conjugate. $$\frac{-6+8 i}{1-i}$$

Step-by-Step Solution

Answer

The complex conjugate is \(-7 - i\).

Step 1: Identify the expression

The expression given is \( \frac{-6 + 8i}{1 - i} \). Our task is to find the complex conjugate of this expression.

Step 2: Simplify the denominator

First, we need to make the denominator a real number. To do this, multiply both the numerator and the denominator by the complex conjugate of the denominator, which is \( 1 + i \). This gives us \( \frac{(-6 + 8i)(1 + i)}{(1 - i)(1 + i)} \).

Step 3: Simplify the denominator further

The denominator \( (1 - i)(1 + i) \) simplifies as follows: \( 1^2 - (i)^2 = 1 - (-1) = 2 \). Thus, the denominator becomes 2.

Step 4: Expand the numerator

Multiply the numerators: \((-6 + 8i)(1 + i) = -6(1) - 6(i) + 8i(1) + 8i^2\). Simplify it to \(-6 + 2i + 8(-1) = -6 + 2i - 8\), which simplifies further to \(-14 + 2i\).

Step 5: Combine terms

Now the expression is \( \frac{-14 + 2i}{2} \). Simplifying by dividing each term by 2 gives us \(-7 + i\).

Step 6: Find the complex conjugate

The complex conjugate of a complex number \(a + bi\) is \(a - bi\). Therefore, the complex conjugate of \(-7 + i\) is \(-7 - i\).

Key Concepts

Complex Numbers

Complex numbers are numbers that have both a real and an imaginary part. Typically, they are expressed in the form \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part. The imaginary unit \( i \) is defined as the square root of \(-1\), which allows for operations involving imaginary numbers. Complex numbers are vital in advanced mathematics and applied fields.

• The real part, \( a \), means the regular number we are accustomed to on the number line.
• The imaginary part, \( bi \), includes a factor of \( i \), representing the value multiplied by \( \sqrt{-1} \).

Drawing parallels between complex numbers and the coordinate plane can help; the real part \( a \) lies on the x-axis, and the imaginary part \( bi \) lies on the y-axis. Understanding this dual nature allows mathematicians to solve equations that do not have real number solutions.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying expressions using mathematical operations and rules. It involves breaking down complex scenarios into simpler parts to find solutions or make sense of equations. When dealing with complex numbers, algebraic manipulation can involve several steps.

• Identify elements within an expression.
• Recognize the need for simplifying parts of the expression, such as separating the real and imaginary components.
• Use conjugates to help eliminate imaginary parts in denominators.

By practicing these steps, you can gain the ability to handle more complicated expressions in a structured manner, ultimately leading to the correct result.

Simplification

Simplification is the process of reducing a mathematical expression to its simplest form. The goal is to make the expression easier to interpret or solve. In the context of complex numbers, simplification often involves eliminating imaginary units from the denominator.

• Combine like terms to reduce expressions.
• In complex fractions, simplify by multiplying by the conjugate to turn the denominator into a real number.
• Finally, divide all terms by any common factor to achieve the simplest form.

This process is critical in complex number manipulation, allowing more efficient computation and interpretation of the expression.

Conjugate Multiplication

Conjugate multiplication is a technique used to eliminate the imaginary part in a denominator. A complex conjugate of a complex number \( a + bi \) is \( a - bi \). It has the effect of turning a complex number into a real number when multiplied appropriately.

• Identify the conjugate of the denominator in complex fractions.
• Multiply both the numerator and denominator by this conjugate.
• This results in a real number in the denominator, simplifying further calculations.

Overall, conjugate multiplication is a powerful tool in algebra and calculus, helping to transform and interpret complex expressions, which leads to easier computation and clearer solutions.