Problem 104 - A Graphical Approach to Precalculus with Limits

Question

Find the complex conjugate. $$\frac{2-i}{2+i}$$

Step-by-Step Solution

Answer

The complex conjugate is \( \frac{3}{5} + \frac{4}{5}i \).

Step 1: Identify the problem

We need to find the complex conjugate of the given expression \( \frac{2-i}{2+i} \). The complex conjugate of a complex number changes the sign of its imaginary part.

Step 2: Multiply numerator and denominator by the complex conjugate of the denominator

The complex conjugate of the denominator \(2+i\) is \(2-i\). We will multiply both the numerator and the denominator by \(2-i\) to eliminate the imaginary part in the denominator: \[\frac{(2-i)(2-i)}{(2+i)(2-i)}\].

Step 3: Simplify the denominator using the difference of squares

The product \((2+i)(2-i)\) can be simplified using the difference of squares formula: \[(a+b)(a-b) = a^2 - b^2\].Apply it: \[ (2)^2 - (i)^2 = 4 - (-1) = 5 \].Thus, the denominator becomes \(5\).

Step 4: Simplify the numerator by expanding

Expand the numerator \((2-i)(2-i)\): \[(2-i)(2-i) = 2^2 - 2 \cdot i - 2 \cdot i + i^2 = 4 - 4i + (-1) = 3 - 4i \].

Step 5: Write the simplified expression

After simplifying both the numerator and the denominator, we have: \[\frac{3 - 4i}{5} = \frac{3}{5} - \frac{4}{5}i\].

Step 6: Find the complex conjugate of the simplified expression

The complex conjugate of \( \frac{3}{5} - \frac{4}{5}i \) is obtained by changing the sign of the imaginary part: \[ \frac{3}{5} + \frac{4}{5}i\].

Key Concepts

Complex Numbers

Complex numbers are an extension of the number system we commonly use. They are composed of two parts: a real part and an imaginary part. The imaginary part is defined as a multiple of the imaginary unit, denoted as \(i\), where \(i\) is the square root of -1. Therefore, any complex number can be expressed in the form \(a + bi\), where \(a\) is the real part, and \(bi\) is the imaginary part, with \(b\) being a real number.
- Real part: The component of the complex number without the imaginary unit.
- Imaginary part: The component that includes \(i\), representing the imaginary aspect of the number.
Understanding complex numbers is crucial when dealing with various mathematical calculations, particularly those involving equations that do not possess a real solution. In our exercise, we are dealing with expressions constructed using these numbers. The key operation required was to find the complex conjugate. This involves changing the sign of the imaginary component. By doing this, we can eliminate the imaginary parts from expressions, thereby simplifying them.

Difference of Squares

The difference of squares is a special algebraic rule that applies to expressions of the form \((a+b)(a-b)\). This simplification results in \(a^2 - b^2\). Recognizing this pattern can greatly simplify calculations involving polynomial expressions.
- Formula: \((a+b)(a-b) = a^2 - b^2\).
- Simplification: Reduces multiplication of binomials to subtraction of squares.
In the provided exercise, this rule was employed to simplify the denominator such that it only contained a real number. By applying the difference of squares, the part of the expression with \((2+i)(2-i)\) transformed to \(4 - (-1)\). Here, \(a = 2\) and \(b = i\). After calculating \(4 - (-1)\), we derived the value 5 for the denominator, removing the imaginary component.

Simplification in Algebra

Simplification in algebra is the process whereby an expression is made easier to handle or understand. It often involves combining like terms, reducing fractions, and using algebraic identities to achieve a more manageable form.
- Like terms: Terms that have the same variable raised to the same power.
- Reducing fractions: Simplifying numerators and denominators by finding common factors.
In this exercise, the goal was to arrive at a simplified form of the given complex fraction. We achieved simplification through several crucial steps:

• Multiplying numerator and denominator by the complex conjugate of the denominator.
• Employing the difference of squares to simplify the denominator.
• Expanding the numerator to simplify its terms.

By simplifying the expression to the form \(\frac{3-4i}{5} = \frac{3}{5} - \frac{4}{5}i\), we stripped it of unnecessary complexity, making it much more straightforward to work with or interpret.