Problem 104 - A Graphical Approach to College Algebra

Question

Divide as indicated. Write each quotient in standard form. $$\frac{2-i}{2+i}$$

Step-by-Step Solution

Answer

\(\frac{3}{5} - \frac{4}{5}i\)

Step 1: Write the Division Expression

We start with the division expression \( \frac{2-i}{2+i} \). Our goal is to divide the numerator by the denominator and simplify it to the standard form \(a + bi\).

Step 2: Multiply by the Conjugate

To eliminate the imaginary component in the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2+i\) is \(2-i\). Thus, multiply:\[\frac{2-i}{2+i} \times \frac{2-i}{2-i} = \frac{(2-i)(2-i)}{(2+i)(2-i)}\]

Step 3: Expand the Numerator and Denominator

Expand both the numerator and denominator using the distributive property (or FOIL method). For the numerator:\[ (2-i)(2-i) = 2*2 - 2i - 2i + i^2 = 4 - 4i + (-1) = 3 - 4i \]For the denominator:\[ (2+i)(2-i) = 2*2 - i^2 = 4 - (-1) = 4 + 1 = 5 \]

Step 4: Simplify the Expression

Now that we have \( \frac{3-4i}{5} \), we divide each term in the numerator by 5:\[\frac{3}{5} - \frac{4}{5}i \]This gives the quotient in standard form \(\frac{3}{5} - \frac{4}{5}i\).

Key Concepts

Standard Form

When dealing with complex numbers, it is important to understand the standard form. The standard form of a complex number is expressed as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.

• The real component is \(a\).
• The imaginary component is represented as \(bi\).

It acts as a standard notation that everyone can follow, making complex numbers easy to add, subtract, multiply, and divide. This form helps you visualize complex numbers as points or vectors on a two-dimensional plane, with the real number being the horizontal component and the imaginary number being the vertical component. To convert a complex division into this form, you multiply the numerator and denominator by the conjugate of the denominator, as shown in the exercise. This helps eliminate imaginary units in the denominator and results in a simple quotient in the \(a + bi\) format.

Imaginary Component

The imaginary component of a complex number involves the imaginary unit \(i\), where \(i\) is defined as the square root of -1. Complex numbers are unique because they have both real and imaginary components.In the division problem \(\frac{2-i}{2+i}\), \(-i\) in the numerator and \(+i\) in the denominator are the imaginary parts. We can't have an imaginary number in the denominator when expressing in standard form, as it needs to be a pure real number.The imaginary component, indicated by \(bi\) in \(a + bi\), represents values that extend perpendicular to the real-number line. Think of it as adding a whole new dimension to numbers! By manipulating the imaginary component with operations such as conjugation, you can simplify complex expressions. In this exercise, multiplying the numerator and denominator by \(2-i\) eliminates the imaginary component in the denominator.

Conjugate

A conjugate in the context of complex numbers is a way to transform or manipulate these numbers. For any complex number \(a + bi\), its conjugate is \(a - bi\). Basically, you switch the sign of the imaginary part. Understanding conjugates is vital because they let us manage the imaginary components in fractions. By multiplying by the conjugate, you can effectively remove imaginary numbers from certain parts of your calculations. For instance, in \(\frac{2-i}{2+i}\), you multiply the transaction by the conjugate of \(2+i\), which is \(2-i\). This strategy turns the denominator into a real number (since \((2+i)(2-i) = 5\)), allowing you to express the problem in the standard form of a complex number!This operation is key in not only division but also simplifying and solving equations with complex numbers, especially when you want to have the expression in an easily understandable form like \(a + bi\).