Problem 73 - Intermediate Algebra

Question

Perform each indicated operation. Write the result in the form \(a+b i\). $$ \frac{2-3 i}{2+i} $$

Step-by-Step Solution

Answer

The result is \(\frac{1}{5} - \frac{8}{5}i\).

Step 1: Identify the Complex Conjugate

To simplify the fraction \(\frac{2-3i}{2+i}\), we need to multiply the numerator and the denominator by the complex conjugate of the denominator, which is \(2-i\).

Step 2: Multiply the Numerator and Denominator

Multiply both the numerator (\(2-3i\)) and the denominator (\(2+i\)) by the complex conjugate \(2-i\): \[\frac{(2-3i)(2-i)}{(2+i)(2-i)}\].

Step 3: Simplify the Numerator

Expand \((2-3i)(2-i)\): Apply the distributive property:\[2 \cdot 2 + 2 \cdot (-i) - 3i \cdot 2 - 3i \cdot (-i)\] Calculate the terms: = \(4 - 2i - 6i + 3i^2\). Since \(i^2 = -1\), this becomes: \(4 - 8i + 3(-1) = 4 - 8i - 3\)= \(1 - 8i\).

Step 4: Simplify the Denominator

Expand \((2+i)(2-i)\): Using the difference of squares formula: \[2^2 - (i)^2 = 4 - (-1) = 4 + 1 = 5\].

Step 5: Write the Result in Simplified Form

The fraction simplifies to:\[\frac{1-8i}{5}\]We can express it in the form \(a + bi\) by dividing each term by 5: \[\frac{1}{5} - \frac{8}{5}i\].So, the final result is \(\frac{1}{5} - \frac{8}{5}i\).

Key Concepts

Understanding the Complex Conjugate

Complex numbers have both a real part and an imaginary part. A complex conjugate is a unique way of transforming a complex number to help in simplifying various expressions. When a complex number is written as \( a+bi \), its complex conjugate is written as \( a-bi \). The complex conjugate essentially flips the sign of the imaginary part, leaving the real part unchanged. For example, the complex conjugate of \(2+i\) is \(2-i\). Using complex conjugates is especially useful in division, as they help eliminate the imaginary part in the denominator. To multiply a complex number by its conjugate, you obtain a purely real number. This is achieved because the imaginary components cancel out, and the result is based on the formula \((a+b)(a-b) = a^2 - b^2\), which exploits the properties of squared terms.

Mastering Multiplying Complex Numbers

When it comes to multiplying complex numbers, the distributive property is your go-to method. Consider two complex numbers: \((2-3i)\) and \((2-i)\). To multiply them, distribute each term in the first complex number across each term in the second.

• First, multiply the real parts: \(2 \times 2 = 4\)
• Next, take the real part times the imaginary part: \(2 \times (-i) = -2i\)
• Then, multiply the imaginary part times the real part: \(-3i \times 2 = -6i\)
• Finally, multiply the imaginary parts: \(-3i \times (-i) = 3i^2\)

Since \(i^2 = -1\), that last part simplifies to \(3 \times -1 = -3\). Simply add up all these results: \(4 - 2i - 6i - 3 = 1 - 8i\). So the product of \((2-3i)(2-i)\) results in the complex number \(1-8i\). When multiplying, always remember the rule of \(i^2 = -1\), as it's crucial for simplifying your results efficiently.

Simplifying Complex Fractions

Complex fractions might initially look complicated, but with the right approach, they become manageable. The key is unraveling the fraction into a form where complex numbers appear as separate terms \(a + bi\). In our example, \(\frac{1-8i}{5}\), each of these terms can be further simplified by dividing through the denominator separately:

• The real component is \(\frac{1}{5}\)
• The imaginary component is \(\frac{-8}{5}i\)

Thus, the expression becomes \(\frac{1}{5} - \frac{8}{5}i\). By separating these terms, you write the complex fraction in the standard complex number form \(a+bi\). This separation not only clarifies the expression's structure but also makes it easier to understand and utilize in further mathematical operations. With practice, simplifying complex fractions will become a reliable tool in your problem-solving toolbox.