Problem 45 - College Algebra and Calculus: An Applied Approach

Question

Write the quotient in standard form. $$\frac{3-i}{3+i}$$

Step-by-Step Solution

Answer

The quotient in standard form is \(0.8 - 0.6i\)

Step 1: Identify the Conjugate

Recognize that the denominator \(3 + i\) is a complex number. The complex conjugate of this number is \(3 - i\). This is obtained by changing the sign of the imaginary part.

Step 2: Multiply by the Conjugate

Multiply both the numerator and the denominator by the conjugate \(3 - i\). This will give: \(\frac{(3-i)(3-i)}{(3+i)(3-i)}\)

Step 3: Expand the Expressions

Expand the expressions in the numerator and the denominator. For the numerator: \((3-i)(3-i) = 9 - 3i - 3i + i^2 = 9 -6i -1 = 8 - 6i\). For the denominator: \((3 + i)(3 - i) = 9 - 3i + 3i - i^2 = 9 + 1 = 10\) since \(i^2 = -1\)

Step 4: Write in Standard Form

Divide the real and the imaginary parts of the numerator by the real denominator to write the result in standard form. This gives \(\frac{8 - 6i}{10} = 0.8 - 0.6i\)

Key Concepts

Complex Conjugate

When working with complex numbers, the concept of a complex conjugate is essential for various operations, especially for simplifying complex fractions. A complex conjugate of a complex number is formed by changing the sign of the imaginary part. For example, if we have a complex number like \( a + bi \), where \( a \) is the real part and \( bi \) is the imaginary part, the complex conjugate is \( a - bi \).

In practice, complex conjugates are used to eliminate the imaginary unit from the denominator of a fraction. This is a critical step because it transforms a complex fraction into a form that is easier to understand and work with. By multiplying the numerator and the denominator of the fraction by the complex conjugate of the denominator, the denominator becomes a real number, simplifying the complex fraction significantly.

Simplifying Complex Fractions

Complex fractions can be intimidating at first, but they follow rules similar to those of real-number fractions. To simplify a complex fraction, the goal is to make the denominator real. One of the most effective methods for achieving this is through multiplication by the complex conjugate, as mentioned earlier.

By multiplying both the numerator and the denominator by the complex conjugate of the denominator, the denominator simplifies to a real number because the product of a complex number and its conjugate is always real. This process effectively removes the imaginary unit from the denominator and simplifies the fraction into an algebraic expression with real coefficients, which can be further reduced to standard form for complex numbers.

Imaginary Unit i

The imaginary unit \( i \) is a fundamental concept in the study of complex numbers. It is defined as the square root of minus one, \( i = \sqrt{-1} \). The significance of \( i \) lies in its ability to represent the square root of a negative number, which is not possible within the real numbers.

The operation with \( i \) must adhere to specific rules, such as \( i^2 = -1 \). These rules must be applied when expanding algebraic expressions involving complex numbers. Understanding the nature of \( i \) helps students to manipulate and simplify expressions with complex numbers and is essential for working with quadratic equations that have no real solutions.

Algebraic Expressions

Algebraic expressions involving complex numbers are not much different from those involving real numbers. The primary difference lies in the inclusion of the imaginary unit \( i \). When dealing with such expressions, it is important to combine like terms, just as you would with real numbers. Real parts are combined with real parts, and imaginary parts with imaginary parts.

When multiplying or expanding expressions involving complex numbers, distribute the terms as usual and apply the definition of \( i \), remembering that \( i^2 = -1 \). This allows the expression to be simplified and if it's a complex fraction, to eventually convert it to standard form, which is written as \( a + bi \) where \( a \) and \( b \) are real numbers. Simplifying algebraic expressions requires careful attention to these rules to arrive at a correct and simplified result.