Problem 93 - Algebra for College Students

Question

Find each of the following quotients and express the answers in the standard form of a complex number. $$\frac{2+6 i}{1+7 i}$$

Step-by-Step Solution

Answer

\( \frac{22}{25} - \frac{4}{25}i \)

Step 1: Convert the Denominator and Numerator

To simplify the quotient \( \frac{2+6i}{1+7i} \), we will multiply both the numerator and the denominator by the conjugate of the denominator \(1+7i\). The conjugate of \(1+7i\) is \(1-7i\), so the expression becomes \( \frac{(2+6i)(1-7i)}{(1+7i)(1-7i)} \).

Step 2: Multiply Numerator

Multiply \((2+6i)\) by \((1-7i)\):- For real parts: \((2 \cdot 1) - (6i \cdot 7i) = 2 - 42(-1) = 2 + 42 = 44\).- For imaginary parts: \((6 \cdot 1) + (2 \cdot -7) = 6 - 14 = -8\). Combine to get the result of multiplying: \(44 - 8i\).

Step 3: Multiply Denominator

Multiply \((1+7i)\) by its conjugate \((1-7i)\):- For real part: \(1 \cdot 1 = 1\).- For imaginary parts: \(7i \cdot -7i = -49(-1) = 49\).Combine to get the result of multiplying: \(1 + 49 = 50\).

Step 4: Simplify to Find the Quotient

The quotient is now simplified to \( \frac{44 - 8i}{50} \). Divide both the real and imaginary parts by 50 to express the complex number in standard form:- Real part: \( \frac{44}{50} = \frac{22}{25} \).- Imaginary part: \( \frac{-8}{50} = \frac{-4}{25} \).So, the quotient in standard form is \( \frac{22}{25} - \frac{4}{25}i \).

Key Concepts

Standard Form of Complex Numbers

Complex numbers are often represented in the standard form, which is typically written as \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part. In this expression, \(i\) stands for the imaginary unit, which is defined by the property \(i^2 = -1\).
When expressing complex numbers in standard form, the goal is to clearly separate the real and imaginary components. For example, if we consider a complex number like \(3 + 4i\), \(3\) is the real part while \(4i\) is the imaginary part.
Understanding how to manipulate and express complex numbers in this form is essential for solving a variety of problems, including those that involve operations like addition, subtraction, and multiplication of complex numbers.

Conjugate of a Complex Number

The conjugate of a complex number is a key concept that helps in simplifying expressions, especially division involving complex numbers. For a complex number written in the form \(a + bi\), its conjugate is \(a - bi\).
This operation essentially flips the sign of the imaginary part. The importance of conjugates comes into play in division, as multiplying a complex number by its conjugate results in a real number. This is because the product of a complex number with its conjugate yields a difference of squares: - \((a + bi)(a - bi) = a^2 - (bi)^2 = a^2 + b^2\).
For example, the conjugate of \(1 + 7i\) is \(1 - 7i\). Using conjugates helps eliminate imaginary parts in the denominator, allowing for easier simplification to standard form.

Multiplying Complex Numbers

Multiplying complex numbers follows a straightforward process similar to expanding algebraic expressions. Using the distributive property, multiplying two complex numbers like \((c + di)(e + fi)\) involves multiplying each part separately:

• Real parts: \(c \cdot e\)
• Cross terms: \(c \cdot fi + di \cdot e\)
• Imaginary parts: \(di \cdot fi\)

Then, the imaginary unit \(i\) can be simplified using the fact that \(i^2 = -1\). For instance, - if multiplying \((2+6i)(1-7i)\), you compute: - Real part: \(2 \cdot 1 + 6(-7) = 44\) - Imaginary part: \(6\cdot 1 - 2 \cdot 7 = -8\), resulting in \(44 - 8i\).
This process helps to consolidate the expression into its simplest form, ready for simplification.

Simplifying Complex Expressions

Simplifying complex expressions is a pivotal step in working with complex numbers, especially in division. After applying operations like multiplication, the resulting expression may still not be in standard form.
To simplify, carefully separate the real and imaginary components. For a quotient like \(\frac{44 - 8i}{50}\), express each part by dividing separately:

• Real part: \(\frac{44}{50} = \frac{22}{25}\)
• Imaginary part: \(\frac{-8}{50} = \frac{-4}{25}\)

This division ensures that each coefficient of the real and imaginary parts is presented as a fraction, or a decimal to reflect the simplest form. Furthermore, reaching a final form like \(\frac{22}{25} - \frac{4}{25}i\) in standard form highlights the distinct separation of real and imaginary parts, essential for clear understanding and easy interpretation.