Problem 17 - Algebra 2 and Trigonometry

Question

In \(3-17,\) find each sum or difference of the complex numbers in \(a+b i\) form. $$ \left(\frac{3}{7}+\frac{1}{3} i\right)+\left(\frac{1}{6}-\frac{7}{8} i\right) $$

Step-by-Step Solution

Answer

The sum of the complex numbers is \( \frac{25}{42} - \frac{13}{24}i \).

Step 1: Identify Real and Imaginary Parts

First, identify the real and imaginary parts of the complex numbers. The first complex number is \( \frac{3}{7} + \frac{1}{3}i \), with real part \( \frac{3}{7} \) and imaginary part \( \frac{1}{3}i \). The second complex number is \( \frac{1}{6} - \frac{7}{8}i \), with real part \( \frac{1}{6} \) and imaginary part \( -\frac{7}{8}i \).

Step 2: Combine Real Parts

Add the real parts of the two complex numbers: \( \frac{3}{7} \) and \( \frac{1}{6} \). Find the common denominator first, which is 42, then add: \[ \frac{3}{7} + \frac{1}{6} = \frac{18}{42} + \frac{7}{42} = \frac{25}{42}. \]

Step 3: Combine Imaginary Parts

Next, add the imaginary parts of the two complex numbers: \( \frac{1}{3}i \) and \( -\frac{7}{8}i \). Find the common denominator, which is 24, then perform the addition: \[ \frac{1}{3}i - \frac{7}{8}i = \frac{8}{24}i - \frac{21}{24}i = -\frac{13}{24}i. \]

Step 4: Write the Result in Standard Form

The sum of the complex numbers is found by combining the results from Step 2 and Step 3. Thus, the sum is \( \frac{25}{42} - \frac{13}{24}i \).

Key Concepts

Complex Addition

Complex numbers extend the real number system by including an imaginary component, denoted as \( i \), where \( i^2 = -1 \). When adding complex numbers, each component—real and imaginary—is treated separately.
Imagine complex numbers as sums of real numbers and imaginary numbers:

• The real part is a typical number, like \( \frac{3}{7} \).
• The imaginary part is a number multiplied by \( i \), like \( \frac{1}{3} i \).

To add two complex numbers, simply add their real parts and imaginary parts separately.
Let's consider the complex numbers: - \( \frac{3}{7} + \frac{1}{3}i \) - \( \frac{1}{6} - \frac{7}{8}i \)
For the real parts, add \( \frac{3}{7} \) and \( \frac{1}{6} \). For the imaginary parts, add \( \frac{1}{3} i \) and \( -\frac{7}{8} i \). By handling each part of the complex numbers separately, complex addition becomes straightforward, allowing you to merge two intricate numbers into one.

Real and Imaginary Parts

Every complex number has two parts.

• The **real part** is akin to any ordinary number without the \( i \), such as \( \frac{3}{7} \) or \( \frac{1}{6} \).
• The **imaginary part** is a number attached to the imaginary unit \( i \), like \( \frac{1}{3} i \) or \( -\frac{7}{8} i \).

Each part follows its own rules in arithmetic operations. Real parts add or subtract just like regular fractions. Imaginary parts do as well, but you must keep the \( i \) factor.
When combining these parts, such as in our exercise, it's essential to handle denominators correctly to make them compatible. Thereby ensuring you perform accurate calculations when adding or subtracting fractions.
Dealing with both fractions and the imaginary unit often intimidates students initially. Yet, mastering these components reveals the elegant structure and surprising applications of complex numbers in mathematics.

Rational Arithmetic

Rational arithmetic involves working with fractions, which is crucial while dealing with the real and imaginary parts of complex numbers equations.
Fractions require you to find a common denominator before adding or subtracting, ensuring each fraction shares the same base.
For example, to add \( \frac{3}{7} \) and \( \frac{1}{6} \), the least common denominator is 42. You convert each fraction: \[ \frac{3}{7} = \frac{18}{42} \quad \text{and} \quad \frac{1}{6} = \frac{7}{42} \] Adding them gives \( \frac{25}{42} \). Rational arithmetic utilizes these conversions, making seemingly daunting operations systematic and manageable.
In the case of adding the imaginary parts \( \frac{1}{3}i \) and \( -\frac{7}{8}i \), the common denominator is 24: \[ \frac{1}{3}i = \frac{8}{24}i \quad \text{and} \quad -\frac{7}{8}i = -\frac{21}{24}i \] Thus, the result, \( -\frac{13}{24}i \), demonstrates how precise handling of fractions within complex arithmetic keeps computations clear and correct. Over time, students will find this method enormously beneficial in mathematical studies.