Problem 24 - A Graphical Approach to Precalculus with Limits

Question

Find the sum of each pair of complex numbers. Express your answer in rectangular form. Do not use a calculator. $$7+6 i, 3 i$$

Step-by-Step Solution

Answer

The sum of \(7+6i\) and \(3i\) is \(7+9i\) in rectangular form.

Step 1: Understand the Problem

We need to find the sum of two complex numbers: \( 7 + 6i \) and \( 3i \). Complex numbers are typically expressed in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part.

Step 2: Identify the Real and Imaginary Parts

For the first complex number \( 7 + 6i \), the real part is \( 7 \) and the imaginary part is \( 6i \). For the second complex number \( 3i \), the real part is \( 0 \) and the imaginary part is \( 3i \).

Step 3: Sum the Real Parts

To find the sum of these complex numbers, we first add their real parts: \( 7 + 0 = 7 \).

Step 4: Sum the Imaginary Parts

Next, add the imaginary parts: \( 6i + 3i = 9i \).

Step 5: Combine the Results

Combine the results from Step 3 and Step 4 to find the sum of the two complex numbers: \( 7 + 9i \).

Key Concepts

Rectangular Form

When dealing with complex numbers, the rectangular form is one of the most common representations. This form expresses complex numbers as the sum of two parts: real and imaginary. The notation typically used is \( a + bi \), where:

• \( a \) represents the real part of the complex number.
• \( bi \) represents the imaginary part, with \( b \) as a real number and \( i \) as the imaginary unit.

Complex numbers in rectangular form easily allow for addition and subtraction. When summing complex numbers, simply add the real and imaginary parts separately. For instance, adding \( 7 + 6i \) and \( 3i \) involves combining the real parts (7 + 0 in this case) and the imaginary parts (6i + 3i). This straightforward approach results in another complex number, maintaining the rectangular form.

Real Part

In the context of complex numbers, the real part is simply the non-imaginary portion. For a complex number expressed as \( a + bi \), the real part is \( a \). This is analogous to the way integers appear on the number line, covering quantities we typically consider without additional dimensions.
For example, with the complex number \( 7 + 6i \), the real part is 7. It can be thought of as the "horizontal" component when visualizing this number on the complex plane. The process of adding real parts focuses only on these horizontal components.

• Example: When adding \( 7 \) (from \( 7 + 6i \)) and \( 0 \) (from \( 3i \)), the result is 7.

Understanding the real part is crucial for performing operations on complex numbers, as it simplifies the arithmetic by treating them individually and without interference from imaginary components.

Imaginary Part

The imaginary part of a complex number adds an intriguing layer beyond just the real line. In the expression \( a + bi \), \( bi \) is the imaginary part, where \( b \) is a real coefficient and \( i \) represents the square root of -1. The imaginary unit, \( i \), introduces a perpendicular dimension to the numeric axis, creating a complex plane.
Understanding and computing the imaginary part involves focusing on the term connected with \( i \). For the complex number \( 7 + 6i \), the imaginary part is \( 6i \). Similarly, in \( 3i \), the entire term \( 3i \) is the imaginary component. When adding imaginary parts, you sum these coefficients separately from the real components.

• Example: Adding \( 6i \) from \( 7 + 6i \) and \( 3i \) yields \( 9i \).

By treating imaginary parts distinctly, one can maneuver through complex arithmetic with clarity and precision, ensuring accurate mathematical solutions.