Problem 45 - A Graphical Approach to College Algebra

Question

Add or subtract as indicated. Write each sum or difference in standard form. $$(3+2 i)+(4-3 i)$$

Step-by-Step Solution

Answer

The sum is \(7 - i\).

Step 1: Identify the Expression Components

We are given the expression \((3 + 2i) + (4 - 3i)\). This expression consists of two complex numbers, where \(3 + 2i\) is the first complex number and \(4 - 3i\) is the second complex number.

Step 2: Separate Real and Imaginary Parts

Separate the real parts and the imaginary parts from both complex numbers. For \(3 + 2i\), the real part is 3, and the imaginary part is 2i. For \(4 - 3i\), the real part is 4, and the imaginary part is -3i.

Step 3: Add the Real Parts Together

Add the real parts of the two complex numbers: \(3 + 4 = 7\).

Step 4: Add the Imaginary Parts Together

Add the imaginary parts of the complex numbers: \(2i + (-3i) = -i\).

Step 5: Combine into Standard Form

Combine the results from steps 3 and 4 to write the sum in standard form. The standard form of a complex number is \(a + bi\), where \(a\) and \(b\) are real numbers. Thus, the sum is \(7 - i\).

Key Concepts

Addition of Complex Numbers

Complex numbers are composed of a real part and an imaginary part, typically written in the form \( a + bi \). To add complex numbers, you simply add their real components together and their imaginary components together. Let's consider the example given:

• The complex numbers are given as \( (3 + 2i) \) and \( (4 - 3i) \).
• To add them, you first look at the real parts: 3 and 4.
• Adding the real parts gives you 7.

Next, focus on the imaginary parts:

• The imaginary components are \( 2i \) and \( -3i \).
• Adding these gives us \( 2i + (-3i) = -i \).

Thus, by combining these results, the final sum of the complex numbers is \( 7 - i \). It's like adding two separate columns: one for the real and one for the imaginary parts.

Imaginary Unit

The imaginary unit, denoted by \( i \), is a fundamental component of complex numbers. Understanding it is crucial to working with complex numbers effectively. By definition:

• \( i \) is the square root of \(-1\), meaning \( i^2 = -1 \).

When dealing with imaginary numbers, the operations may seem unusual at first but are quite logical:

• \( 0i \) represents no imaginary part, simplifying to zero.
• Multiplying by \( i \) rotates a point on the complex plane 90 degrees, which gives it a unique property in geometry and engineering.

In our problem, \( i \) appears in expressions \( 2i \) and \(-3i\). Here, it's treated as a regular number in arithmetic operations within the context of these expressions.

Standard Form

Writing complex numbers in standard form ensures consistency and clarity. In standard form, a complex number is expressed as \( a + bi \), where:

• \( a \) is the real part.
• \( b \) is the coefficient of the imaginary part, \( i \).

This format is beneficial as it separates the real and imaginary components clearly, making it easier to perform further mathematical operations like addition, subtraction, and multiplication.
In our solution, after calculating the sum of the given complex numbers, we find that the sum \( 7 - i \) is already in standard form. Here:

• 7 is the real part, represented by \( a \).
• -1 is the imaginary coefficient, represented by \( b \) (the number before \( i \)).

Expressing complex solutions in standard form simplifies interpretation and application in various mathematical contexts.