Problem 13 - Algebra for College Students

Question

Add or subtract as indicated. $$(3+2 i)-(5+7 i)$$

Step-by-Step Solution

Answer

\(-2 - 5i\)

Step 1: Identify the Real and Imaginary Parts

In the complex numbers given, identify the real and imaginary parts. For \(3 + 2i\), the real part is 3 and the imaginary part is 2. For \(5 + 7i\), the real part is 5 and the imaginary part is 7.

Step 2: Subtract the Real Parts

Subtract the real part of the second complex number from the real part of the first complex number: \(3 - 5 = -2\).

Step 3: Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the imaginary part of the first complex number: \(2 - 7 = -5\).

Step 4: Combine the Results

Combine the results of the real and imaginary subtractions to find the resulting complex number: \(-2 - 5i\).

Key Concepts

Real and Imaginary Parts

To fully understand complex numbers, it's crucial to recognize their components. Every complex number has a real part and an imaginary part. Consider the complex number notation as \(a + bi\), where \(a\) is the real part, and \(b\) is the imaginary part, linked with the imaginary unit \(i\), where \(i^2 = -1\).
Identifying these parts is the first step in performing any arithmetic operation involving complex numbers. For example, in the complex number \(3 + 2i\), the real part is \(3\), and the imaginary part is \(2\). Likewise, in \(5 + 7i\), the real part is \(5\) while the imaginary part is \(7\).
Understanding this distinction helps in visualizing complex numbers on a plane called the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part.

Subtracting Complex Numbers

Subtracting complex numbers is performed by separately subtracting their real and imaginary components. Imagine you have the complex numbers \(3 + 2i\) and \(5 + 7i\) and you want to subtract the latter from the former.

• First, subtract the real parts: \(3 - 5 = -2\).
• Next, subtract the imaginary parts: \(2 - 7 = -5\).

The result of this subtraction is another complex number that combines both results: \(-2 - 5i\). This method ensures that all operations respect the unique structure of complex numbers, maintaining the correct mathematical relationships between their parts.

Algebraic Operations

Algebraic operations on complex numbers work similarly to those on real numbers but require careful handling of their imaginary parts. The primary operations on complex numbers are addition, subtraction, multiplication, and division. Each follows specific rules:

• **Addition/Subtraction**: Combine like parts—real with real, imaginary with imaginary.
• **Multiplication**: Distribute the terms as you would in algebraic expressions, applying \(i^2 = -1\) when squaring \(i\).
• **Division**: Involves multiplying by the conjugate of the denominator to remove the imaginary part from the denominator.

These operations maintain the balance between the real and imaginary parts. Practicing these operations with simple examples helps in gaining confidence, making the process intuitive, even for complex expressions. Remember, the key to mastering operations with complex numbers lies in understanding how the imaginary unit \(i\) interacts with the arithmetic rules we use.