Problem 14 - Algebra for College Students

Question

Add or subtract as indicated. $$(1+3 i)-(4+9 i)$$

Step-by-Step Solution

Answer

The result is \(-3 - 6i\).

Step 1: Identify the Real Parts

Observe the problem, \((1 + 3i) - (4 + 9i)\). We first identify the real parts from each complex number. The real part of the first number is 1, and for the second number, it is 4.

Step 2: Subtract the Real Parts

Next, subtract the real part of the second complex number from the first: \(1 - 4 = -3\).

Step 3: Identify the Imaginary Parts

Now identify the imaginary parts from each complex number. The imaginary part of the first number is \(3i\), and for the second number, it is \(9i\).

Step 4: Subtract the Imaginary Parts

Subtract the imaginary part of the second complex number from the first: \(3i - 9i = -6i\).

Step 5: Combine the Results

Finally, combine the results of the real and imaginary part subtractions to get the final answer: \(-3 - 6i\).

Key Concepts

Real and Imaginary Parts

Complex numbers are made up of two essential components: the real part and the imaginary part. A complex number can be expressed in the form \(a + bi\), where \(a\) represents the real part, and \(bi\) represents the imaginary part. The imaginary unit \(i\) is defined as the square root of \(-1\). This concept allows us to handle quantities that involve the square root of negative numbers, which cannot be resolved using only real numbers.
Understanding real and imaginary parts is crucial in manipulating complex numbers. When dealing with arithmetic operations such as addition or subtraction, we treat these parts separately. Identifying the real and imaginary components correctly sets the foundation for all further calculations. This separation allows us to work on each part independently, ensuring that complex number operations are straightforward and manageable.

Complex Number Subtraction

Subtracting complex numbers involves dealing with both their real and imaginary components separately. Suppose you want to subtract one complex number from another, such as in \((1 + 3i) - (4 + 9i)\). Here’s how you proceed:
**Identify and Subtract Real Parts**
First, focus on the real parts: 1 and 4 from each complex number. Subtract the real part of the second number from the first: \(1 - 4 = -3\).
**Identify and Subtract Imaginary Parts**
Next, look at the imaginary parts: \(3i\) and \(9i\). Again, subtract the imaginary part of the second number from the first: \(3i - 9i = -6i\).
**Combine the Results**
Once both subtractions are complete, combine the results to get the final answer: \(-3 - 6i\).
This method ensures that both components of the complex numbers are handled correctly and independently, providing a structured and accurate approach to subtraction.

Step by Step Solution

It's important to take the process step by step to avoid any confusion when performing operations on complex numbers. Here's a clear breakdown of the subtraction example:

• **Identify the Real Parts:** Focus first on the real numbers. For example, from \((1 + 3i) - (4 + 9i)\), take the real parts, 1 and 4.
• **Subtract the Real Parts:** Next, subtract these real numbers: \(1 - 4 = -3\).
• **Identify the Imaginary Parts:** Move on to the imaginary parts, which are \(3i\) and \(9i\).
• **Subtract the Imaginary Parts:** Perform the subtraction: \(3i - 9i = -6i\).
• **Combine Real and Imaginary Results:** Finally, combine the results for a complete solution: \(-3 - 6i\).

This procedural approach ensures accuracy and helps reinforce the understanding of complex numbers operations. By separating each stage, it prevents errors and assists in gaining a clearer comprehension of complex arithmetic.