Problem 20 - Algebra for College Students

Question

Add or subtract as indicated. $$(12-9 i)-(14-6 i)$$

Step-by-Step Solution

Answer

The result is \(-2 - 3i\).

Step 1: Identify Like Terms

Break down the expression into its real and imaginary components. The expression is \((12 - 9i) - (14 - 6i)\), which consists of real numbers 12 and 14, and imaginary numbers -9i and -6i.

Step 2: Subtract Real Parts

Subtract the real parts: \(12 - 14 = -2\). This gives us the real part of the resultant complex number.

Step 3: Subtract Imaginary Parts

Subtract the imaginary parts: \(-9i - (-6i) = -9i + 6i = -3i\). This gives us the imaginary part of the resultant complex number.

Step 4: Combine Results

Combine the results from Steps 2 and 3. The new complex number is \(-2 - 3i\).

Key Concepts

Imaginary Numbers

Imaginary numbers are a fundamental part of complex numbers. They are numbers that can be represented as a multiple of the imaginary unit, denoted as \(i\), where \(i\) is defined as the square root of -1. Thus, an imaginary number is any value that can be expressed in the form of \(bi\) where \(b\) is a real number. Here are some key points about imaginary numbers:

• Like real numbers, imaginary numbers can be added, subtracted, multiplied, and divided.
• When squared, the imaginary unit \(i\) gives \(-1\), i.e., \(i^2 = -1\).
• Imaginary numbers do not have a position on the real number line; instead, they exist perpendicular to it in the complex plane.

In our exercise, the imaginary components are \(-9i\) and \(-6i\). Understanding these numbers is essential for handling operations involving complex numbers.

Real Numbers

Real numbers are the set of all rational and irrational numbers, which can be found on the number line. They are what we typically think of as 'regular' numbers, such as 1, 2.5, -3, etc. Real numbers can:

• Be positive, negative, or zero.
• Be whole numbers or decimals, and can include fractions.
• Represent physical quantities like distance, height, and money.

In the context of this exercise, the real numbers used are 12 and 14. These form the non-imaginary components in the complex number expression: \((12 - 9i) - (14 - 6i)\). Understanding how to handle these numbers is crucial for performing arithmetic with complex numbers.

Subtraction of Complex Numbers

Subtraction of complex numbers is quite straightforward. A complex number is often represented in the form \(a + bi\), where \(a\) and \(b\) are real numbers. Here's how subtraction works:1. **Identify Like Terms:** Separate the real and imaginary parts.2. **Subtract Real Parts:** Subtract the real components of the complex numbers.3. **Subtract Imaginary Parts:** Subtract the imaginary components.4. **Combine Parts:** Combine the results to form the new complex number.In our example, the operation to be performed is \((12 - 9i) - (14 - 6i)\). First, we subtract the real parts: \(12 - 14 = -2\). Then, we apply subtraction to the imaginary parts \(-9i - (-6i) = -9i + 6i = -3i\). Finally, the results are combined to form the complex number: \(-2 - 3i\). This step-by-step method ensures accurate results and better comprehension of complex number operations.