Problem 18 - Algebra and Trigonometry

Question

Evaluate the expression and write the result in the form a bi. $$ (3-2 i)+\left(-5-\frac{1}{1} i\right) $$

Step-by-Step Solution

Answer

The expression evaluates to \(-2 - 3i\).

Step 1: Simplify the expression

First, we need to simplify the given expression by combining like terms. This involves adding the real parts together and the imaginary parts together separately. The simplified expression is \((3 - 2i) + (-5 - 1i)\).

Step 2: Add the real parts together

Add the real parts of the numbers together. Here, the real parts are 3 and -5.\[3 + (-5) = -2\]

Step 3: Add the imaginary parts together

Now, add the imaginary parts together. The imaginary parts are -2i and -1i.\[-2i + (-1i) = -3i\]

Step 4: Write the final expression

Combine the results from Step 2 and Step 3 to write the final expression in the form \(a + bi\):\[-2 - 3i\].

Key Concepts

Real and Imaginary Parts

Complex numbers are fascinating because they include both real and imaginary parts. They take the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part. Understanding how to distinguish and work with these parts is essential.

• **Real Part**: This is represented by the number \(a\) in the complex number \(a + bi\). It behaves just like numbers you are already comfortable with.
• **Imaginary Part**: This is represented by \(bi\), where \(b\) is a real number and \(i\) is the imaginary unit. The imaginary unit \(i\) is defined such that \(i^2 = -1\).

In the expression \(3 - 2i\), 3 is the real part, and \(-2i\) is the imaginary part. Recognizing this will help you manipulate these numbers effectively.

Addition of Complex Numbers

Adding complex numbers might seem daunting at first, but it becomes simple once you know the rule of combining like terms. When adding two complex numbers, you separately add the real parts and the imaginary parts.
For our example, consider the expression \((3 - 2i) + (-5 - 1i)\). Here's how you add them:

• Identify the real parts: 3 and -5. Add them together: \(3 + (-5) = -2\).
• Identify the imaginary parts: \(-2i\) and \(-1i\). Add them together: \(-2i + (-1i) = -3i\).

This process is exactly like combining like terms in algebra—just keep real and imaginary parts separate and you'll have your correct answer.

Simplifying Expressions

Simplifying expressions involving complex numbers is essential to keep calculations manageable. The aim is to combine the components of the complex number to express the result in the standard form \(a + bi\).
Let's revisit our complex expression \((3 - 2i) + (-5 - 1i)\). Here are the steps to simplify:

• First, add the real parts: 3 and -5, which gives \(-2\).
• Next, add the imaginary parts: \(-2i\) and \(-1i\), giving \(-3i\).

Now combine them to get the simplified expression \(-2 - 3i\).
The goal is to express it as \(a + bi\), facilitating understanding and further operations. With practice, this process becomes second nature, enabling you to solve complex number problems efficiently.