Problem 93 - Elementary Algebra

Question

Perform the operation. $$ (2-5 i)+(3+4 i) $$

Step-by-Step Solution

Answer

The result of the operation is \(5 - i\).

Step 1: Identify the Real Parts

First, identify the real parts of the given complex numbers. In the expression \((2-5i) + (3+4i)\), the real parts are \(2\) from the first complex number and \(3\) from the second complex number.

Step 2: Identify the Imaginary Parts

Next, identify the imaginary parts of the complex numbers. In the expression, the imaginary parts are \(-5i\) from the first complex number and \(4i\) from the second complex number.

Step 3: Add the Real Parts

Now, add the real parts together: \(2 + 3 = 5\). This gives us the real part of the result.

Step 4: Add the Imaginary Parts

Similarly, add the imaginary parts together: \(-5i + 4i = -1i\). This gives us the imaginary part of the result.

Step 5: Combine the Results

Combine the sum of the real parts and the sum of the imaginary parts to form the new complex number: \(5 - 1i\).

Key Concepts

Addition of Complex Numbers

Complex numbers often appear in the form of \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. To add two complex numbers, such as \((2-5i)\) and \((3+4i)\), we combine their real parts and imaginary parts separately.

Here’s a simple way to visualize it:

• Identify and add the real components: \(2 + 3 = 5\).
• Identify and add the imaginary components: \(-5i + 4i = -1i\).

By following these steps, the result is a new complex number: \(5 - 1i\).
Breaking down the addition into separate components makes the process straightforward and helps in understanding how complex numbers interact.

Real Part

The real part of a complex number is the component without the imaginary unit \(i\). In expressions like \(2 - 5i\) or \(3 + 4i\), the numbers \(2\) and \(3\) are the real parts. These are simply the ordinary numbers you're already familiar with.

When performing operations like addition, it's crucial to isolate and handle these real parts separately:

1. Look at both complex numbers and find their real components.2. Add these components to get the real part of the result.

In our example, we found the sum of the real parts to be \(5\). Understanding this concept will help make the whole process of working with complex numbers more intuitive.

Imaginary Part

The imaginary part of a complex number is the term that involves the imaginary unit \(i\). For example, in \(-5i\) and \(4i\), the parts \(-5\) and \(4\) are coefficients of \(i\). These coefficients are crucial for operations involving complex numbers.

Here's how you handle them in addition:

• Identify these coefficients in each complex number.
• Add these coefficients like regular numbers: \(-5 + 4 = -1\).

Then, append \(i\) to this result to get \(-1i\). Recognizing imaginary parts and treating them carefully will allow you to master complex number operations.

Complex Number Operations

Operations with complex numbers, such as addition, subtraction, multiplication, and division, require a clear understanding of both their real and imaginary components. By building on the basic principles, you can apply these operations in broader contexts.

For addition, follow these key steps:

• Separate the complex numbers into real and imaginary parts.
• Combine like terms: real with real, imaginary with imaginary.
• Write the resulting complex number as \(a + bi\).

This approach ensures clarity and accuracy when performing more advanced operations that involve complex numbers, allowing you to build upon a solid foundation of complex number arithmetic.