Problem 15 - College Algebra with Corequisite Support

Question

For the following exercises, perform the indicated operation and express the result as a simplified complex number. $$ (3+2 i)+(5-3 i) $$

Step-by-Step Solution

Answer

The simplified complex number is \(8 - i\).

Step 1: Identify Real and Imaginary Parts

First, identify the real and imaginary parts of both complex numbers. For \((3 + 2i)\), the real part is 3 and the imaginary part is \(2i\). For \((5 - 3i)\), the real part is 5 and the imaginary part is \(-3i\).

Step 2: Add Real Parts Together

Add the real parts of both complex numbers together. This means adding 3 and 5, which gives: \(3 + 5 = 8\).

Step 3: Add Imaginary Parts Together

Add the imaginary parts of both complex numbers. This means adding \(2i\) and \(-3i\), which gives: \(2i - 3i = -i\).

Step 4: Combine Results

Combine the results of the real and imaginary parts to form the new complex number: \(8 - i\).

Key Concepts

Real and Imaginary Parts

In complex numbers, understanding real and imaginary parts is crucial. A complex number is typically written as \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
For \(3 + 2i\), \(3\) is the real part, while \(2i\) represents the imaginary part.
Similarly, for the complex number \(5 - 3i\), \(5\) is the real part and \(-3i\) is the imaginary part.

Here are points to remember about real and imaginary parts:

• Real parts are simply real numbers; they have no imaginary element.
• Imaginary parts include "\(i\)", the square root of -1, creating a non-real component.
• The combination of these parts forms a complex number, representing points on a complex plane.

Understanding these components allows seamless manipulation of complex numbers.

Complex Number Addition

Complex number addition follows the basic principle of combining like terms.
This means adding the real parts together and the imaginary parts together.
For example, with \((3 + 2i) + (5 - 3i)\):

• Add the real parts: \(3 + 5 = 8\).
• Add the imaginary parts: \(2i - 3i = -i\).

Just like with regular algebra, group the numbers to make the operation clear. Remember that it’s similar to combining terms like \(3x + 5x\) in algebra, but here we do it with real and imaginary components.
Addition is fundamental in working with complex numbers, ensuring accurate combinations and problem-solving.

Simplified Complex Form

Once you've added the real and imaginary components of two complex numbers, the result should be expressed in a simplified complex form.
A simplified complex form, like \(8 - i\), is the standard way of writing complex numbers.

• The simplified form displays the real part first, followed by the imaginary part, like \(a + bi\).
• This form helps in identifying the magnitude and direction of the number quickly on the complex plane.

Having your result in simplified form is not just about convention; it helps in further calculations and provides clarity in your final answer.
Ensuring your complex numbers are in this form maintains consistency and eases further mathematical operations.