Problem 44 - Precalculus Student Solutions Manual 5th

Question

Find each sum or difference. Write the answer in standard form. $$(4-i)+(8+5 i)$$

Step-by-Step Solution

Answer

12 + 4i

Step 1 - Identify the real and imaginary parts

Separate the real and imaginary parts of each complex number. For (4 - i), the real part is 4 and the imaginary part is -i. For (8 + 5i), the real part is 8 and the imaginary part is 5i.

Step 2 - Add the real parts

Add the real parts together: 4 + 8. This gives: 12.

Step 3 - Add the imaginary parts

Add the imaginary parts together: -i + 5i. This gives: (5 - 1)i = 4i.

Step 4 - Combine the results

Combine the results from Steps 2 and 3 to write the answer in standard form: 12 + 4i.

Key Concepts

Addition of Complex Numbers

Addition of complex numbers is a simple and fundamental operation. When adding complex numbers, you need to combine their real parts and their imaginary parts separately.

For example, consider the complex numbers (4 - i) and (8 + 5i).

The process involves these steps:

• Identify the real parts: For (4 - i), the real part is 4. For (8 + 5i), the real part is 8.
• Identify the imaginary parts: For (4 - i), the imaginary part is -i. For (8 + 5i), the imaginary part is 5i.
• Add the real parts: 4 + 8 = 12
• Add the imaginary parts: (-1)i + 5i = 4i
• Combine the results: 12 (real part) + 4i (imaginary part).

So, when you add (4 - i) and (8 + 5i), the result is $12\; +\; 4i$. This method ensures you get the correct sum by treating real and imaginary parts independently.

Real and Imaginary Parts

Understanding real and imaginary parts is crucial in complex number operations.

Real parts of a complex number are the numbers without the imaginary unit 'i'.

For example:

• In 4 - i, the real part is 4.
• In 8 + 5i, the real part is 8.

Identifying these parts helps in managing the addition and subtraction.

Imaginary parts include the numbers with the imaginary unit 'i'. The imaginary unit 'i' is defined as \[i = \sqrt{-1}\].

For example:

• In 4 - i, the imaginary part is -i.
• In 8 + 5i, the imaginary part is 5i.

By separating and handling real and imaginary parts correctly, complex numbers' operations become straightforward and systematic.

Standard Form of Complex Numbers

A complex number in standard form is represented as a + bi, where a is the real part and bi is the imaginary part.

This form makes it easy to visualize and perform operations on complex numbers.

For instance, after adding (4 - i) and (8 + 5i), the result was 12 + 4i.

In this form:

• 12 is the real part 'a'
• 4i is the imaginary part 'bi'

The standard form helps to clearly identify and differentiate between the real and imaginary components. This clarity is vital for further complex number operations like multiplication and division, as well as in applications across engineering and physics.