Problem 9 - Algebra for College Students

Question

Add or subtract as indicated. $$(6+3 i)+(4+5 i)$$

Step-by-Step Solution

Answer

The sum is \(10 + 8i\).

Step 1: Identify the Complex Numbers

In this exercise, we have two complex numbers: \(6 + 3i\) and \(4 + 5i\). We are tasked with adding these two numbers.

Step 2: Add the Real Parts

The real parts of the complex numbers are 6 and 4. Add these together: \(6 + 4 = 10\).

Step 3: Add the Imaginary Parts

The imaginary parts of the complex numbers are 3i and 5i. Add these together: \(3i + 5i = 8i\).

Step 4: Combine the Results

Combine the results from Steps 2 and 3 to form the new complex number: \(10 + 8i\).

Key Concepts

Addition of Complex Numbers

Adding complex numbers is as simple as performing addition with their real and imaginary components separately. Each complex number is a sum of a real part and an imaginary part. To find the sum of two complex numbers:

• First, add their real parts together.
• Then, add their imaginary parts.

For example, in the given problem, we have two complex numbers: \(6 + 3i\) and \(4 + 5i\). Start by adding the real parts, which are 6 and 4, to get 10. Next, add the imaginary parts, 3i and 5i, to get 8i. Combining these results, you get the final sum: \(10 + 8i\). Just remember to keep the real and imaginary parts separate until you're ready to combine them into a single complex sum.

Real and Imaginary Parts

In complex numbers, you'll encounter two main components: the real part and the imaginary part. Understanding each is key to grasping complex number operations.

• The real part of a complex number is the number that appears before the imaginary unit, \(i\).
• The imaginary part is the number that multiplies \(i\).

For the complex number \(6 + 3i\), the real part is 6, and the imaginary part is 3i. Notice that the imaginary part includes the imaginary unit \(i\), which indicates that it is not a real number, but an imaginary one. When performing any operation with complex numbers, handle these parts individually until it is time to combine them together as a complex number result.

Complex Number Operations

Complex number operations often include addition, subtraction, multiplication, and division. Each operates with the separate handling of real and imaginary parts.With addition and subtraction, follow a straightforward rule:

• Add or subtract real parts separately from imaginary parts.
• Combine real and imaginary parts after completing the operations.

Multiplication and division are slightly more complex. For multiplication, apply the distributive property and remember that \(i^2 = -1\). For division, you'll typically multiply by the conjugate to simplify.Regardless of the operation, the essence remains: treat real and imaginary parts with their respective rules. This ensures accuracy and clarity as you work through complex number problems.