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Chapter 7: Problem 28

Add or subtract and simplify. Write each answer in the form \(a+b i\). $$ (5-6 i)+(8+9 i) $$

Short Answer

Expert verified
13 + 3i

Step by step solution

01

Identify Real and Imaginary Parts

Identify the real and imaginary parts of each complex number. For \(5-6i\), the real part is \5\ and the imaginary part is \ -6i\. For \(8+9i\), the real part is \8\ and the imaginary part is \9i\.
02

Add Real Parts

Add the real parts of the two complex numbers: \(5+8\).
03

Add Imaginary Parts

Add the imaginary parts of the two complex numbers: \(-6i + 9i\).
04

Simplify the Sum

Simplify the results of the addition: \(5+8 = 13\) and \(-6i + 9i = 3i\). Therefore, the resulting complex number is \(13 + 3i\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Real Parts
In any complex number, the real part is the number that does not have the imaginary unit 'i' attached to it. In our example, we have two complex numbers: (5-6i) and (8+9i). The real parts here are 5 and 8, respectively. To find the sum of the real parts, you simply add these together. Therefore, 5 + 8 gives us 13. This is the first component of our final result. Always remember to identify and separate out the real part before performing any operations.
Imaginary Parts
The imaginary part of a complex number is the component that is multiplied by the imaginary unit 'i'. For the numbers (5-6i) and (8+9i), the imaginary parts are -6i and 9i, respectively. When adding or subtracting complex numbers, you should treat the imaginary parts just like regular numbers, keeping in mind that they are coupled with 'i'. Here, we add -6i and 9i to get 3i. This becomes the second component of our resulting complex number after addition.
Simplification
After separately calculating the sums of the real parts and the imaginary parts, you combine them to form one complex number. From our previous steps, we found the real part sum to be 13 and the imaginary part sum to be 3i. So the simplified form of our initial expression (5 - 6i) + (8 + 9i) is 13 + 3i. Simplification makes it easier to understand and work with complex numbers. By breaking the problem down into smaller steps鈥攊dentifying real and imaginary parts, adding them separately, and then combining the results鈥攕implification ensures clarity and correctness.

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