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Chapter 8: Problem 31

Add or subtract as indicated. Write your answers in the form \(a+b i .\) $$ (5-i)+(-5+i) $$

Short Answer

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Step by step solution

01

Identify the Real Parts

The real parts of the complex numbers are 5 and -5. Identify these real parts for addition.
02

Identify the Imaginary Parts

The imaginary parts of the complex numbers are -i and i. Identify these imaginary parts for addition.
03

Add the Real Parts

Add the real parts together: 5 + (-5). The result is 0.
04

Add the Imaginary Parts

Add the imaginary parts together: -i + i. The result is 0.
05

Combine the Results

Combine the results from adding the real parts and the imaginary parts. So, the result is 0 + 0i, which is just 0.

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Key Concepts

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

Complex Numbers
Complex numbers are an essential part of algebra and mathematics, representing quantities that have both a real part and an imaginary part. A complex number is usually written in the form a + bi, where a is the real part and bi is the imaginary part.
Here, i is the imaginary unit, which is defined as the square root of -1. Therefore, i虏 = -1.
Complex numbers can be added, subtracted, multiplied, and divided just like real numbers, but you need to follow specific rules when working with the imaginary part.
Real and Imaginary Parts
Understanding the real and imaginary parts of a complex number is crucial. The real part is straightforward; it's just the number without the imaginary unit i.
Similarly, the imaginary part is the number that is multiplied by the imaginary unit i. For example, in the complex number 3 + 4i, 3 is the real part and 4 is the imaginary part.
When performing operations like addition or subtraction on complex numbers, you handle the real parts and imaginary parts separately. Always keep track of each part to avoid mistakes.
Algebraic Operations
Adding and subtracting complex numbers follows a specific set of rules. Let's use the given exercise example, \( (5 - i) + (-5 + i) \), to understand these rules:
  • First, identify the real parts: 5 and -5.
  • Next, identify the imaginary parts: -i and i.
  • Now, add the real parts separately: 5 + (-5) = 0.
  • Then, add the imaginary parts: -i + i = 0.

Combine the results to get the final answer, which for this example is 0 + 0i. Therefore, the sum is 0.
By following these steps, you can easily add or subtract any complex numbers you encounter.

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