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Chapter 9: Problem 13

Perform the operations. $$ (-8-3 i)+(5+2 i) $$

Short Answer

Expert verified
The result is \(-3 - i\).

Step by step solution

01

Break Down the Terms

Identify the real and imaginary components of each complex number in the given operation. The first number is (-8 - 3i), where the real part is -8 and the imaginary part is -3i. The second number is (5 + 2i), where the real part is 5 and the imaginary part is 2i.
02

Add the Real Parts

Add the real parts of the complex numbers together. Combine -8 (from the first number) and 5 (from the second number): \(-8 + 5 = -3\).
03

Add the Imaginary Parts

Add the imaginary parts of the complex numbers together. Combine -3i (from the first number) and 2i (from the second number): \(-3i + 2i = -i\).
04

Combine the Results

Combine the results from the previous steps to form the final complex number: The real part is -3 and the imaginary part is -i, so the final result is \(-3 - i\).

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Key Concepts

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

Real Part of Complex Numbers
Every complex number can be broken down into two main components: the real part and the imaginary part. When we talk about the real part, we refer to the number that you can place on the number line without any imaginary component.
For example, in the complex number \(-8 - 3i\), the real part is \(-8\). It is the portion of the complex number that resembles a traditional number without an \(i\) attached to it. Similarly, in the complex number \(5 + 2i\), the real part is \(5\).
Understanding this helps in various mathematical operations such as addition, where the real parts are handled separately from the imaginary parts.
Imaginary Part of Complex Numbers
The imaginary part of a complex number is associated with the letter \(i\), which represents the square root of \(-1\). This part can't be plotted on a normal number line but is crucial to forming complex numbers.
For instance, the complex number \(-8 - 3i\) has an imaginary part of \(-3i\). It defines the imaginary component tied to the square root of a negative number. Similarly, in \(5 + 2i\), the imaginary part is \(2i\).
In complex number calculations, treating the imaginary part separately is essential, as they often lead to unique mathematical explorations.
Addition of Complex Numbers
To add complex numbers, the process involves combining like terms: real parts with real parts, and imaginary parts with imaginary parts. This makes the addition seamless from the traditional arithmetic of numbers.
Let's see how the addition of \(-8 - 3i\) and \(5 + 2i\) works:
  • First, add the real parts: \(-8 + 5 = -3\).
  • Next, add the imaginary parts: \(-3i + 2i = -i\).
  • Combine these to form a new complex number: \(-3 - i\).
This method allows for a step-by-step approach that is clear and straightforward, ensuring that each part of the complex number is handled properly.

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