/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 20 \((15-8 i)+(2-4 i)\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 20

\((15-8 i)+(2-4 i)\)

Short Answer

Expert verified
17 - 12i

Step by step solution

01

Identify the Real and Imaginary Parts

Separate the real and imaginary parts of each complex number. For the first number (15 - 8i): the real part is 15 and the imaginary part is -8i. For the second number (2 - 4i): the real part is 2 and the imaginary part is -4i.
02

Add the Real Parts

Add the real parts of the two complex numbers. Real part: 15 + 2 = 17.
03

Add the Imaginary Parts

Add the imaginary parts of the two complex numbers. Imaginary part: -8i + (-4i) = -12i.
04

Combine the Results

Combine the sum of the real parts and the sum of the imaginary parts back into a single complex number. The result is: 17 - 12i.

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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 numbers that have two parts: a real part and an imaginary part. They are 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 satisfies the equation \(i^2 = -1\). Complex numbers expand our number system to include solutions to equations that don't have real solutions.
real and imaginary parts
Any complex number can be broken down into two components: the real part and the imaginary part. Looking at the number \(15 - 8i\), the real part is 15, and the imaginary part is -8i. Similarly, for the complex number \(2 - 4i\), the real part is 2, and the imaginary part is -4i. Knowing how to distinguish and work with these parts is essential for operations involving complex numbers.
adding real parts
To add the real parts of two complex numbers, simply find their numerical sum. In our example, we start with the real parts 15 and 2. Adding these, we get:
\[ \text{Real part: } 15 + 2 = 17 \ \].
This new number (17) represents the sum of the real components of our original complex numbers.
adding imaginary parts
Next, we add the imaginary parts of our complex numbers. The imaginary parts are -8i and -4i. Adding these, we get:
\[ \text{Imaginary part: } -8i + (-4i) = -12i \ \].
This gives us the imaginary part of our result. Combining this with the real part from the previous step, we get the final result:
\[ 17 - 12i \ \].
This is the sum of the original complex numbers.

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\((15+8 i)+(2+4 i)\)

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