Problem 26 Perform the operation(s) and wri... [FREE SOLUTION]

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Elementary Algebra Within Reach

Ron Larson, Kimberly Nolting

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 10: Problem 26

Perform the operation(s) and write the result in standard form. $$20 i+(9-26 i)$$

Short Answer

Expert verified

The result of the operation is $9 - 6i$

Step by step solution

Identify the real and imaginary parts of the complex numbers

In the complex number $20i$, the real part is 0 and the imaginary part is 20. In the complex number $9-26i$, the real part is 9 and the imaginary part is -26.

Add the real parts together and add the imaginary parts together

Add the real parts 0 and 9 together to get 9. Add the imaginary parts 20 and -26 together to get -6.

Write the Result

Combine the summed real and imaginary parts to form the final complex number. The result is $9 - 6i$, which is the standard form of a complex number.

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Key Concepts

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

Imaginary Numbers

Imaginary numbers are fascinating! They introduce us to a part of mathematics beyond the real number line. Unlike real numbers, imaginary numbers are all about multiples of the unit 'i'. But what is 'i'? It is defined as the square root of -1.
This means when you multiply 'i' by itself, you get -1. This is different from any real number, as no real number squared gives a negative number.

For example, in our exercise, we have numbers like 20i and -26i. Here, these parts indicate imaginary components.
Operations with imaginary numbers are similar to variables like adding or subtracting 'x'. Simply combine the coefficients of 'i'.

Real Numbers

Real numbers are those we often use in everyday life. They include all the numbers on the number line that you can think of, including whole numbers, fractions, and decimals. They are either positive or negative and include zero.
In the context of complex numbers, real numbers act as the 'static' part of complex expressions. They don't involve the imaginary unit 'i'.

In our exercise example, '9' is a real number from the complex number (9 - 26i).
Real parts are simply added together in operations with complex numbers.

Standard Form of a Complex Number

The standard form of a complex number is a crucial way to express these numbers neatly. It is written as $ a + bi $, where:

$ a $ is the real part.
$ bi $ is the imaginary part, with 'b' being the coefficient of 'i'.

For example, in the given operation, after adding and simplifying, the complex number was expressed in the standard form as $9 - 6i$.
This form separates the real and imaginary components clearly, allowing us to easily identify and perform further operations.
By sticking to this format, we ensure clarity when dealing with any complex numbers. This method helps in mathematical operations such as addition, subtraction, multiplication, and even division of complex numbers.

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91影视

Perform the operation(s) and write the result in standard form. $$20 i+(9-26 i)$$

Short Answer

Step by step solution

Identify the real and imaginary parts of the complex numbers

Add the real parts together and add the imaginary parts together

Write the Result

Key Concepts

Imaginary Numbers

Real Numbers

Standard Form of a Complex Number

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