Problem 12 Identify each number as real, co... [FREE SOLUTION]

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Precalculus Student Solutions Manual 5th

Margaret L. Lial, John Hornsby, David I. Schneider

$Math Studyset 91影视 Explanations$ Math

5 Edition

Chapter 1: Problem 12

Identify each number as real, complex, pure imaginary, or nonreal complex. (More than one of these descriptions will apply. ) $$-6-2 i$$

Short Answer

Expert verified

-6-2i is a complex number and a nonreal complex number.

Step by step solution

Identify the real part

The real part of the number is the component without the imaginary unit. For the number {-6-2i}, the real part is {-6.}

Identify the imaginary part

The imaginary part of the number is the component with the imaginary unit, {i}. For the number {-6-2i}, the imaginary part is {-2i.}

Recognize the complex number

A complex number is a number that has both a real part and an imaginary part. Since {-6-2i} has both, it is indeed a complex number.

Determine if the number is pure imaginary

A pure imaginary number has no real part, only an imaginary part. Since {-6-2i} has a real part of {-6}, it is not a pure imaginary number.

Identify if the number is nonreal complex

A nonreal complex number has a non-zero imaginary part. Since {-6-2i} has an imaginary part of {-2i}, it is a nonreal complex number.

Recognize the number as real

Even though {-6-2i} has a real part, it also has a non-zero imaginary part. Therefore, it cannot be classified as a real number alone.

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Key Concepts

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

Real Part

The real part of a complex number is the portion that doesn鈥檛 involve the imaginary unit, which is denoted as $i$. Imaginary units represent the square root of -1. In the expression of a complex number like $-6 - 2i$, the real part is the number that stands alone without the $i$.

For example:

In $5 + 4i$, the real part is 5.
In $-3 - 7i$, the real part is -3.
For the number given in our exercise, $-6 - 2i$, the real part is -6.

Remember, the real part is always a real number, be it positive, negative, or zero.

Imaginary Part

The imaginary part of a complex number is the portion that does involve the imaginary unit, $i$. It is paired with a real number coefficient. In the expression $-6 - 2i$, the imaginary part includes the number coefficient and the $i$.

Examples include:

For $5 + 4i$, the imaginary part is 4i.
For $-3 - 7i$, the imaginary part is -7i.
In the case of our exercise, with $-6 - 2i$, the imaginary part is -2i.

If the imaginary part is zero, the number becomes a real number. Therefore, in order to classify correctly, always identify the real and imaginary parts separately.

Pure Imaginary Number

A pure imaginary number is one that has no real part, only an imaginary part. When we talk about pure imaginary numbers, we are focusing on expressions like $0 + bi$, where $b$ is not zero.

To clarify:

The number $3i$ is a pure imaginary number because it can be written as $0 + 3i$.
Similarly, $-5i$ is pure imaginary as it is $0 - 5i$.
The given number, $-6 - 2i$, has a real part of -6, meaning it is not a pure imaginary number.

Recognizing the presence of a real part disqualifies the number from being pure imaginary. Make sure that for a number to be classified as pure imaginary, the real part has to be exactly zero.

Nonreal Complex Number

Nonreal complex numbers have both a real part and an imaginary part, where the imaginary part is non-zero. Essentially, this means it fits the form $a + bi$ with both $a$ (the real part) and $bi$ (the imaginary part) present.

Examples are:

$4 + 5i$ is a nonreal complex number because both 4 (real part) and 5i (imaginary part) are present.
Another example is $-3 - 7i$, which has real part -3 and imaginary part -7i.
In our exercise, $-6 - 2i$ is a nonreal complex number as it includes real part -6 and imaginary part -2i.

A number is termed nonreal if it has an imaginary part that is not zero. Hence, when identifying nonreal complex numbers, always check the presence of a non-zero imaginary part.

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91影视

Identify each number as real, complex, pure imaginary, or nonreal complex. (More than one of these descriptions will apply. ) $$-6-2 i$$

Short Answer

Step by step solution

Identify the real part

Identify the imaginary part

Recognize the complex number

Determine if the number is pure imaginary

Identify if the number is nonreal complex

Recognize the number as real

Key Concepts

Real Part

Imaginary Part

Pure Imaginary Number

Nonreal Complex Number

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