Table of contents

0. Review of Algebra4h 18m
1. Equations & Inequalities3h 18m
2. Graphs of Equations43m
3. Functions2h 17m
4. Polynomial Functions1h 44m
5. Rational Functions1h 23m
6. Exponential & Logarithmic Functions2h 28m
7. Systems of Equations & Matrices4h 6m
8. Conic Sections2h 23m
9. Sequences, Series, & Induction1h 22m
10. Combinatorics & Probability1h 45m

1. Equations & Inequalities

The Imaginary Unit

College Algebra

1. Equations & Inequalities

The Imaginary Unit

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0:41 minutes

Problem 16

Textbook Question

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

Verified step by step guidance

Identify the components of the number: -6 - 2i.

Recognize that the number is in the form a + bi, where a = -6 and b = -2.

Understand that a complex number is any number of the form a + bi, where a and b are real numbers.

Note that if b is not zero, the number is a complex number and also a pure imaginary number if a is zero.

Since a = -6 (not zero) and b = -2 (not zero), the number is a complex number and a real number.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Real Numbers

Real numbers include all the numbers that can be found on the number line, encompassing both rational numbers (like integers and fractions) and irrational numbers (like √2 and π). They can be positive, negative, or zero. In the context of the question, -6 is a real number because it is a whole number that lies on the number line.

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where 'a' is the real part and 'b' is the coefficient of the imaginary unit 'i'. The imaginary unit 'i' is defined as the square root of -1. In the question, -2i is a complex number where the real part is 0 and the imaginary part is -2.

Imaginary Numbers

Imaginary numbers are a subset of complex numbers where the real part is zero, and they are expressed in the form bi, where 'b' is a real number. These numbers cannot be represented on the traditional number line. In the question, -2i is classified as a pure imaginary number because it has no real component, only an imaginary one.