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

Complex Numbers

College Algebra

1. Equations & Inequalities

Complex Numbers

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1:27 minutes

Problem 20

Textbook Question

Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) √-36

Verified step by step guidance

Recognize that the expression \( \sqrt{-36} \) involves taking the square root of a negative number.

Recall that the square root of a negative number is not a real number, and it involves the imaginary unit \( i \), where \( i = \sqrt{-1} \).

Express \( \sqrt{-36} \) as \( \sqrt{36} \times \sqrt{-1} \).

Calculate \( \sqrt{36} \), which is 6, and multiply it by \( i \) to get \( 6i \).

Conclude that \( 6i \) is a pure imaginary number, which is also a type of complex 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 do not include imaginary or complex numbers, which involve the square root of negative values.

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', which is defined as √-1. Complex numbers can represent solutions to equations that have no real solutions.

Imaginary Numbers

Imaginary numbers are a subset of complex numbers that can be expressed as bi, where 'b' is a real number and 'i' is the imaginary unit. They arise when taking the square root of negative numbers, such as √-1, and are essential in extending the concept of numbers to solve equations that do not have real solutions.