Ch. 1 - Equations and Inequalities

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 20a

Chapter 2, Problem 20a

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 involves the square root of a negative number: $\sqrt{-36}$.

Recall that the square root of a negative number is not a real number, but can be expressed using imaginary numbers: $\sqrt{-a} = i\sqrt{a}$ where $a > 0$ and $i$ is the imaginary unit with $i^2 = -1$.

Rewrite $\sqrt{-36}$ as $\sqrt{36} \times \sqrt{-1}$, which simplifies to \$6i$.

Since \$6i$ has no real part and a nonzero imaginary part, it is classified as a pure imaginary number.

Because it is expressed in terms of $i$, it is also a complex number (all imaginary numbers are complex), but it is not a real number.

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Key Concepts

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

Complex Numbers

Complex numbers consist of a real part and an imaginary part and are written in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. They include all real numbers and imaginary numbers.

Imaginary and Pure Imaginary Numbers

Imaginary numbers are multiples of the imaginary unit i. A pure imaginary number has no real part and is expressed as bi, where b ≠ 0. For example, √-36 equals 6i, which is pure imaginary.

Classification of Numbers (Real, Complex, Nonreal Complex)

Real numbers have no imaginary part, while complex numbers include both real and imaginary parts. Nonreal complex numbers have a nonzero imaginary part. Since √-36 = 6i has no real part but a nonzero imaginary part, it is both pure imaginary and nonreal complex.

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