Ch. 1 - Equations and Inequalities

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

Lial 13th Edition

Ch. 1 - Equations and Inequalities

Problem 16

Chapter 2, Problem 16

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

Verified step by step guidance

Start by identifying the given number: \(-6 - 2i\).

Recall that a complex number is generally written as \(a + bi\), where \(a\) and \(b\) are real numbers and \(i\) is the imaginary unit with \(i^2 = -1\).

In this case, \(a = -6\) and \(b = -2\), so the number has both a real part and an imaginary part.

Since the number has a nonzero real part (\(-6\)) and a nonzero imaginary part (\(-2i\)), it is a complex number but not purely imaginary.

Because it has a real part, it is not purely imaginary, and since it is expressed in the form \(a + bi\), it is a complex number; it is also a real number only if the imaginary part is zero, which it is not, so 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.

Real Numbers

Real numbers include all the numbers that can be found on the number line, such as integers, fractions, and irrational numbers. They do not have an imaginary component and can be positive, negative, or zero.

Complex Numbers

Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where 'a' and 'b' are real numbers and 'i' is the imaginary unit with the property i² = -1. They encompass all real and imaginary numbers.

Pure Imaginary Numbers

Pure imaginary numbers are a subset of complex numbers where the real part is zero and the number is expressed as bi, with b ≠ 0. For example, -2i is pure imaginary because it has no real component.

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