Identify each number as real, complex, pure imaginary, or nonreal com-plex. (More than one of these descriptions will apply.) -7i

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Recognize that the number given is \(-7i\), where \(i\) is the imaginary unit defined by \(i^2 = -1\).

Recall the definitions: a real number has no imaginary part, a complex number is of the form \(a + bi\) where \(a\) and \(b\) are real numbers, a pure imaginary number has zero real part and a nonzero imaginary part, and a nonreal complex number is a complex number with a nonzero imaginary part and zero or nonzero real part but not purely real.

Identify the real part (\(a\)) and the imaginary part (\(b\)) of the number \(-7i\). Here, \(a = 0\) and \(b = -7\).

Since \(a = 0\) and \(b eq 0\), the number \(-7i\) is a pure imaginary number.

Because every pure imaginary number is also a complex number, \(-7i\) is both pure imaginary and complex, but not real.

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Key Concepts

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

Real and Complex Numbers

Real numbers include all rational and irrational numbers that can be found on the number line. 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 i² = -1.

Imaginary and Pure Imaginary Numbers

Imaginary numbers are multiples of the imaginary unit i. A pure imaginary number has no real part and is written as bi, where b is a nonzero real number. For example, -7i is pure imaginary because its real part is zero.

Nonreal Complex Numbers

Nonreal complex numbers are complex numbers whose real part is zero or nonzero but have a nonzero imaginary part, making them not purely real. Pure imaginary numbers are a subset of nonreal complex numbers since their real part is zero but imaginary part is nonzero.

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