Question

categorize each number as real, complex, or imaginary. (4 points)
real complex imaginary
$4 + 2i$
$sqrt{3}$
$9 - 4i$
$-2.5i$

Explanation:

Step1: Recall definitions

• Real number: A number that can be expressed as a decimal (including integers, fractions, radicals that simplify to real, etc.), no imaginary unit \(i\).
• Complex number: A number of the form \(a + bi\), where \(a,b\in\mathbb{R}\) (real numbers), \(i=\sqrt{-1}\). All real and imaginary numbers are complex (since real numbers can be written as \(a + 0i\), imaginary as \(0+bi\)).
• Imaginary number: A complex number where \(a = 0\) and \(b

eq0\) (i.e., of the form \(bi\), \(b
eq0\)).

Step2: Categorize \(4 + 2i\)

It is in the form \(a+bi\) with \(a = 4\), \(b = 2\) (both real, \(b
eq0\)). So it is a complex number (and not real or purely imaginary since \(a
eq0\)).

Step3: Categorize \(\sqrt{3}\)

It can be written as \(\sqrt{3}+0i\), so it is a real number (and thus also complex, but real is more specific here).

Step4: Categorize \(9 - 4i\)

Form \(a + bi\) with \(a = 9\), \(b=-4\) (real, \(b
eq0\)). So complex number.

Step5: Categorize \(-2.5i\)

Form \(0+(-2.5)i\) ( \(a = 0\), \(b=-2.5
eq0\) ). So it is an imaginary number (and also complex, since imaginary numbers are a subset of complex).

Answer:

• \(4 + 2i\): Complex (select the circle under Complex)
• \(\sqrt{3}\): Real (select the circle under Real)
• \(9 - 4i\): Complex (select the circle under Complex)
• \(-2.5i\): Imaginary (select the circle under Imaginary)