To determine the correct form of a complex number, let's begin by recalling the definition of a complex number. A complex number is typically expressed in the form [tex]\(a + bi\)[/tex], where:
- [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are real numbers,
- [tex]\(i\)[/tex] is the imaginary unit, which satisfies [tex]\(i^2 = -1\)[/tex].
Given the options provided:
1. [tex]\(i^2\)[/tex]
2. [tex]\(a b i\)[/tex]
3. [tex]\(a + b i\)[/tex]
4. [tex]\(\frac{a}{b i}\)[/tex]
We will analyze each option:
1. [tex]\(i^2\)[/tex]
- [tex]\(i^2\)[/tex] represents the imaginary unit squared. Since [tex]\(i^2 = -1\)[/tex], this is purely a real number and not in the standard form of a complex number.
2. [tex]\(a b i\)[/tex]
- [tex]\(a b i\)[/tex] is a product of real numbers [tex]\(a\)[/tex], [tex]\(b\)[/tex], and the imaginary unit [tex]\(i\)[/tex]. This expression is purely imaginary (assuming [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are non-zero) but does not constitute the standard representation of a complex number.
3. [tex]\(a + b i\)[/tex]
- [tex]\(a + b i\)[/tex] combines the real number [tex]\(a\)[/tex] and the imaginary number [tex]\(b i\)[/tex]. This is the correct standard form of a complex number.
4. [tex]\(\frac{a}{b i}\)[/tex]
- [tex]\(\frac{a}{b i}\)[/tex] represents a fraction where the denominator is an imaginary number. We could simplify this expression further, but it is not in the standard form [tex]\(a + b i\)[/tex].
After reviewing each option, we can conclude that the correct form of a complex number is:
[tex]\[ \boxed{a + b i} \][/tex]
