Let's classify each case to determine if the expression [tex]\(a + bi\)[/tex] represents a real number or a non-real complex number. The key point is to identify the value of [tex]\(b\)[/tex]:
- If [tex]\(b = 0\)[/tex], the expression [tex]\(a + bi\)[/tex] simplifies to [tex]\(a\)[/tex], which is a real number.
- If [tex]\(b \neq 0\)[/tex], the expression [tex]\(a + bi\)[/tex] remains a complex number that is not purely real because it includes an imaginary component.
Now let's apply this to each provided case:
1. Case: [tex]\(a = 0; b = 0\)[/tex]
- The expression becomes [tex]\(0 + 0i = 0\)[/tex].
- Since the imaginary part [tex]\(b\)[/tex] is 0, the expression is a real number.
- Classification: Real Number
2. Case: [tex]\(a = 0; b \neq 0\)[/tex]
- The expression becomes [tex]\(0 + bi\)[/tex], where [tex]\(b\)[/tex] is not 0.
- Since there is a non-zero imaginary part, the expression is not purely real and is a non-real complex number.
- Classification: Non-Real Complex Number
3. Case: [tex]\(a \neq 0; b = 0\)[/tex]
- The expression becomes [tex]\(a + 0i = a\)[/tex], where [tex]\(a\)[/tex] is not 0.
- Since the imaginary part [tex]\(b\)[/tex] is 0, the expression is a real number.
- Classification: Real Number
4. Case: [tex]\(a \neq 0; b \neq 0\)[/tex]
- The expression becomes [tex]\(a + bi\)[/tex], where both [tex]\(a\)[/tex] and [tex]\(b\)[/tex] are not 0.
- Since there is a non-zero imaginary part, the expression is not purely real and is a non-real complex number.
- Classification: Non-Real Complex Number
Therefore, the classification of each case is as follows:
1. [tex]\(a = 0; b = 0\)[/tex]: Real Number
2. [tex]\(a = 0; b \neq 0\)[/tex]: Non-Real Complex Number
3. [tex]\(a \neq 0; b = 0\)[/tex]: Real Number
4. [tex]\(a \neq 0; b \neq 0\)[/tex]: Non-Real Complex Number
