Answer :
To accurately categorize each expression into either purely real numbers or non-real complex numbers, we will examine if each number has an imaginary component or not.
### Purely Real Numbers
Purely real numbers are those that do not have any imaginary component. They only exist on the real number line and can be both positive, negative, or zero. Based on the given data, the following expressions are purely real numbers:
- [tex]\(5\)[/tex]
- [tex]\(-8.3\)[/tex]
- [tex]\(0\)[/tex]
- [tex]\(2.718\)[/tex]
### Non-real Complex Numbers
Non-real complex numbers, also known as imaginary or complex numbers, have both a real part and an imaginary part, or purely an imaginary part. These numbers include an 'i' or 'j' component, where [tex]\(i\)[/tex] or [tex]\(j\)[/tex] is the imaginary unit ([tex]\(i^2 = -1\)[/tex] or [tex]\(j^2 = -1\)[/tex]). Based on the given data, the following expressions are non-real complex numbers:
- [tex]\(3 + 4j\)[/tex] (real part: 3, imaginary part: 4j)
- [tex]\(-7j\)[/tex] (purely imaginary, real part: 0, imaginary part: -7j)
- [tex]\(1 - 2j\)[/tex] (real part: 1, imaginary part: -2j)
- [tex]\(0.5j\)[/tex] (purely imaginary, real part: 0, imaginary part: 0.5j)
Now, we can fill in the table accordingly:
[tex]\[ \begin{tabular}{|l|l|} \hline \textbf{Purely Real Number} & \textbf{Non-real Complex Number} \\ \hline 5 & 3 + 4j \\ -8.3 & -7j \\ 0 & 1 - 2j \\ 2.718 & 0.5j \\ \hline \end{tabular} \][/tex]
Thus, the expressions have been sorted into the appropriate categories.
### Purely Real Numbers
Purely real numbers are those that do not have any imaginary component. They only exist on the real number line and can be both positive, negative, or zero. Based on the given data, the following expressions are purely real numbers:
- [tex]\(5\)[/tex]
- [tex]\(-8.3\)[/tex]
- [tex]\(0\)[/tex]
- [tex]\(2.718\)[/tex]
### Non-real Complex Numbers
Non-real complex numbers, also known as imaginary or complex numbers, have both a real part and an imaginary part, or purely an imaginary part. These numbers include an 'i' or 'j' component, where [tex]\(i\)[/tex] or [tex]\(j\)[/tex] is the imaginary unit ([tex]\(i^2 = -1\)[/tex] or [tex]\(j^2 = -1\)[/tex]). Based on the given data, the following expressions are non-real complex numbers:
- [tex]\(3 + 4j\)[/tex] (real part: 3, imaginary part: 4j)
- [tex]\(-7j\)[/tex] (purely imaginary, real part: 0, imaginary part: -7j)
- [tex]\(1 - 2j\)[/tex] (real part: 1, imaginary part: -2j)
- [tex]\(0.5j\)[/tex] (purely imaginary, real part: 0, imaginary part: 0.5j)
Now, we can fill in the table accordingly:
[tex]\[ \begin{tabular}{|l|l|} \hline \textbf{Purely Real Number} & \textbf{Non-real Complex Number} \\ \hline 5 & 3 + 4j \\ -8.3 & -7j \\ 0 & 1 - 2j \\ 2.718 & 0.5j \\ \hline \end{tabular} \][/tex]
Thus, the expressions have been sorted into the appropriate categories.
