Certainly! Let's evaluate each number and determine the correct labels based on their mathematical properties.
1. [tex]\( \frac{1}{5} \)[/tex]:
- Real: Yes, it is a real number.
- Rational: No, it cannot be expressed as an integer or simple fraction.
- Irrational: Yes, since it is not a rational number.
- Integer: No, it is not a whole number.
- Whole: No, it is not a non-negative integer.
- Counting: No, it is not a positive integer.
2. [tex]\(-100.5\)[/tex]:
- Real: Yes, it is a real number.
- Rational: No, it cannot be simply expressed as a fraction.
- Irrational: Yes, since it is not a rational number.
- Integer: No, it is not a whole number.
- Whole: No, it is not a non-negative integer.
- Counting: No, it is not a positive integer.
3. [tex]\(22\)[/tex]:
- Real: Yes, it is a real number.
- Rational: Yes, it is a rational number (can be expressed as [tex]\(\frac{22}{1}\)[/tex]).
- Irrational: No, it is a rational number.
- Integer: Yes, it is an integer.
- Whole: Yes, it is a non-negative integer.
- Counting: Yes, it is a positive integer.
4. [tex]\(2\sqrt{6}\)[/tex]:
- Real: Yes, it is a real number.
- Rational: No, it cannot be expressed as a simple fraction.
- Irrational: Yes, since it is not a rational number.
- Integer: No, it is not an integer.
- Whole: No, it is not a non-negative integer.
- Counting: No, it is not a positive integer.
5. [tex]\(-0.45311\)[/tex]:
- Real: Yes, it is a real number.
- Rational: No, it cannot be simply expressed as a fraction.
- Irrational: Yes, since it is not a rational number.
- Integer: No, it is not an integer.
- Whole: No, it is not a non-negative integer.
- Counting: No, it is not a positive integer.
6. [tex]\(-\frac{10}{3}\)[/tex]:
- Real: Yes, it is a real number.
- Rational: No, it cannot be expressed as a simple fraction.
- Irrational: Yes, since it is not a rational number.
- Integer: No, it is not an integer.
- Whole: No, it is not a non-negative integer.
- Counting: No, it is not a positive integer.
Now, let's fill in the table accordingly:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
& Real & Rational & Irrational & Integer & Whole & Counting \\
\hline
[tex]$1 / 5$[/tex] & ✓ & & ✓ & & & \\
\hline
-100.5 & ✓ & & ✓ & & & \\
\hline
22 & ✓ & ✓ & & ✓ & ✓ & ✓ \\
\hline
[tex]$2 \sqrt{6}$[/tex] & ✓ & & ✓ & & & \\
\hline
[tex]$-0.45311$[/tex] & ✓ & & ✓ & & & \\
\hline
[tex]$-10 / 3$[/tex] & ✓ & & ✓ & & & \\
\hline
\end{tabular}
