This worksheet features a table where students identify number sets—such as natural, whole, integer, rational, irrational, and real—for a list of twenty values.

To solve the problem, we need to identify which of the given numbers belong to each of the following sets: Natural Numbers, Whole Numbers, Integers, Rational Numbers, and Irrational Numbers. Let's go through each number step by step.

Definitions Recap:

1. Natural Numbers (ℕ): Positive integers starting from 1. Examples: \(1, 2, 3, \ldots\).
2. Whole Numbers: Non-negative integers starting from 0. Examples: \(0, 1, 2, 3, \ldots\).
3. Integers (ℤ): All whole numbers and their negatives. Examples: \(\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\).
4. Rational Numbers (ℚ): Numbers that can be expressed as a ratio of two integers \(\frac{p}{q}\), where \(q \neq 0\). Examples: \(\frac{1}{2}, 0.5, -3, 0.333\ldots\).
5. Irrational Numbers: Numbers that cannot be expressed as a ratio of two integers. They have non-repeating, non-terminating decimal expansions. Examples: \(\sqrt{2}, \pi, e\).

Analysis of Each Number:

#### 1. \( \sqrt{25} \)
- Simplify: \( \sqrt{25} = 5 \)
- Natural Number: Yes
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(5 = \frac{5}{1}\))
- Irrational Number: No

#### 2. \(-7\)
- Natural Number: No (negative)
- Whole Number: No (negative)
- Integer: Yes
- Rational Number: Yes (since \(-7 = \frac{-7}{1}\))
- Irrational Number: No

#### 3. \(0\)
- Natural Number: No (0 is not included in natural numbers)
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(0 = \frac{0}{1}\))
- Irrational Number: No

#### 4. \(\emptyset\) (Empty Set)
- This is not a number but a set. It does not belong to any of the number sets.
- Natural Number: No
- Whole Number: No
- Integer: No
- Rational Number: No
- Irrational Number: No

#### 5. \(0.23\)
- Natural Number: No (decimal)
- Whole Number: No (decimal)
- Integer: No (decimal)
- Rational Number: Yes (since \(0.23 = \frac{23}{100}\))
- Irrational Number: No

#### 6. \(4.\overline{8}\) (Repeating Decimal)
- Natural Number: No (decimal)
- Whole Number: No (decimal)
- Integer: No (decimal)
- Rational Number: Yes (repeating decimals are rational)
- Irrational Number: No

#### 7. \(10\)
- Natural Number: Yes
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(10 = \frac{10}{1}\))
- Irrational Number: No

#### 8. \(\frac{1}{2}\)
- Natural Number: No (fraction)
- Whole Number: No (fraction)
- Integer: No (fraction)
- Rational Number: Yes (since it is a fraction)
- Irrational Number: No

#### 9. \(\sqrt{121}\)
- Simplify: \( \sqrt{121} = 11 \)
- Natural Number: Yes
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(11 = \frac{11}{1}\))
- Irrational Number: No

#### 10. \(\frac{6}{3}\)
- Simplify: \( \frac{6}{3} = 2 \)
- Natural Number: Yes
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(2 = \frac{2}{1}\))
- Irrational Number: No

#### 11. \(4\)
- Natural Number: Yes
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(4 = \frac{4}{1}\))
- Irrational Number: No

#### 12. \(\sqrt{49}\)
- Simplify: \( \sqrt{49} = 7 \)
- Natural Number: Yes
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(7 = \frac{7}{1}\))
- Irrational Number: No

#### 13. \(\frac{1}{3}\)
- Natural Number: No (fraction)
- Whole Number: No (fraction)
- Integer: No (fraction)
- Rational Number: Yes (since it is a fraction)
- Irrational Number: No

#### 14. \(-107\)
- Natural Number: No (negative)
- Whole Number: No (negative)
- Integer: Yes
- Rational Number: Yes (since \(-107 = \frac{-107}{1}\))
- Irrational Number: No

#### 15. \(\sqrt{16}\)
- Simplify: \( \sqrt{16} = 4 \)
- Natural Number: Yes
- Whole Number: Yes
- Integer: Yes
- Rational Number: Yes (since \(4 = \frac{4}{1}\))
- Irrational Number: No

#### 16. \(0.18\)
- Natural Number: No (decimal)
- Whole Number: No (decimal)
- Integer: No (decimal)
- Rational Number: Yes (since \(0.18 = \frac{18}{100}\))
- Irrational Number: No

#### 17. \(-10^2\)
- Simplify: \( -10^2 = -100 \)
- Natural Number: No (negative)
- Whole Number: No (negative)
- Integer: Yes
- Rational Number: Yes (since \(-100 = \frac{-100}{1}\))
- Irrational Number: No

#### 18. \(\frac{1}{\sqrt{2}}\)
- Simplify: \( \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2} \)
- Natural Number: No (not an integer)
- Whole Number: No (not an integer)
- Integer: No (not an integer)
- Rational Number: No (involves \(\sqrt{2}\), which is irrational)
- Irrational Number: Yes

Final Answer:

\[
\boxed{
\begin{array}{c|c|c|c|c|c}
\text{#} & \text{Natural Numbers} & \text{Whole Numbers} & \text{Integers} & \text{Rational Numbers} & \text{Irrational Numbers} \\
\hline
1 & \checkmark & \checkmark & \checkmark & \checkmark & \times \\
2 & \times & \times & \checkmark & \checkmark & \times \\
3 & \times & \checkmark & \checkmark & \checkmark & \times \\
4 & \times & \times & \times & \times & \times \\
5 & \times & \times & \times & \checkmark & \times \\
6 & \times & \times & \times & \checkmark & \times \\
7 & \checkmark & \checkmark & \checkmark & \checkmark & \times \\
8 & \times & \times & \times & \checkmark & \times \\
9 & \checkmark & \checkmark & \checkmark & \checkmark & \times \\
10 & \checkmark & \checkmark & \checkmark & \checkmark & \times \\
11 & \checkmark & \checkmark & \checkmark & \checkmark & \times \\
12 & \checkmark & \checkmark & \checkmark & \checkmark & \times \\
13 & \times & \times & \times & \checkmark & \times \\
14 & \times & \times & \checkmark & \checkmark & \times \\
15 & \checkmark & \checkmark & \checkmark & \checkmark & \times \\
16 & \times & \times & \times & \checkmark & \times \\
17 & \times & \times & \checkmark & \checkmark & \times \\
18 & \times & \times & \times & \times & \checkmark \\
\end{array}
}
\]