Comprehensive worksheet designed to help students practice classifying numbers within the real number system hierarchy.

Problem Analysis and Solution

The task involves classifying numbers into categories of the real number system and determining the truthfulness of statements about these numbers. Let's break it down step by step.

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#### Part 1: Classifying Numbers

We need to classify each number under the following categories:
- Real: All numbers on the number line.
- Rational: Numbers that can be expressed as a ratio of two integers (\( \frac{p}{q} \), where \( q \neq 0 \)).
- Irrational: Numbers that cannot be expressed as a ratio of two integers and have non-repeating, non-terminating decimal expansions.
- Integer: Whole numbers and their negatives (e.g., ..., -2, -1, 0, 1, 2, ...).
- Whole: Non-negative integers (0, 1, 2, 3, ...).
- Counting: Positive integers (1, 2, 3, ...).

##### Numbers to Classify:
1. \(-19\)
2. \(\pi\)
3. \(-\frac{2}{3}\)
4. \(42\)
5. \(\sqrt{8}\)
6. \(0.582\)
7. \(0\)
8. \(2.51821...\) (repeating or non-repeating?)
9. \(\sqrt{25}\)

##### Classification:

1. \(-19\):
- Real: Yes
- Rational: Yes (it is an integer, which is a rational number)
- Irrational: No
- Integer: Yes
- Whole: No (negative)
- Counting: No (negative)

2. \(\pi\):
- Real: Yes
- Rational: No (it is an irrational number)
- Irrational: Yes
- Integer: No
- Whole: No
- Counting: No

3. \(-\frac{2}{3}\):
- Real: Yes
- Rational: Yes (it is a fraction)
- Irrational: No
- Integer: No
- Whole: No
- Counting: No

4. \(42\):
- Real: Yes
- Rational: Yes (it is an integer)
- Irrational: No
- Integer: Yes
- Whole: Yes
- Counting: Yes

5. \(\sqrt{8}\):
- Real: Yes
- Rational: No (since \( \sqrt{8} = 2\sqrt{2} \), and \( \sqrt{2} \) is irrational)
- Irrational: Yes
- Integer: No
- Whole: No
- Counting: No

6. \(0.582\):
- Real: Yes
- Rational: Yes (it is a terminating decimal, which can be expressed as a fraction)
- Irrational: No
- Integer: No
- Whole: No
- Counting: No

7. \(0\):
- Real: Yes
- Rational: Yes (it can be expressed as \( \frac{0}{1} \))
- Irrational: No
- Integer: Yes
- Whole: Yes
- Counting: No (not positive)

8. \(2.51821...\):
- Real: Yes
- Rational: Depends on whether the decimal repeats or not.
- If it repeats (e.g., \(2.51821821...\)), it is rational.
- If it does not repeat (non-terminating, non-repeating), it is irrational.
- Irrational: Only if it does not repeat.
- Integer: No
- Whole: No
- Counting: No

9. \(\sqrt{25}\):
- Real: Yes
- Rational: Yes (\( \sqrt{25} = 5 \), which is an integer)
- Irrational: No
- Integer: Yes
- Whole: Yes
- Counting: Yes

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#### Part 2: Determining Truth of Statements

We need to determine if the given statements are true or false and provide examples if false.

##### Statement 5: The square root of a non-perfect square integer is always irrational.
- True: The square root of a non-perfect square integer (e.g., \( \sqrt{2}, \sqrt{3}, \sqrt{8} \)) is always irrational because it cannot be expressed as a ratio of two integers.

##### Statement 6: A never-ending decimal is always irrational.
- False: A never-ending decimal can be either rational or irrational.
- Example: \(0.3333...\) (repeating decimal) is rational (\( \frac{1}{3} \)).
- Counterexample: \(0.1010010001...\) (non-repeating, non-terminating) is irrational.

##### Statement 7: Integers are always rational.
- True: Every integer can be expressed as a fraction with a denominator of 1 (e.g., \(5 = \frac{5}{1}\)).

##### Statement 8: The square root of a perfect square integer is always rational.
- True: The square root of a perfect square integer (e.g., \( \sqrt{4} = 2, \sqrt{9} = 3 \)) is always an integer, which is rational.

##### Statement 9: All fractions are rational.
- True: By definition, a fraction is a number that can be expressed as \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \), which fits the definition of a rational number.

##### Statement 10: Whole numbers are always rational.
- True: Whole numbers are non-negative integers, and every integer can be expressed as a fraction with a denominator of 1 (e.g., \(5 = \frac{5}{1}\)).

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Final Answer

#### Classification Table:
| Number | Real | Rational | Irrational | Integer | Whole | Counting |
|--------------|------|----------|------------|---------|-------|----------|
| \(-19\) | √ | √ | | √ | | |
| \(\pi\) | √ | | √ | | | |
| \(-\frac{2}{3}\)| √ | √ | | | | |
| \(42\) | √ | √ | | √ | √ | √ |
| \(\sqrt{8}\) | √ | | √ | | | |
| \(0.582\) | √ | √ | | | | |
| \(0\) | √ | √ | | √ | √ | |
| \(2.51821...\)| √ | Depends | Depends | | | |
| \(\sqrt{25}\)| √ | √ | | √ | √ | √ |

#### Statements:
1. Statement 5: True
2. Statement 6: False (Example: \(0.3333...\) is rational)
3. Statement 7: True
4. Statement 8: True
5. Statement 9: True
6. Statement 10: True

Final Answer Boxed:
\[
\boxed{
\begin{array}{c}
\text{See classification table above.} \\
\text{Statements: 5 (True), 6 (False), 7 (True), 8 (True), 9 (True), 10 (True).}
\end{array}
}
\]