This worksheet helps students master the real number system by classifying values like pi and square roots and testing their knowledge with true/false statements.

Let's solve this step by step.

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Part 1: Classify each number

We need to classify each number under the correct categories:

- Real: All numbers on the number line (includes rational and irrational).
- Rational: Can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers, $q \ne 0$. Includes terminating and repeating decimals.
- Irrational: Cannot be expressed as a fraction; non-repeating, non-terminating decimals.
- Integer: Whole numbers and their negatives: ..., -2, -1, 0, 1, 2, ...
- Whole: Non-negative integers: 0, 1, 2, 3, ...
- Counting: Positive integers: 1, 2, 3, ...

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#### 1. -19
- Real: ✔
- Rational: ✔ (can be written as $-\frac{19}{1}$)
- Irrational: ✘
- Integer: ✔
- Whole: ✘ (negative)
- Counting: ✘

✔ Real, Rational, Integer

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#### 2. π (pi)
- Real: ✔
- Rational: ✘ (π is irrational)
- Irrational: ✔
- Integer: ✘
- Whole: ✘
- Counting: ✘

✔ Real, Irrational

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#### 3. $-\frac{2}{3}$
- Real: ✔
- Rational: ✔ (it's a fraction)
- Irrational: ✘
- Integer: ✘ (not a whole number)
- Whole: ✘
- Counting: ✘

✔ Real, Rational

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#### 4. 42
- Real: ✔
- Rational: ✔ ($\frac{42}{1}$)
- Irrational: ✘
- Integer: ✔
- Whole: ✔ (non-negative integer)
- Counting: ✔ (positive integer)

✔ Real, Rational, Integer, Whole, Counting

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#### 5. $\sqrt{8}$
- Simplify: $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2} \approx 2.828...$
- This is irrational because $\sqrt{2}$ is irrational.
- Real: ✔
- Rational: ✘
- Irrational: ✔
- Integer: ✘
- Whole: ✘
- Counting: ✘

✔ Real, Irrational

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#### 6. 0.582
- Terminating decimal → can be written as a fraction: $\frac{582}{1000} = \frac{291}{500}$
- Real: ✔
- Rational: ✔
- Irrational: ✘
- Integer: ✘
- Whole: ✘
- Counting: ✘

✔ Real, Rational

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#### 7. 0
- Real: ✔
- Rational: ✔ ($\frac{0}{1}$)
- Irrational: ✘
- Integer: ✔
- Whole: ✔ (whole numbers include 0)
- Counting: ✘ (counting starts at 1)

✔ Real, Rational, Integer, Whole

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#### 8. 2.51821...
- The notation "..." suggests it’s a non-repeating, non-terminating decimal.
- If it's non-repeating, it’s irrational.
- But note: we don’t know for sure unless specified. However, since it’s written with ellipsis and no pattern, assume it’s non-repeating → irrational.
- Real: ✔
- Rational: ✘
- Irrational: ✔
- Integer: ✘
- Whole: ✘
- Counting: ✘

✔ Real, Irrational

> ⚠️ Note: If it were repeating (e.g., 2.51821821...), it would be rational. But here, it's just shown as "2.51821..." — likely meant to represent an irrational number.

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#### 9. $\sqrt{25}$
- $\sqrt{25} = 5$, which is a whole number.
- Real: ✔
- Rational: ✔
- Irrational: ✘
- Integer: ✔
- Whole: ✔
- Counting: ✔

✔ Real, Rational, Integer, Whole, Counting

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✔ Final Table:

| Number | Real | Rational | Irrational | Integer | Whole | Counting |
|--------------|------|----------|------------|---------|-------|----------|
| -19 | ✔ | ✔ | ✘ | ✔ | ✘ | ✘ |
| π | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| $-\frac{2}{3}$ | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
| 42 | ✔ | ✔ | ✘ | ✔ | ✔ | ✔ |
| $\sqrt{8}$ | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| 0.582 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
| 0 | ✔ | ✔ | ✘ | ✔ | ✔ | ✘ |
| 2.51821... | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| $\sqrt{25}$ | ✔ | ✔ | ✘ | ✔ | ✔ | ✔ |

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Part 2: True or False Questions

#### 5. The square root of a non-perfect square integer is always irrational.
- True
Example: $\sqrt{2}, \sqrt{3}, \sqrt{5}$ — all irrational.
There is no non-perfect square integer whose square root is rational.

✔ True

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#### 6. A never-ending decimal is always irrational.
- False
Counterexample: $0.\overline{3} = \frac{1}{3}$ — it's repeating, so rational.
Only non-repeating, non-terminating decimals are irrational.

✘ False
Example: $0.\overline{3}$ is a never-ending decimal but rational.

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#### 7. Integers are always rational.
- True
Any integer $n$ can be written as $\frac{n}{1}$ → rational.

✔ True

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#### 8. The square root of a perfect square integer is always rational.
- True
Perfect squares: 1, 4, 9, 16, 25, ...
$\sqrt{1}=1$, $\sqrt{4}=2$, etc. — all integers → rational.

✔ True

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#### 9. All fractions are rational.
- False
Wait — only if they are ratios of integers.
But “fraction” in math usually means $\frac{a}{b}$ where $a,b$ are integers, $b \ne 0$ → that’s the definition of rational.
However, if someone says "fraction" like $\frac{\pi}{2}$, then it's not rational.

But in standard math education, fractions refer to rational numbers.

So:
- If "fraction" means ratio of two integers → then yes, all such fractions are rational.
- But if "fraction" includes things like $\frac{\sqrt{2}}{2}$, then it's not rational.

But in this context, "fraction" implies rational number.

✔ So: True — assuming "fraction" means ratio of integers.

However, some might argue: "All fractions are rational" is true only if we define fractions properly.

But in standard math curriculum, all fractions (of integers) are rational.

✔ True

> ⚠️ But let’s clarify: If a fraction has irrational numerator or denominator, it may not be rational. But typically, "fraction" in basic math means rational number.

So, True.

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#### 10. Whole numbers are always rational.
- Whole numbers: 0, 1, 2, 3, ...
- Each can be written as $\frac{n}{1}$ → rational.

✔ True

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✔ Final Answers for #5–10:

| Question | Answer | Explanation |
|--------|--------|-----------|
| 5 | True | Square roots of non-perfect squares are irrational. |
| 6 | False | Repeating decimals (like 0.333...) are never-ending but rational. |
| 7 | True | Integers can be written as fractions (e.g., $5 = \frac{5}{1}$). |
| 8 | True | $\sqrt{25} = 5$, rational. |
| 9 | True | Fractions (ratios of integers) are rational. |
| 10 | True | Whole numbers are integers → rational. |

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✔ Summary:

Table completed as above.
Statements:
5. True
6. False (example: $0.\overline{3}$)
7. True
8. True
9. True
10. True

Let me know if you'd like this formatted as a printable answer sheet!