Practice worksheet for understanding the real number system, including classification of numbers and true/false statements about number properties.

Let’s go step by step to solve each part of this worksheet.

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Part I: Classify each number

We need to check which categories apply to each number:

- Real: All numbers on the number line — includes rational and irrational.
- Rational: Can be written as a fraction (p/q, where q ≠ 0). Includes integers, terminating decimals, repeating decimals.
- Irrational: Cannot be written as a fraction. Non-repeating, non-terminating decimals.
- Integer: Whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Whole: Non-negative integers: 0, 1, 2, 3, ...
- Counting: Positive integers starting from 1: 1, 2, 3, ... (sometimes called “natural numbers”)

Now let’s classify each:

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1. –19

- Real? → Yes (all these are real unless specified otherwise)
- Rational? → Yes (can write as -19/1)
- Irrational? → No
- Integer? → Yes (negative whole number)
- Whole? → No (whole numbers start at 0 and go up; no negatives)
- Counting? → No (counting starts at 1; no negatives)

✔ Checks: Real, Rational, Integer

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2. π (pi)

- Real? → Yes
- Rational? → No (it’s a famous irrational number)
- Irrational? → Yes
- Integer? → No
- Whole? → No
- Counting? → No

✔ Checks: Real, Irrational

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3. –2/3

- Real? → Yes
- Rational? → Yes (it’s a fraction)
- Irrational? → No
- Integer? → No (not a whole number)
- Whole? → No
- Counting? → No

✔ Checks: Real, Rational

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4. 42

- Real? → Yes
- Rational? → Yes (42/1)
- Irrational? → No
- Integer? → Yes
- Whole? → Yes (non-negative integer)
- Counting? → Yes (positive integer ≥1)

✔ Checks: Real, Rational, Integer, Whole, Counting

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5. √8

First, simplify: √8 = √(4×2) = 2√2 ≈ 2.828... — not a perfect square root.

- Real? → Yes
- Rational? → No (√2 is irrational, so 2√2 is also irrational)
- Irrational? → Yes
- Integer? → No
- Whole? → No
- Counting? → No

✔ Checks: Real, Irrational

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6. 0.582

This is a terminating decimal.

- Real? → Yes
- Rational? → Yes (can write as 582/1000)
- Irrational? → No
- Integer? → No
- Whole? → No
- Counting? → No

✔ Checks: Real, Rational

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7. 0

- Real? → Yes
- Rational? → Yes (0/1)
- Irrational? → No
- Integer? → Yes
- Whole? → Yes (includes 0)
- Counting? → No (counting starts at 1)

✔ Checks: Real, Rational, Integer, Whole

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8. 2.51821...

The "..." means it goes on forever. But does it repeat or not?

Since it says “2.51821...” without indicating repetition, we assume it’s non-repeating → likely meant to be irrational.

But wait — if it were repeating, it would be rational. The problem doesn’t specify, but in context with other problems like √8 and π, this is probably intended to be an example of a non-repeating, non-terminating decimal → irrational.

However, technically, without more info, we can’t be sure. But since it’s listed alongside known irrationals and has ellipsis, we’ll treat it as irrational.

- Real? → Yes
- Rational? → No (assumed non-repeating)
- Irrational? → Yes
- Integer? → No
- Whole? → No
- Counting? → No

✔ Checks: Real, Irrational

*(Note: If the student later learns that some patterns repeat even if not obvious, they might reconsider — but for now, based on typical worksheet design, this is irrational.)*

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9. √25

Simplify: √25 = 5

- Real? → Yes
- Rational? → Yes (5/1)
- Irrational? → No
- Integer? → Yes
- Whole? → Yes
- Counting? → Yes

✔ Checks: Real, Rational, Integer, Whole, Counting

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Part II: True or False?

For each, decide if true or false. If false, give an example.

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5. The square root of a non-perfect square integer is always irrational.

→ TRUE.

Why? Because if you take the square root of an integer that isn’t a perfect square (like 2, 3, 5, 8), you get a number that cannot be written as a fraction — it’s irrational. Example: √2, √3, √8 — all irrational.

✔ Answer: True

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6. A never-ending decimal is always irrational.

→ FALSE.

Counterexample: 0.333... (which is 1/3) never ends, but it repeats — so it’s rational.

Another: 0.121212... = 12/99 = 4/33 — still rational.

Only non-repeating, never-ending decimals are irrational.

✔ Answer: False — Example: 0.333... = 1/3 (rational)

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7. Integers are always rational.

→ TRUE.

Any integer n can be written as n/1 → which fits definition of rational number.

Example: -5 = -5/1, 0 = 0/1, 100 = 100/1 — all rational.

✔ Answer: 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, 36...

Their roots: 1, 2, 3, 4, 5, 6 — all integers → which are rational.

Even negative roots? Like √(-4)? Not real — but the statement says “perfect square integer”, which implies positive perfect squares (since negatives aren’t perfect squares in reals). So yes, always rational.

✔ Answer: True

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9. All fractions are rational.

→ TRUE.

By definition, a rational number is any number that can be expressed as a fraction p/q where p and q are integers and q ≠ 0.

So yes — every fraction (with integer numerator and denominator, denominator ≠ 0) is rational.

Exception? What about fractions with irrational parts? Like √2 / 2? That’s not considered a “fraction” in the basic sense — usually “fraction” here means ratio of two integers.

In middle/high school math, “fraction” typically means p/q with integers → so yes, all such fractions are rational.

✔ Answer: True

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10. Whole numbers are always rational.

→ TRUE.

Whole numbers: 0, 1, 2, 3, ...

Each can be written as itself over 1 → e.g., 0 = 0/1, 5 = 5/1 → rational.

✔ Answer: True

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Final Answers Summary:

Part I Table:

| Number | Real | Rational | Irrational | Integer | Whole | Counting |
|------------|------|----------|------------|---------|-------|----------|
| –19 | ✓ | ✓ | | ✓ | | |
| π | ✓ | | ✓ | | | |
| –2/3 | ✓ | ✓ | | | | |
| 42 | ✓ | ✓ | | ✓ | ✓ | ✓ |
| √8 | ✓ | | ✓ | | | |
| 0.582 | ✓ | ✓ | | | | |
| 0 | ✓ | ✓ | | ✓ | ✓ | |
| 2.51821... | ✓ | | ✓ | | | |
| √25 | ✓ | ✓ | | ✓ | ✓ | ✓ |

Part II:

5. True
6. False — Example: 0.333... = 1/3 (rational)
7. True
8. True
9. True
10. True

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Final Answer:
See completed table above for Part I. For Part II: 5=True, 6=False (example: 0.333...=1/3), 7=True, 8=True, 9=True, 10=True.