Classifying & Ordering Real Numbers-Part 1 interactive worksheet ... - Free Printable

Let's solve each part of this worksheet step by step.

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Part 1: Classifying & Ordering Real Numbers

We are given a list of numbers and need to place them in the correct regions of the Venn diagram based on their classification:

Numbers:
- -19
- $ 1.\overline{2} $ (repeating decimal)
- 0
- 3
- $ \sqrt{10} $
- $ -\sqrt{81} $
- 3.456
- $ -\frac{6}{11} $
- -1.4828... (non-repeating, non-terminating)
- -44
- $ \pi $

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#### Step 1: Understand the Categories

- Natural Numbers: Positive integers starting from 1 → {1, 2, 3, ...}
- Whole Numbers: Natural numbers + 0 → {0, 1, 2, 3, ...}
- Integers: Whole numbers and their negatives → {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Rational Numbers: Can be written as a fraction $ \frac{p}{q} $, where $ p, q $ are integers and $ q \ne 0 $. Includes terminating or repeating decimals.
- Irrational Numbers: Cannot be expressed as a fraction. Non-repeating, non-terminating decimals (e.g., $ \sqrt{2}, \pi $).
- Real Numbers: All rational and irrational numbers.

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Now classify each number:

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1. -19
- Integer (yes) → not whole, not natural
- Rational (yes) → can be written as $ -\frac{19}{1} $
- So: Integer, Rational

2. $ 1.\overline{2} $
- Repeating decimal → Rational
- Not an integer → not whole or natural
- So: Rational

3. 0
- Whole number → also integer, rational
- Not natural (natural starts at 1)
- So: Whole, Integer, Rational

4. 3
- Natural → also whole, integer, rational
- So: Natural, Whole, Integer, Rational

5. $ \sqrt{10} $
- $ \sqrt{10} \approx 3.162... $, non-repeating, non-terminating → Irrational
- So: Irrational

6. $ -\sqrt{81} $
- $ \sqrt{81} = 9 $, so $ -\sqrt{81} = -9 $
- Integer, rational
- So: Integer, Rational

7. 3.456
- Terminating decimal → Rational
- Not an integer → not whole or natural
- So: Rational

8. $ -\frac{6}{11} $
- Fraction → Rational
- Not an integer → not whole or natural
- So: Rational

9. -1.4828...
- The dots suggest non-repeating, non-terminating → likely irrational
- If it's truly non-repeating, then Irrational
- So: Irrational

10. -44
- Integer, rational
- Not whole or natural
- So: Integer, Rational

11. $ \pi $
- Famous irrational number → Irrational

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Venn Diagram Placement

Now place each number:

#### NATURAL:
→ 3

#### WHOLE:
→ 0, 3

#### INTEGERS:
→ -19, 0, 3, -9 (from $-\sqrt{81}$), -44

#### RATIONAL:
→ -19, $1.\overline{2}$, 0, 3, 3.456, $-\frac{6}{11}$, -9, -44

#### IRRATIONAL:
→ $ \sqrt{10} $, -1.4828..., $ \pi $

Note: $ -\sqrt{81} = -9 $, which is rational.

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Part 2: Fill in the Table

| Number | Natural | Whole | Integer | Rational | Irrational | Real |
|--------|---------|-------|---------|----------|------------|------|
| a) $ \sqrt{81} $ | YES | YES | YES | YES | NO | YES |
| b) $ 1.\overline{2} $ | NO | NO | NO | YES | NO | YES |
| c) 0 | NO | YES | YES | YES | NO | YES |
| d) 13 | YES | YES | YES | YES | NO | YES |

#### Explanation:
- a) $ \sqrt{81} = 9 $ → natural, whole, integer, rational
- b) $ 1.\overline{2} $ → rational, not integer, not whole/natural
- c) 0 → whole, integer, rational, not natural
- d) 13 → natural, whole, integer, rational

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Part 3: Select a number that is an integer, but not a whole number.

Answer: Any negative integer.

Examples:
- -19
- -44
- -9 ($-\sqrt{81}$)

✔ Choose: -19

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Part 4: Select a Rational number that is not an Integer.

Any rational number that is not a whole number or negative whole number.

Examples:
- $ 1.\overline{2} $
- 3.456
- $ -\frac{6}{11} $
- $ \frac{1}{2} $ (not listed, but any fraction)

✔ Choose: $ 1.\overline{2} $ or 3.456 or $ -\frac{6}{11} $

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✔ Final Answers:

#### Part 1: Venn Diagram
- NATURAL: 3
- WHOLE: 0, 3
- INTEGERS: -19, 0, 3, -9, -44
- RATIONAL: -19, $1.\overline{2}$, 0, 3, 3.456, $-\frac{6}{11}$, -9, -44
- IRRATIONAL: $ \sqrt{10} $, -1.4828..., $ \pi $

> Note: $ -\sqrt{81} = -9 $ → rational and integer.

#### Part 2: Table

| Number | Natural | Whole | Integer | Rational | Irrational | Real |
|--------|---------|-------|---------|----------|------------|------|
| a) $ \sqrt{81} $ | YES | YES | YES | YES | NO | YES |
| b) $ 1.\overline{2} $ | NO | NO | NO | YES | NO | YES |
| c) 0 | NO | YES | YES | YES | NO | YES |
| d) 13 | YES | YES | YES | YES | NO | YES |

#### Part 3: Integer but not whole number
→ -19 (or -44, or -9)

#### Part 4: Rational but not integer
→ $ 1.\overline{2} $ (or 3.456, or $-\frac{6}{11}$)

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Let me know if you'd like a visual version of the Venn diagram!