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

The Venn diagram shows:
- Real Numbers → split into Rational and Irrational
- Rational: Can be written as a fraction $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $. Includes terminating decimals, repeating decimals, integers, whole numbers, natural numbers.
- Irrational: Cannot be written as a fraction. Non-repeating, non-terminating decimals.
- Examples: $ \sqrt{2}, \pi, e $
- Integers: Whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Whole Numbers: 0, 1, 2, 3, ...
- Natural Numbers: 1, 2, 3, ... (positive integers)

So the hierarchy is:
> Natural ⊂ Whole ⊂ Integers ⊂ Rational ⊂ Real
> Irrational ⊂ Real

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Now classify each number:

1. -19
- Integer? Yes
- Whole? No (negative)
- Natural? No
- Rational? Yes (can write as $ -\frac{19}{1} $)
→ Integer, Rational

2. $ 1.\overline{2} $ (repeating decimal)
- Repeating decimals are rational
- Not an integer, not whole, not natural
→ Rational only

3. 0
- Integer? Yes
- Whole? Yes
- Natural? No (natural starts at 1)
- Rational? Yes
→ Whole, Integer, Rational

4. 3
- Natural? Yes
- Whole? Yes
- Integer? Yes
- Rational? Yes
→ Natural, Whole, Integer, Rational

5. $ \sqrt{10} $
- $ \sqrt{10} \approx 3.162... $, non-repeating, non-terminating
- Not a perfect square → irrational
→ Irrational

6. $ -\sqrt{81} $
- $ \sqrt{81} = 9 $, so $ -\sqrt{81} = -9 $
- -9 is an integer
→ Integer, Rational

7. 3.456
- Terminating decimal → rational
- Not an integer
→ Rational only

8. $ -\frac{6}{11} $
- Fraction → rational
- Not an integer
→ Rational only

9. -1.4828... (non-repeating, non-terminating)
- Given as non-repeating → irrational
→ Irrational

10. -44
- Integer? Yes
- Whole? No
- Natural? No
- Rational? Yes
→ Integer, Rational

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

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#### Venn Diagram Placement

Now assign each number to the correct region:

- NATURAL: 3
- WHOLE: 0, 3
- INTEGERS: -19, 0, 3, -9 (from $-\sqrt{81}$), -44
- RATIONAL (not in integers): $1.\overline{2}$, 3.456, $-\frac{6}{11}$
- IRRATIONAL: $ \sqrt{10} $, -1.4828..., $ \pi $

Note: $ -\sqrt{81} = -9 $, which is an integer, so it goes in Integers.

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Part 2: Fill in the Table

| Number | Natural | Whole | Integer | Rational | Irrational | Real |
|--------|---------|-------|---------|----------|------------|------|
| a) $ \sqrt{81} $ | | | | | | |
| b) $ 1.\overline{2} $ | | | | | | |
| c) 0 | | | | | | |
| d) 13 | | | | | | |

Let’s evaluate each:

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a) $ \sqrt{81} $
- $ \sqrt{81} = 9 $
- 9 is natural, whole, integer, rational, real
- Not irrational (since it's rational)

→ YES for: Natural, Whole, Integer, Rational, Real
→ NO for: Irrational

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b) $ 1.\overline{2} $
- Repeating decimal → rational
- Not an integer, not whole, not natural
- But rational and real

→ YES: Rational, Real
→ NO: Natural, Whole, Integer, Irrational

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c) 0
- Whole number
- Integer
- Rational
- Real
- Not natural (natural numbers start from 1)
- Not irrational

→ YES: Whole, Integer, Rational, Real
→ NO: Natural, Irrational

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d) 13
- Natural: yes
- Whole: yes
- Integer: yes
- Rational: yes
- Real: yes
- Irrational: no

→ YES: Natural, Whole, Integer, Rational, Real
→ NO: Irrational

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Final Table:

| | 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 |

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

- Whole numbers: 0, 1, 2, 3, ...
- Integers include negative numbers: ..., -3, -2, -1, 0, 1, 2, ...

So any negative integer is an integer but not a whole number.

From the list:
- -19, -44, -9 ($-\sqrt{81}$) → all are integers but not whole

✔ Example: -19 or -44 or $-\sqrt{81}$

Answer: -19 (or any negative integer)

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

Rational numbers include fractions, terminating/repeating decimals.

Examples from list:
- $ 1.\overline{2} $ → rational, not integer
- 3.456 → rational, not integer
- $ -\frac{6}{11} $ → rational, not integer

Any of these work.

✔ Answer: $ 1.\overline{2} $ (or 3.456, or $-\frac{6}{11}$)

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✔ Final Answers:

#### Part 1: Venn Diagram (Placement Summary)

- Natural: 3
- Whole: 0, 3
- Integers: -19, 0, 3, -9, -44
- Rational (non-integers): $1.\overline{2}$, 3.456, $-\frac{6}{11}$
- Irrational: $ \sqrt{10} $, -1.4828..., $ \pi $

> Note: $ -\sqrt{81} = -9 $, so it belongs in Integers, not irrational.

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#### Part 2: Table

| | 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 |

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#### Part 3:
A number that is an integer but not a whole number:
👉 -19 (or -44 or $-\sqrt{81}$)

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#### Part 4:
A rational number that is not an 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!