Real numbers - Free Printable

Let's solve the problem step by step based on the image you provided. The task involves classifying numbers into Rational Numbers and Irrational Numbers, using a Venn diagram that shows:

- Integers (subset of Rational Numbers)
- Whole Numbers (subset of Integers)
- Rational Numbers (includes integers, fractions, terminating/repeating decimals)
- Irrational Numbers (non-repeating, non-terminating decimals like √2 or π)

We are given a list of numbers to classify and place in the correct region of the diagram.

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Step 1: Understand the Categories

- Whole Numbers: {0, 1, 2, 3, ...} — Non-negative integers
- Integers: {..., -2, -1, 0, 1, 2, ...}
- Rational Numbers: Can be written as a fraction $\frac{p}{q}$ where $p$ and $q$ are integers and $q \neq 0$. Includes:
- Terminating decimals (e.g., 0.5)
- Repeating decimals (e.g., 0.333...)
- All integers and fractions
- Irrational Numbers: Cannot be expressed as a fraction. Examples: √2, π, √3, etc.

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Step 2: Analyze Each Number

#### 1) √7
- √7 ≈ 2.645... — Non-repeating, non-terminating
- Not a perfect square → Irrational

✔ Irrational Numbers

#### 2) 14.2
- Decimal terminates → can be written as $ \frac{142}{10} = \frac{71}{5} $
- Rational

✔ Rational Numbers (not an integer or whole number)

#### 3) 8
- Integer, whole number
- Can be written as $ \frac{8}{1} $ → rational

✔ Whole Numbers (also in Integers and Rational Numbers)

#### 4) √16
- √16 = 4 → Perfect square
- 4 is an integer, whole number, rational

✔ Whole Numbers

#### 5) 2
- Whole number, integer, rational

✔ Whole Numbers

#### 6) √16
- Same as #4 → √16 = 4

✔ Whole Numbers

#### 7) 0
- Whole number, integer, rational

✔ Whole Numbers

#### 8) 0.32
- Terminating decimal → $ \frac{32}{100} = \frac{8}{25} $
- Rational

✔ Rational Numbers (not whole or integer)

#### 9) 7.66...
- Repeating decimal? The notation "..." suggests it might be repeating.
- But written as "7.66" with dots → likely means 7.666... (repeating 6)
- Repeating decimals are rational

So, 7.666... = $ 7 + \frac{2}{3} = \frac{23}{3} $ → rational

✔ Rational Numbers

#### 10) 2/5
- Fraction → already rational

✔ Rational Numbers

#### 11) All whole numbers are rational numbers.
- True. Every whole number $ n $ can be written as $ \frac{n}{1} $
- So they are rational

✔ True

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Step 3: Place Numbers in Diagram

Now, let’s assign each number to the correct region in the Venn diagram:

- Whole Numbers (Innermost circle): 8, √16 (which is 4), 2, 0
- Integers (next ring): includes whole numbers and negative integers, but our list has no negatives, so same as above
- Rational Numbers (outer oval): all numbers except irrational ones
- Irrational Numbers (separate blue box): only √7

Wait — check: √16 = 4 → whole number → belongs in Whole Numbers

But note: √7 is not rational → goes in Irrational Numbers

Let’s list final placements:

| Number | Classification |
|--------|----------------|
| √7 | Irrational Numbers |
| 14.2 | Rational Numbers (not integer) |
| 8 | Whole Numbers |
| √16 | Whole Numbers |
| 2 | Whole Numbers |
| √16 | Whole Numbers |
| 0 | Whole Numbers |
| 0.32 | Rational Numbers |
| 7.66...| Rational Numbers |
| 2/5 | Rational Numbers |

Note: √16 appears twice — both times it's 4 → whole number

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Final Answers:

1) √7 → Irrational Numbers
2) 14.2 → Rational Numbers
3) 8 → Whole Numbers
4) √16 → Whole Numbers
5) 2 → Whole Numbers
6) √16 → Whole Numbers
7) 0 → Whole Numbers
8) 0.32 → Rational Numbers
9) 7.66... → Rational Numbers
10) 2/5 → Rational Numbers
11) All whole numbers are rational numbers → True

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Explanation for #11:

Yes, all whole numbers are rational numbers because any whole number $ n $ can be written as $ \frac{n}{1} $, which is a ratio of two integers. Therefore, they are rational.

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Summary:

- Whole Numbers: 8, √16 (4), 2, 0
- Rational Numbers (but not whole): 14.2, 0.32, 7.66..., 2/5
- Irrational Numbers: √7

✔ All classifications are correct.

Let me know if you'd like a visual representation or help drawing the diagram!