Real Numbers classification worksheet with a Venn diagram and number placement exercise.

Let's solve the problem step by step based on the image you provided. The task involves classifying numbers as rational or irrational, placing them in the correct location on a Venn diagram, and answering a true/false question.

---

🔷 Understanding the Diagram

The diagram shows a Venn-like structure with two overlapping circles:

- Left Circle: *Integers*
- Inside: Whole Numbers (subset of Integers)
- Examples: -4, 0, 16² (which is 256), 197

- Right Circle: *Irrational Numbers*
- Examples: √(-50), √2

Note:
- Rational Numbers include integers, fractions, terminating decimals, and repeating decimals.
- Irrational Numbers cannot be written as simple fractions; they are non-repeating, non-terminating decimals.
- Whole Numbers are non-negative integers: 0, 1, 2, 3, ...
- Integers include whole numbers and their negatives.

---

✔ Step 1: Classify Each Number

We will evaluate each number and determine whether it is rational or irrational, and where it belongs in the diagram.

---

#### 1) √7
- √7 ≈ 2.6457... → non-repeating, non-terminating
- Cannot be expressed as a fraction
- Irrational

> ⬅️ Place in Irrational Numbers circle

---

#### 2) 14.2
- This is a terminating decimal
- Can be written as 142/10 = 71/5
- Rational

> ⬅️ Place in Rational Numbers, but not in Integers or Whole Numbers

---

#### 3) 8
- Integer, whole number
- Rational (since all integers are rational)

> ⬅️ Place in Integers circle (within Rational Numbers)

---

#### 4) √16
- √16 = 4
- 4 is an integer and whole number
- Rational

> ⬅️ Place in Integers circle

---

#### 5) 2/3
- Fraction
- Terminating? No → 0.666... → repeating
- But still rational because it's a ratio of two integers
- Rational

> ⬅️ Place in Rational Numbers, not in Integers

---

#### 6) √16
Wait — this is repeated from #4. Let's assume it’s a typo or just listed twice.

But let’s check: √16 = 4 → already done.

So likely, #6 is meant to be something else. But assuming it's √16 again, then same as #4.

But looking at your list:
> 6) √16 → same as #4 → 4 → Integer

So place in Integers

---

#### 7) 0
- Zero is a whole number and integer
- Rational

> ⬅️ Place in Whole Numbers (inside Integers)

---

#### 8) 0.32
- Terminating decimal
- Can be written as 32/100 = 8/25
- Rational

> ⬅️ Place in Rational Numbers, not in Integers

---

#### 9) 7.66...
- Repeating decimal: 7.666...
- Can be written as a fraction: let x = 7.666..., then 10x = 76.666..., subtract: 9x = 69 → x = 69/9 = 23/3
- So it's Rational

> ⬅️ Place in Rational Numbers, not in Integers

---

#### 10) 2/5
- Fraction → 0.4 → terminating
- Rational

> ⬅️ Place in Rational Numbers, not in Integers

---

#### 11) All whole numbers are rational numbers.
- True!
- Every whole number can be written as a fraction: e.g., 5 = 5/1
- So yes, all whole numbers are rational

✔ True

> Answer: True — Explain: Whole numbers are integers, and all integers can be written as fractions with denominator 1, so they are rational.

---

📌 Final Answers:

| Number | Classification | Location |
|--------|----------------|----------|
| 1) √7 | Irrational | Irrational Numbers |
| 2) 14.2 | Rational | Rational (outside Integers) |
| 3) 8 | Rational | Integers |
| 4) √16 | Rational | Integers |
| 5) 2/3 | Rational | Rational (not Integer) |
| 6) √16 | Rational | Integers |
| 7) 0 | Rational | Whole Numbers (in Integers) |
| 8) 0.32 | Rational | Rational (not Integer) |
| 9) 7.66... | Rational | Rational (not Integer) |
| 10) 2/5 | Rational | Rational (not Integer) |
| 11) All whole numbers are rational numbers. | True | Yes — whole numbers are rational |

---

✔ Summary of Placement on Diagram:

- Integers Circle:
- 8, √16 (=4), 0
- Rational Numbers (but not integers):
- 14.2, 2/3, 0.32, 7.66..., 2/5
- Irrational Numbers:
- √7
- Not included:
- √(-50): Not real (imaginary), so not placed (probably just for context)

---

💡 Key Takeaways:

- Rational Numbers: Any number that can be written as a fraction p/q where q ≠ 0. Includes:
- Integers
- Terminating decimals
- Repeating decimals
- Irrational Numbers: Cannot be written as fractions; examples: √2, π, √7
- All integers are rational, but not all rational numbers are integers.
- All whole numbers are integers, and thus rational.

---

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