Solved Name: Period: Number Classification Worksheet 1) | Chegg.com - Free Printable

Let’s solve each part of this Number Classification Worksheet step by step, explaining the reasoning behind each classification.

---

## 🔢 1) Re-write each number in the Venn Diagram where it belongs.

We are given a nested Venn diagram:

```
REAL NUMBERS
├── RATIONAL
│ └── INTEGERS
│ └── WHOLE
│ └── NATURAL
└── IRRATIONAL
```

Definitions to remember:

- Natural Numbers (ℕ): Positive integers starting from 1 → {1, 2, 3, ...}
- Whole Numbers: Natural numbers + 0 → {0, 1, 2, 3, ...}
- Integers (ℤ): Whole numbers + negatives → {..., -2, -1, 0, 1, 2, ...}
- Rational Numbers (ℚ): Any number that can be written as a fraction `a/b` where `a` and `b` are integers and `b ≠ 0`. Includes terminating or repeating decimals.
- Irrational Numbers: Real numbers that cannot be expressed as a simple fraction — non-repeating, non-terminating decimals.
- Real Numbers (ℝ): All rational and irrational numbers combined.

---

Let's classify each number:

#### ✔ `-19`
→ Integer, Rational, Real
*(Negative integer → not whole or natural)*

#### ✔ `1.2̅` (which means 1.222...)
→ Rational (repeating decimal), Real
*(Can be written as a fraction: 1.2̅ = 11/9)*

#### ✔ `0`
→ Whole, Integer, Rational, Real
*(Not natural — natural numbers start at 1)*

#### ✔ `3`
→ Natural, Whole, Integer, Rational, Real

#### ✔ `√10`
→ Irrational, Real
*(√10 ≈ 3.162… — non-repeating, non-terminating; not a perfect square)*

#### ✔ `√81`
→ √81 = 9 → Natural, Whole, Integer, Rational, Real

#### ✔ `3.456`
→ Rational (terminating decimal), Real
*(Can be written as 3456/1000)*

#### ✔ `-⁴√11` (negative fourth root of 11)
→ Irrational, Real
*(Fourth root of 11 is irrational; negative doesn’t change that)*

#### ✔ `-1.48298……` (with ellipsis — implies non-repeating, non-terminating)
→ Irrational, Real
*(Assuming it’s not repeating — if it were repeating, it would say so like 1.2̅)*

#### ✔ `π + 3`
→ Irrational, Real
*(π is irrational; adding rational number (3) to irrational gives irrational)*

#### ✔ `-44`
→ Integer, Rational, Real
*(Negative integer → not whole or natural)*

---

📌 Where to place them in the Venn diagram:

You’d write each number in the smallest set it belongs to, but since it’s a Venn diagram, you can put them in the appropriate region.

> NATURAL: 3, √81 (since √81=9)

> WHOLE (but not natural): 0

> INTEGERS (but not whole): -19, -44

> RATIONAL (but not integer): 1.2̅, 3.456

> IRRATIONAL: √10, -⁴√11, -1.48298..., π + 3

*(Note: All numbers here are real — no complex numbers!)*

---

## 🧩 2) List all classifications of the number.

a) `√10`

→ Irrational, Real

*(Cannot be expressed as a fraction; decimal is non-repeating/non-terminating)*

---

b) `-44`

→ Integer, Rational, Real

*(Negative whole number → integer; all integers are rational and real)*

---

c) `3`

→ Natural, Whole, Integer, Rational, Real

*(Smallest positive integer → fits all these categories)*

---

d) `-⁴√11`

→ Irrational, Real

*(Fourth root of 11 is irrational; negative sign doesn’t make it rational)*

---

## ✔ 3) Check all boxes that apply to the number.

We’ll go one by one.

---

a) `√81`

→ √81 = 9

So:
- ✔️ Natural (yes, 9 is a counting number)
- ✔️ Whole (yes, includes 0 and positives)
- ✔️ Integer (yes)
- ✔️ Rational (yes, 9 = 9/1)
- ✘ Irrational (no, it’s rational)
- ✔️ Real (yes, all above are real)

✔ Check: Natural, Whole, Integer, Rational, Real

---

b) `1.2̅` (1.222...)

→ Repeating decimal → Rational

- ✘ Natural (not a whole number)
- ✘ Whole (not an integer)
- ✘ Integer
- ✔️ Rational (yes, repeating decimal = rational)
- ✘ Irrational
- ✔️ Real

✔ Check: Rational, Real

---

c) `0`

→ Whole number, but not natural.

- ✘ Natural (natural starts at 1)
- ✔️ Whole (yes)
- ✔️ Integer (yes)
- ✔️ Rational (0 = 0/1)
- ✘ Irrational
- ✔️ Real

✔ Check: Whole, Integer, Rational, Real

---

d) `13`

→ Positive integer

- ✔️ Natural
- ✔️ Whole
- ✔️ Integer
- ✔️ Rational
- ✘ Irrational
- ✔️ Real

✔ Check: Natural, Whole, Integer, Rational, Real

---

## 🎯 Final Answers Summary:

---

1) Venn Diagram Placement (by category):

- Natural: 3, √81
- Whole (non-natural): 0
- Integer (non-whole): -19, -44
- Rational (non-integer): 1.2̅, 3.456
- Irrational: √10, -⁴√11, -1.48298..., π + 3

*(All are real numbers.)*

---

2) Classifications:

a) `√10` → Irrational, Real
b) `-44` → Integer, Rational, Real
c) `3` → Natural, Whole, Integer, Rational, Real
d) `-⁴√11` → Irrational, Real

---

3) Checkboxes:

| Number | Natural | Whole | Integer | Rational | Irrational | Real |
|-------------|---------|-------|---------|----------|------------|------|
| a) √81 | ✔️ | ✔️ | ✔️ | ✔️ | ✘ | ✔️ |
| b) 1.2̅ | ✘ | ✘ | ✘ | ✔️ | ✘ | ✔️ |
| c) 0 | ✘ | ✔️ | ✔️ | ✔️ | ✘ | ✔️ |
| d) 13 | ✔️ | ✔️ | ✔️ | ✔️ | ✘ | ✔️ |

---

✔ You’re all set! This worksheet reinforces understanding of number sets — crucial for algebra and higher math. Let me know if you want a visual sketch of the Venn diagram with placements!