Real Number System worksheet for classifying numbers and understanding their relationships.

Let's solve the problem step by step.

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Task:

Classify each number as Natural, Whole, Integer, Rational, or Irrational.
A number can belong to more than one set.

We are given a list of numbers and need to place them in the correct region of the Real Number System Venn Diagram.

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Definitions (Quick Review):

- Natural Numbers: Positive integers starting from 1 → {1, 2, 3, ...}
- Whole Numbers: Natural numbers plus zero → {0, 1, 2, 3, ...}
- Integers: Whole numbers and their negatives → {..., -3, -2, -1, 0, 1, 2, 3, ...}
- Rational Numbers: Numbers that can be written as a fraction $\frac{a}{b}$ where $a$ and $b$ are integers and $b \neq 0$. Includes terminating and repeating decimals.
- Irrational Numbers: Numbers that cannot be written as fractions. Non-repeating, non-terminating decimals (e.g., $\pi$, $\sqrt{2}$).

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Now let’s classify each number:

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#### 1) -19

- Negative integer → Integer
- Not natural, not whole (negative)
- Can be written as $-\frac{19}{1}$ → Rational
- ✔ Integer, Rational

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#### 2) $\frac{1}{4}$

- Fraction → Rational
- Not an integer (not a whole number)
- Not natural
- ✔ Rational

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#### 3) 64

- Positive whole number → Natural, Whole, Integer
- Can be written as $\frac{64}{1}$ → Rational
- ✔ Natural, Whole, Integer, Rational

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#### 4) $\sqrt{2}$

- Square root of 2 is irrational (non-repeating, non-terminating decimal)
- Cannot be expressed as a fraction
- Not an integer, not rational
- ✔ Irrational

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#### 5) -8

- Negative integer → Integer
- Not natural, not whole
- Can be written as $-\frac{8}{1}$ → Rational
- ✔ Integer, Rational

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#### 6) 6.171717...

- Repeating decimal → Rational
- This is $6.\overline{17}$ → can be written as a fraction
- Not an integer
- ✔ Rational

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#### 7) √16

- $\sqrt{16} = 4$
- 4 is a natural number → Natural, Whole, Integer, Rational
- ✔ Natural, Whole, Integer, Rational

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#### 8) -18

- Negative integer → Integer
- Not natural, not whole
- Can be written as $-\frac{18}{1}$ → Rational
- ✔ Integer, Rational

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#### 9) 1,000

- Large positive whole number → Natural, Whole, Integer
- Can be written as $\frac{1000}{1}$ → Rational
- ✔ Natural, Whole, Integer, Rational

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#### 10) The following numbers when placing on the Real Numbers Diagram:

List:
a. 86
b. 0.25
c. 1
d. √8
e. $\frac{1}{2}$
f. -14
g. 2π, 2.838383...
h. √8 (again?)
i. √8 (duplicate?)
j. √8
Wait — it seems like √8 appears multiple times.

But let's go through each:

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##### a. 86

- Natural, Whole, Integer, Rational
✔ Natural, Whole, Integer, Rational

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##### b. 0.25

- Terminating decimal → can be written as $\frac{1}{4}$ → Rational
- Not integer, not natural
✔ Rational

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##### c. 1

- Natural, Whole, Integer, Rational
✔ Natural, Whole, Integer, Rational

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##### d. √8

- $\sqrt{8} = \sqrt{4 \cdot 2} = 2\sqrt{2}$ → irrational (since $\sqrt{2}$ is irrational)
- So Irrational
✔ Irrational

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##### e. $\frac{1}{2}$

- Fraction → Rational
✔ Rational

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##### f. -14

- Negative integer → Integer, Rational
✔ Integer, Rational

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##### g. 2π, 2.838383...

This seems to be two numbers:
- 2π: $2 \times \pi$ → π is irrational, so 2π is also irrational
- 2.838383...: repeating decimal → Rational

So:
- 2π → Irrational
- 2.838383... → Rational

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##### h. √8 → already done → Irrational

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##### i. √8 → same → Irrational

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##### j. √8 → same → Irrational

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✔ Final Classification Summary:

| Number | Classification |
|-------|----------------|
| 1) -19 | Integer, Rational |
| 2) $\frac{1}{4}$ | Rational |
| 3) 64 | Natural, Whole, Integer, Rational |
| 4) $\sqrt{2}$ | Irrational |
| 5) -8 | Integer, Rational |
| 6) 6.171717... | Rational |
| 7) $\sqrt{16}$ | Natural, Whole, Integer, Rational |
| 8) -18 | Integer, Rational |
| 9) 1,000 | Natural, Whole, Integer, Rational |
| 10a) 86 | Natural, Whole, Integer, Rational |
| 10b) 0.25 | Rational |
| 10c) 1 | Natural, Whole, Integer, Rational |
| 10d) $\sqrt{8}$ | Irrational |
| 10e) $\frac{1}{2}$ | Rational |
| 10f) -14 | Integer, Rational |
| 10g) 2π | Irrational |
| 10g) 2.838383... | Rational |
| 10h,i,j) $\sqrt{8}$ | Irrational |

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📌 Placement in Venn Diagram:

The diagram has nested circles:
- Natural ⊂ Whole ⊂ Integer ⊂ Rational
- Irrational is separate from Rational, both under Real Numbers

So:

- Natural: 64, 86, 1, 1000, $\sqrt{16}$ (which is 4)
- Whole: 64, 86, 1, 1000, $\sqrt{16}$, 0 (if included), but not negative numbers
- Integer: -19, -8, -18, -14, 64, 86, 1, 1000, $\sqrt{16}$
- Rational: All except irrational ones → includes all integers, fractions, terminating/repeating decimals
- Irrational: $\sqrt{2}$, $\sqrt{8}$, 2π

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✔ Final Answer (for the chart):

Fill in the blanks accordingly:

```
1) -19 → Integer, Rational
2) 1/4 → Rational
3) 64 → Natural, Whole, Integer, Rational
4) √2 → Irrational
5) -8 → Integer, Rational
6) 6.171717... → Rational
7) √16 → Natural, Whole, Integer, Rational
8) -18 → Integer, Rational
9) 1,000 → Natural, Whole, Integer, Rational
```

And for the second part:

- 86 → Natural, Whole, Integer, Rational
- 0.25 → Rational
- 1 → Natural, Whole, Integer, Rational
- √8 → Irrational
- 1/2 → Rational
- -14 → Integer, Rational
- 2π → Irrational
- 2.838383... → Rational
- √8 (repeated) → Irrational

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✔ How to Place in Diagram:

- Natural: 86, 1, 1000, 64, $\sqrt{16}$ (4)
- Whole: Same as above + 0 (but not listed)
- Integer: Add -19, -8, -18, -14
- Rational: All except √2, √8, 2π
- Irrational: √2, √8, 2π

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Let me know if you'd like this filled into a printable version!