Real Number System Worksheet with classification exercises and number line graphing.

Let's solve this Real Number System Worksheet step by step.

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Part 1: Classify Each Number

We are to circle all classifications that apply to each number. The categories are:

- Real: All numbers on the number line.
- Rational: Can be written as a fraction $ \frac{p}{q} $, where $ p $ and $ q $ are integers and $ q \neq 0 $. Includes terminating or repeating decimals.
- Irrational: Cannot be expressed as a fraction; non-repeating, non-terminating decimals (e.g., $ \pi $, $ \sqrt{2} $).
- Integer: Whole numbers and their negatives: ..., -3, -2, -1, 0, 1, 2, 3, ...
- Whole: Non-negative integers: 0, 1, 2, 3, ...
- Natural: Positive integers: 1, 2, 3, ...

Now let’s go through each one:

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#### 1. 0.8
- Real: Yes ✔
- Rational: Yes — it's $ \frac{4}{5} $ ✔
- Irrational: No ✘
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Rational

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#### 2. $ \sqrt{64} $
- $ \sqrt{64} = 8 $
- Real: Yes ✔
- Rational: Yes — 8 is an integer ✔
- Irrational: No ✘
- Integer: Yes ✔
- Whole: Yes ✔
- Natural: Yes ✔

✔ Real, Rational, Integer, Whole, Natural

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#### 3. 0
- Real: Yes ✔
- Rational: Yes — $ \frac{0}{1} $ ✔
- Irrational: No ✘
- Integer: Yes ✔
- Whole: Yes ✔
- Natural: No — natural numbers start at 1 (some definitions include 0, but standard math curriculum says no) ✘

✔ Real, Rational, Integer, Whole

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#### 4. -19
- Real: Yes ✔
- Rational: Yes — $ \frac{-19}{1} $ ✔
- Irrational: No ✘
- Integer: Yes ✔
- Whole: No ✘ (whole numbers are non-negative)
- Natural: No ✘

✔ Real, Rational, Integer

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#### 5. $ -\sqrt{100} $
- $ \sqrt{100} = 10 $, so $ -\sqrt{100} = -10 $
- Real: Yes ✔
- Rational: Yes ✔
- Irrational: No ✘
- Integer: Yes ✔
- Whole: No ✘
- Natural: No ✘

✔ Real, Rational, Integer

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#### 6. 2.343443444...
- This decimal does not repeat in a pattern, and digits keep changing with more 4s added.
- It's non-repeating, non-terminating, so irrational.
- Real: Yes ✔
- Rational: No ✘
- Irrational: Yes ✔
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Irrational

> Note: Even though it starts with repeating digits, the pattern changes (2.343443444...) — this is not periodic → irrational.

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#### 7. $ \frac{3}{7} $
- Fraction of two integers → rational
- Real: Yes ✔
- Rational: Yes ✔
- Irrational: No ✘
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Rational

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#### 8. $ \sqrt{75} $
- $ \sqrt{75} = \sqrt{25 \cdot 3} = 5\sqrt{3} $ → irrational
- Real: Yes ✔
- Rational: No ✘
- Irrational: Yes ✔
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Irrational

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#### 9. $ 6\frac{2}{3} $
- Mixed number = $ \frac{20}{3} $
- Real: Yes ✔
- Rational: Yes ✔
- Irrational: No ✘
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Rational

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#### 10. $ 12.6\overline{7} $
- Repeating decimal → rational
- Can be written as a fraction
- Real: Yes ✔
- Rational: Yes ✔
- Irrational: No ✘
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Rational

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#### 11. $ \frac{12}{5} $
- Fraction → rational
- Equals 2.4
- Real: Yes ✔
- Rational: Yes ✔
- Irrational: No ✘
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Rational

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#### 12. $ \pi $
- Famous irrational number
- Real: Yes ✔
- Rational: No ✘
- Irrational: Yes ✔
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Irrational

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✔ Summary Table for Part 1:

| # | Number | Real | Rational | Irrational | Integer | Whole | Natural |
|---|----------------|------|----------|------------|---------|-------|---------|
| 1 | 0.8 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
| 2 | √64 = 8 | ✔ | ✔ | ✘ | ✔ | ✔ | ✔ |
| 3 | 0 | ✔ | ✔ | ✘ | ✔ | ✔ | ✘ |
| 4 | -19 | ✔ | ✔ | ✘ | ✔ | ✘ | ✘ |
| 5 | -√100 = -10 | ✔ | ✔ | ✘ | ✔ | ✘ | ✘ |
| 6 | 2.343443444... | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| 7 | 3/7 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
| 8 | √75 | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |
| 9 | 6 2/3 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
|10 | 12.6̅7 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
|11 | 12/5 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
|12 | π | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |

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Part 2: Graph and Label on Number Line

We are to plot these values on the number line from -5 to 5.

Let’s evaluate each:

#### A. 0.75
- Between 0 and 1 → closer to 1
- Place at 0.75

#### B. $ \sqrt{3} $
- $ \sqrt{3} \approx 1.732 $
- Between 1 and 2 → closer to 2

#### C. $ \sqrt{9} $
- $ \sqrt{9} = 3 $

#### D. $ -2\frac{1}{2} = -2.5 $
- Between -3 and -2 → midpoint

#### E. $ -\frac{15}{10} = -1.5 $
- Between -2 and -1 → midpoint

#### F. $ 4.2 $
- Between 4 and 5 → closer to 4

#### G. $ -\sqrt{2} $
- $ \sqrt{2} \approx 1.414 $, so $ -\sqrt{2} \approx -1.414 $
- Between -2 and -1 → closer to -1.4

#### H. $ \pi \approx 3.14 $
- Between 3 and 4 → closer to 3

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📊 Number Line Labels:

```
<---|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|------>
-5 -4 -3 -2 -1 0 1 2 3 4 5
D(-2.5) E(-1.5) G(-1.414) A(0.75) B(1.732) C(3) H(3.14) F(4.2)
```

But we need to label each with the letter.

So:

- A. 0.75 → place between 0 and 1, mark A
- B. √3 ≈ 1.732 → between 1 and 2, mark B
- C. √9 = 3 → exactly at 3, mark C
- D. -2.5 → halfway between -3 and -2, mark D
- E. -1.5 → halfway between -2 and -1, mark E
- F. 4.2 → slightly past 4, mark F
- G. -√2 ≈ -1.414 → just left of -1.5, mark G
- H. π ≈ 3.14 → just right of 3, mark H

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✔ Final Answer: Graph Instructions

Label the number line as follows:

- At -2.5: D
- At -1.5: E
- At -1.414: G
- At 0.75: A
- At 1.732: B
- At 3: C
- At 3.14: H
- At 4.2: F

You can draw small dots and label them accordingly.

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✔ Final Notes:

- Make sure to circle all applicable classifications for each number in Part 1.
- For Part 2, plot each point accurately and label with the corresponding letter.

Let me know if you'd like a visual sketch or printable version!