Solved Real Number System Worksheet Graph and label each | Chegg.com - Free Printable

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 ✔
- 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 from 1 (some include 0, but standard definition excludes it)

✔ 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 ≥ 0)
- Natural: No ✘

✔ Real, Rational, Integer

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#### 5. $ -\sqrt{100} $
- $ \sqrt{100} = 10 $ → $ -\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 is a non-repeating, non-terminating decimal with increasing pattern of 4s.
- It does not repeat regularly, so it's irrational.
- Real: Yes ✔
- Rational: No ✘
- Irrational: Yes ✔
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Irrational

> Note: Even though it looks like a pattern, unless it's a repeating cycle (like 2.343434...), it's not rational.

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#### 7. $ \frac{3}{7} $
- Fraction of two integers.
- 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} $
- $ \sqrt{3} $ is irrational → so $ \sqrt{75} $ is 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: $ 12.67777... $
- Repeating decimals are rational
- Real: Yes ✔
- Rational: Yes ✔
- Irrational: No ✘
- Integer: No ✘
- Whole: No ✘
- Natural: No ✘

✔ Real, Rational

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#### 11. $ \frac{12}{5} $
- $ = 2.4 $, which is a terminating decimal
- 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:

| # | 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 | 6⅔ = 20/3 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
|10 | 12.6̅7 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
|11 | 12/5 = 2.4 | ✔ | ✔ | ✘ | ✘ | ✘ | ✘ |
|12 | π | ✔ | ✘ | ✔ | ✘ | ✘ | ✘ |

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Part 2: Graph and Label on Number Line

We are given:

A. 0.75
B. $ \sqrt{3} $ ≈ 1.732
C. $ \sqrt{9} = 3 $
D. $ -2\frac{1}{2} = -2.5 $
E. $ -\frac{15}{10} = -1.5 $
F. $ 4.2 $
G. $ -\sqrt{2} $ ≈ -1.414
H. $ \pi $ ≈ 3.1416

Let’s place them on the number line from -5 to 5.

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#### Approximate Values:
- A. 0.75 → between 0 and 1
- B. $ \sqrt{3} $ ≈ 1.732 → between 1 and 2
- C. $ \sqrt{9} = 3 $ → exactly at 3
- D. -2.5 → halfway between -3 and -2
- E. -1.5 → halfway between -2 and -1
- F. 4.2 → slightly past 4
- G. $ -\sqrt{2} $ ≈ -1.414 → just left of -1.5? Wait: -1.414 is greater than -1.5, so it's between -1.5 and -1
- H. $ \pi $ ≈ 3.14 → just after 3

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#### Plotting on Number Line:

```
<----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|---->
-5 -4 -3 -2 -1 0 1 2 3 4 5
```

Label positions:

- D. -2.5 → midpoint between -3 and -2
- E. -1.5 → midpoint between -2 and -1
- G. -√2 ≈ -1.414 → very close to -1.5, but slightly to the right (closer to -1)
- A. 0.75 → 3/4 of the way from 0 to 1
- B. √3 ≈ 1.732 → between 1 and 2, closer to 2
- C. 3 → exactly at 3
- H. π ≈ 3.14 → just after 3
- F. 4.2 → slightly past 4

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Final Labeled Number Line (approximate):

```
<----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|---->
-5 -4 -3 -2 -1 0 1 2 3 4 5
D E G A B C H F
```

More accurately:

- D (-2.5) → between -3 and -2
- E (-1.5) → between -2 and -1
- G (-1.414) → just to the right of E (closer to -1)
- A (0.75) → near 1, but before it
- B (1.732) → between 1 and 2
- C (3) → exactly at 3
- H (π ≈ 3.14) → just after 3
- F (4.2) → just after 4

So the labeled line should look like:

```
<----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|---->
-5 -4 -3 -2 -1 0 1 2 3 4 5
D E G A B C H F
```

But more precisely:

- D at -2.5
- E at -1.5
- G at ~ -1.41 (so just right of E)
- A at 0.75
- B at ~1.73
- C at 3
- H at ~3.14
- F at 4.2

So labeling:

```
D E G A B C H F
↑ ↑ ↑ ↑ ↑ ↑ ↑ ↑
-2.5 -1.5 -1.41 0.75 1.73 3 3.14 4.2
```

On the number line:

```
<----|-----|-----|-----|-----|-----|-----|-----|-----|-----|-----|---->
-5 -4 -3 -2 -1 0 1 2 3 4 5
D E G A B C H F
```

Place labels accordingly.

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✔ Final Answer Summary:

#### Classification Answers:

1. 0.8 → Real, Rational
2. √64 → Real, Rational, Integer, Whole, Natural
3. 0 → Real, Rational, Integer, Whole
4. -19 → Real, Rational, Integer
5. -√100 → Real, Rational, Integer
6. 2.343443444... → Real, Irrational
7. 3/7 → Real, Rational
8. √75 → Real, Irrational
9. 6⅔ → Real, Rational
10. 12.6̅7 → Real, Rational
11. 12/5 → Real, Rational
12. π → Real, Irrational

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#### Graphing:

Label the number line with:

- D at -2.5
- E at -1.5
- G at ≈ -1.41 (just right of E)
- A at 0.75
- B at ≈ 1.73
- C at 3
- H at ≈ 3.14
- F at 4.2

Plot these points accordingly.

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