Classifying and Ordering real Numbers worksheet - Free Printable

Let's solve each part of the problem step by step.

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Problem 1: Classify the numbers into the Venn Diagram

We are given the following numbers:

- $\sqrt{19}$
- $\pi$
- $\frac{2}{3}$
- $-\sqrt{16}$
- $0$
- $-\frac{12}{6}$
- $8$
- $\frac{6}{45}$
- $\sqrt{53}$

We need to classify them as Rational or Irrational, and further classify rational numbers into Integers, Whole Numbers, and Natural Numbers.

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#### Step 1: Understand the number sets

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

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#### Now analyze each number:

1. $\sqrt{19}$
- $\sqrt{19} \approx 4.358...$, not a perfect square → Irrational

2. $\pi$
- Well-known irrational number → Irrational

3. $\frac{2}{3}$
- Fraction → Rational
- Not an integer → not in Integers, Whole, or Natural

4. $-\sqrt{16}$
- $\sqrt{16} = 4$, so $-\sqrt{16} = -4$
- Integer → Rational, and specifically Integer, but not whole or natural (negative)

5. $0$
- Is an integer, whole number, but not natural → Rational, Integer, Whole Number

6. $-\frac{12}{6}$
- Simplifies to $-2$ → Integer → Rational, Integer, not whole or natural

7. $8$
- Positive integer → Rational, Integer, Whole Number, Natural Number

8. $\frac{6}{45}$
- Simplify: $\frac{6}{45} = \frac{2}{15} \approx 0.133...$, repeating decimal → Rational
- Not an integer → only Rational

9. $\sqrt{53}$
- $\sqrt{53} \approx 7.28$, not a perfect square → Irrational

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#### Classification Summary:

| Number | Type |
|--------|------|
| $\sqrt{19}$ | Irrational |
| $\pi$ | Irrational |
| $\frac{2}{3}$ | Rational |
| $-\sqrt{16} = -4$ | Rational, Integer |
| $0$ | Rational, Integer, Whole Number |
| $-\frac{12}{6} = -2$ | Rational, Integer |
| $8$ | Rational, Integer, Whole Number, Natural Number |
| $\frac{6}{45} = \frac{2}{15}$ | Rational |
| $\sqrt{53}$ | Irrational |

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#### Venn Diagram Placement:

- Irrational Numbers:
- $\sqrt{19}$
- $\pi$
- $\sqrt{53}$

- Rational Numbers:
- $\frac{2}{3}$
- $-4$
- $0$
- $-2$
- $8$
- $\frac{2}{15}$

- Integers (subset of Rational):
- $-4$, $0$, $-2$, $8$

- Whole Numbers (subset of Integers):
- $0$, $8$

- Natural Numbers (subset of Whole Numbers):
- $8$

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✔ Answer for Problem 1:

Irrational Numbers:
- $\sqrt{19}$
- $\pi$
- $\sqrt{53}$

Rational Numbers (not integers):
- $\frac{2}{3}$
- $\frac{6}{45}$

Integers (but not whole):
- $-\sqrt{16} = -4$
- $-\frac{12}{6} = -2$

Whole Numbers (but not natural):
- $0$

Natural Numbers:
- $8$

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Problem 2: Order from greatest to least

Numbers:
$83,\ -9.3,\ 12,\ \frac{25}{26},\ \sqrt{91}$

First, evaluate each:

- $83$ → 83
- $-9.3$ → -9.3
- $12$ → 12
- $\frac{25}{26} \approx 0.9615$
- $\sqrt{91} \approx 9.539$

Now order from greatest to least:

1. $83$
2. $12$
3. $\sqrt{91} \approx 9.54$
4. $\frac{25}{26} \approx 0.96$
5. $-9.3$

✔ Ordered:
$$
83,\ 12,\ \sqrt{91},\ \frac{25}{26},\ -9.3
$$

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Problem 3: Order from greatest to least

Numbers:
$-25,\ \frac{72}{9},\ \sqrt{14},\ 6.5,\ \frac{21}{6}$

Evaluate:

- $-25$ → -25
- $\frac{72}{9} = 8$
- $\sqrt{14} \approx 3.74$
- $6.5$ → 6.5
- $\frac{21}{6} = 3.5$

Now order from greatest to least:

1. $\frac{72}{9} = 8$
2. $6.5$
3. $\sqrt{14} \approx 3.74$
4. $\frac{21}{6} = 3.5$
5. $-25$

✔ Ordered:
$$
\frac{72}{9},\ 6.5,\ \sqrt{14},\ \frac{21}{6},\ -25
$$

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✔ Final Answers:

#### 1) Classification:

- Irrational Numbers: $\sqrt{19},\ \pi,\ \sqrt{53}$
- Rational Numbers:
- Integers: $-\sqrt{16} = -4,\ 0,\ -\frac{12}{6} = -2,\ 8$
- Whole Numbers: $0,\ 8$
- Natural Numbers: $8$
- Non-integer Rationals: $\frac{2}{3},\ \frac{6}{45}$

#### 2) Order from greatest to least:
$$
83,\ 12,\ \sqrt{91},\ \frac{25}{26},\ -9.3
$$

#### 3) Order from greatest to least:
$$
\frac{72}{9},\ 6.5,\ \sqrt{14},\ \frac{21}{6},\ -25
$$

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