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. $\sqrt{19}$
- 19 is not a perfect square → $\sqrt{19}$ is irrational.
- ✔ Irrational

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

3. $\frac{2}{3}$
- Fraction of two integers → rational.
- Not an integer → not in Integers, Whole, or Natural.
- ✔ Rational (not in inner circles)

4. $-\sqrt{16}$
- $\sqrt{16} = 4$, so $-\sqrt{16} = -4$
- $-4$ is an integer.
- ✔ Rational, Integer (but not whole or natural since it's negative)

5. $0$
- Integer, whole number, but not natural.
- ✔ Rational, Integer, Whole Number

6. $-\frac{12}{6}$
- Simplify: $-\frac{12}{6} = -2$
- Integer.
- ✔ Rational, Integer

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

8. $\frac{6}{45}$
- Simplify: $\frac{6}{45} = \frac{2}{15}$
- Fraction → rational.
- Not an integer.
- ✔ Rational (only in Rational, not in inner sets)

9. $\sqrt{53}$
- 53 is not a perfect square → irrational.
- ✔ Irrational

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#### Final Classification:

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

Rational Numbers:
- $\frac{2}{3}$
- $-\sqrt{16} = -4$
- $0$
- $-\frac{12}{6} = -2$
- $8$
- $\frac{6}{45} = \frac{2}{15}$

Now place them in the Venn diagram:

- Natural Numbers: $8$
- Whole Numbers: $0, 8$
- Integers: $-4, -2, 0, 8$
- Rational Numbers (not integers): $\frac{2}{3}, \frac{2}{15}$
- Irrational Numbers: $\sqrt{19}, \pi, \sqrt{53}$

> Note: The Venn diagram shows nested sets:
> - Natural ⊂ Whole ⊂ Integers ⊂ Rational
> - Irrational is separate from Rational.

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Problem 2: Order from greatest to least

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

#### Step 1: Approximate values

- $83$: already known
- $-9.3$: negative
- $12$: positive
- $\frac{25}{26} \approx 0.9615$
- $\sqrt{91}$: since $9^2 = 81$, $10^2 = 100$, so $\sqrt{91} \approx 9.54$

So we have:

- $83$
- $12$
- $\sqrt{91} \approx 9.54$
- $\frac{25}{26} \approx 0.96$
- $-9.3$

#### Order from greatest to least:

$$
83 > 12 > \sqrt{91} > \frac{25}{26} > -9.3
$$

✔ Final order:
1. $83$
2. $12$
3. $\sqrt{91}$
4. $\frac{25}{26}$
5. $-9.3$

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Problem 3: Order from greatest to least

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

#### Step 1: Simplify

- $\frac{72}{9} = 8$
- $\frac{21}{6} = 3.5$
- $\sqrt{14}$: between $3^2=9$ and $4^2=16$, so $\sqrt{14} \approx 3.74$
- $6.5$ remains
- $-25$ remains

So values:
- $-25$
- $8$
- $\sqrt{14} \approx 3.74$
- $6.5$
- $3.5$

Now order from greatest to least:

- $8 > 6.5 > \sqrt{14} \approx 3.74 > 3.5 > -25$

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

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

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✔ Summary Answers

#### 1) Classification:

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

Rational Numbers:
- $\frac{2}{3}$ → Rational only
- $-\sqrt{16} = -4$ → Integer
- $0$ → Whole, Integer
- $-\frac{12}{6} = -2$ → Integer
- $8$ → Natural, Whole, Integer
- $\frac{6}{45} = \frac{2}{15}$ → Rational only

In Venn Diagram:
- Natural: $8$
- Whole: $0, 8$
- Integers: $-4, -2, 0, 8$
- Rational (non-integer): $\frac{2}{3}, \frac{2}{15}$
- Irrational: $\sqrt{19}, \pi, \sqrt{53}$

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

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

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