Rational and Irrational Numbers exercise - Free Printable

Problem Overview:

The task is to classify the given numbers into two categories: Rational and Irrational.

- Rational Numbers: These are numbers that can be expressed as a ratio of two integers (i.e., in the form \( \frac{p}{q} \), where \( p \) and \( q \) are integers, and \( q \neq 0 \)). Rational numbers include integers, fractions, terminating decimals, and repeating decimals.

- Irrational Numbers: These are real numbers that cannot be expressed as a simple fraction. They have non-terminating, non-repeating decimal expansions. Examples include \( \pi \), \( e \), and square roots of non-perfect squares.

Given Numbers:

We need to classify the following numbers:
1. \( 3.14 \)
2. \( 0.444 \)
3. \( \sqrt{121} \)
4. \( \sqrt{1000} \)
5. \( \pi \)
6. \( \frac{\pi}{7} \)
7. \( 0.333\ldots \)
8. \( 1.12313\ldots \)
9. \( 0.\overline{12} \)
10. \( e \)
11. \( \sqrt[3]{9} \)
12. \( 2.\overline{9} \)
13. \( 2e \)
14. \( 1.234 \)
15. \( \sqrt[3]{8} \)

Step-by-Step Classification:

#### 1. \( 3.14 \)
- This is a terminating decimal.
- It can be written as \( \frac{314}{100} \).
- Classification: Rational

#### 2. \( 0.444 \)
- This is a terminating decimal.
- It can be written as \( \frac{444}{1000} \).
- Classification: Rational

#### 3. \( \sqrt{121} \)
- \( \sqrt{121} = 11 \), which is an integer.
- Classification: Rational

#### 4. \( \sqrt{1000} \)
- \( \sqrt{1000} \approx 31.6227766\ldots \), which is a non-terminating, non-repeating decimal.
- Classification: Irrational

#### 5. \( \pi \)
- \( \pi \approx 3.14159265\ldots \), which is a well-known irrational number.
- Classification: Irrational

#### 6. \( \frac{\pi}{7} \)
- Since \( \pi \) is irrational, dividing it by 7 (a rational number) results in an irrational number.
- Classification: Irrational

#### 7. \( 0.333\ldots \)
- This is a repeating decimal, which can be written as \( \frac{1}{3} \).
- Classification: Rational

#### 8. \( 1.12313\ldots \)
- The ellipsis indicates a non-repeating, non-terminating decimal.
- Classification: Irrational

#### 9. \( 0.\overline{12} \)
- This is a repeating decimal, which can be expressed as a fraction.
- Classification: Rational

#### 10. \( e \)
- \( e \approx 2.718281828\ldots \), which is a well-known irrational number.
- Classification: Irrational

#### 11. \( \sqrt[3]{9} \)
- The cube root of 9 is a non-terminating, non-repeating decimal.
- Classification: Irrational

#### 12. \( 2.\overline{9} \)
- This is a repeating decimal. Note that \( 2.\overline{9} = 3 \) (since \( 0.\overline{9} = 1 \)).
- Classification: Rational

#### 13. \( 2e \)
- Since \( e \) is irrational, multiplying it by 2 (a rational number) results in an irrational number.
- Classification: Irrational

#### 14. \( 1.234 \)
- This is a terminating decimal.
- It can be written as \( \frac{1234}{1000} \).
- Classification: Rational

#### 15. \( \sqrt[3]{8} \)
- \( \sqrt[3]{8} = 2 \), which is an integer.
- Classification: Rational

Final Classification:

- Rational: \( 3.14, 0.444, \sqrt{121}, 0.333\ldots, 0.\overline{12}, 2.\overline{9}, 1.234, \sqrt[3]{8} \)
- Irrational: \( \sqrt{1000}, \pi, \frac{\pi}{7}, 1.12313\ldots, e, \sqrt[3]{9}, 2e \)

Boxed Answer:

\[
\boxed{
\begin{array}{c|c}
\text{Rational} & \text{Irrational} \\
\hline
3.14 & \sqrt{1000} \\
0.444 & \pi \\
\sqrt{121} & \frac{\pi}{7} \\
0.333\ldots & 1.12313\ldots \\
0.\overline{12} & e \\
2.\overline{9} & \sqrt[3]{9} \\
1.234 & 2e \\
\sqrt[3]{8} & \\
\end{array}
}
\]