Question

Match the number with the correct description.(5 points) Rational or irrational and why?
101−−−√
0.424242424...
0.133134135...
16−−√
0.789

Answer 1

Let's match each of the given numbers with its correct description (rational or irrational) and explain why.

1. **\( \sqrt{101} \)**
- **Description**: Irrational
- **Why**: \( \sqrt{101} \) cannot be expressed as a fraction of two integers because 101 is not a perfect square, and its decimal representation is non-repeating and non-terminating.

2. **0.424242424...**
- **Description**: Rational
- **Why**: This number has a repeating decimal (the "42" repeats), which can be expressed as a fraction (specifically \( \frac{42}{99} \)).

3. **0.133134135...**
- **Description**: Irrational
- **Why**: The decimal appears to be non-repeating and non-terminating. As such, it cannot be expressed as a fraction of two integers.

4. **\( \sqrt{16} \)**
- **Description**: Rational
- **Why**: \( \sqrt{16} = 4 \), which is an integer and can be expressed as a fraction (\( \frac{4}{1} \)).

5. **0.789**
- **Description**: Rational
- **Why**: This number has a finite decimal representation, which can be expressed as a fraction (\( \frac{789}{1000} \)).

To summarize:
- \( \sqrt{101} \): Irrational
- 0.424242424...: Rational
- 0.133134135...: Irrational
- \( \sqrt{16} \): Rational
- 0.789: Rational