Match the number with the correct description.(5 points) Rational or irrational and why?

Question

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