Let's evaluate each number to determine which are rational.

1. 0.4141141114…

   • This is a non-repeating, non-terminating decimal, meaning it cannot be expressed as a ratio of two integers.
   • Not rational.

2. √5

   • The square root of 5 is an irrational number because it cannot be expressed as a fraction of two integers and it has a non-repeating, non-terminating decimal expansion.
   • Not rational.

3. 12/13

   • This is a ratio of two integers (a fraction), and it can be expressed exactly as $\frac{12}{13}$.
   • Rational.

4. −6

   • This is an integer, and all integers are rational because they can be expressed as a ratio (e.g., $-6 = \frac{-6}{1}$).
   • Rational.

5. 5.2481

   • This is a terminating decimal, and terminating decimals are always rational because they can be expressed as fractions (e.g., $5.2481 = \frac{52481}{10000}$).
   • Rational.

6. 0.6565565556…

   • This is a non-repeating, non-terminating decimal, so it cannot be expressed as a ratio of two integers.
   • Not rational.

Final Answer:

• Rational numbers: 12/13, −6, 5.2481
• Not rational numbers: 0.4141141114…, √5, 0.6565565556…

Would you like more details or clarification?

Relative questions:

1. What makes a number irrational?
2. Can a repeating decimal be considered rational?
3. How do we convert terminating decimals into fractions?
4. Why is the square root of non-perfect squares irrational?
5. What is the difference between rational and irrational numbers?

Tip:

Remember, any decimal that either terminates or repeats can always be expressed as a fraction, making it rational.