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To solve the task, we need to identify the rational numbers from the provided list. Here's how to determine which numbers are rational:

- **Rational numbers** are numbers that can be expressed as a fraction \( \frac{a}{b} \), where \( a \) and \( b \) are integers, and \( b \neq 0 \).

Let’s go through each option:

1. \( \frac{1}{3} \) - This is a fraction, so it is a **rational number**.
2. \( 0.\overline{9} \) (repeating decimal) - This is equivalent to \( 1 \), so it is also a **rational number**.
3. \( \sin x \) - This represents a trigonometric function, which is not always rational for arbitrary \( x \). So, **not rational**.
4. \( \frac{2}{5} \) - This is a fraction, so it is a **rational number**.
5. \( 3 \cdot 10^{16} \) - This is a large integer, and integers are always **rational numbers**.
6. \( -1 \) - This is an integer, and integers are **rational numbers**.
7. \( \pi \) - This is an irrational number (it cannot be written as a fraction), so it is **not rational**.
8. \( i \) - This is an imaginary number, so it is **not rational**.

### Correct answers:
- \( \frac{1}{3} \) (checked)
- \( 0.\overline{9} \) (should be checked)
- \( \frac{2}{5} \) (checked)
- \( 3 \cdot 10^{16} \) (should be checked)
- \( -1 \) (checked)

The correct selections should be:  
- \( \frac{1}{3} \)
- \( 0.\overline{9} \)
- \( \frac{2}{5} \)
- \( 3 \cdot 10^{16} \)
- \( -1 \)

Do you have any questions or need further details?

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Here are 5 related questions to expand the concept:
1. What is the difference between rational and irrational numbers?
2. Can \( \sin x \) ever be a rational number?
3. Why is \( \pi \) considered an irrational number?
4. How can repeating decimals be converted into fractions?
5. Are all integers rational numbers?

**Tip:** Every integer can be written as a fraction with denominator 1, which makes it a rational number.

Zaškrtněte všechna racionální čísla (Check all rational numbers)

This exercise helps students identify rational numbers from a set of options. The rational numbers include fractions, integers, and repeating decimals. Learn to distinguish between rational and irrational numbers in a simple, interactive task.

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