Make your statement about one or two of these number sets: real numbers, rational numbers, irrational numbers, integers, or natural numbers.

Let's explore **rational numbers** and **irrational numbers**.

- **Rational numbers** are numbers that can be expressed as the ratio of two integers, where the denominator is not zero. This includes numbers like \( \frac{1}{2}, 5, -3, 0.75 \), and repeating decimals like \( 0.\overline{3} \). Rational numbers form a dense set, meaning between any two rational numbers, there is another rational number.

- **Irrational numbers**, on the other hand, cannot be written as a ratio of two integers. Their decimal expansions are non-repeating and non-terminating. Famous examples include \( \sqrt{2}, \pi, e \), and numbers like \( 0.1010010001\ldots \). The set of irrational numbers is also dense, and combined with rational numbers, they make up the set of **real numbers**.

Would you like more details on this, or have any questions?

Here are five relative questions:
1. How can you prove that \( \sqrt{2} \) is irrational?
2. What is the difference between terminating and repeating decimals in rational numbers?
3. Why is \( \pi \) considered an irrational number?
4. Can the sum of a rational number and an irrational number ever be rational?
5. What is the relationship between real numbers and complex numbers?

**Tip**: Every integer is a rational number, but not every rational number is an integer!

Explore the properties of rational and irrational numbers, learn how they differ, and see examples such as \(\frac{1}{2}\), \(\pi\), and \(\sqrt{2}\). This article covers essential mathematical concepts and is suitable for high school students in Grades 9-12.

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