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!