An irrational number is a type of real number that cannot be expressed as a fraction (or ratio) of two integers. In other words, it cannot be written in the form $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$.

Characteristics of Irrational Numbers:

• Non-repeating, non-terminating decimals: The decimal representation of an irrational number goes on forever without repeating any pattern. For example, $\pi$ (pi) is approximately 3.14159..., but its decimal expansion never repeats or terminates.
• Cannot be expressed as a fraction: Unlike rational numbers, which can be written as a fraction like $\frac{1}{2}$, irrational numbers do not have such a representation.

Examples of Irrational Numbers:

• $\pi$: The ratio of the circumference of a circle to its diameter.
• $e$: The base of the natural logarithm.
• $\sqrt{2}$: The square root of 2, which cannot be simplified into a fraction.

Why are they irrational?

An irrational number's decimal expansion is infinite and does not settle into a repeating pattern. This makes it impossible to represent as a simple ratio of integers.

Would you like more examples or deeper insights? Here are some related questions to explore:

1. How do you prove that a number like $\sqrt{2}$ is irrational?
2. Can an irrational number ever be the solution to a simple equation like $x^2 = 2$?
3. What is the difference between rational and irrational numbers?
4. Can the sum or product of two irrational numbers ever be rational?
5. How are irrational numbers used in real-world applications like geometry or physics?

Tip: Many irrational numbers, like $\pi$ and $\sqrt{2}$, show up in geometry and measurements related to circles and diagonals, making them very common in mathematical applications.