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.

Explore the concept of irrational numbers, including their definition and examples such as pi, e, and the square root of 2. Learn why irrational numbers cannot be expressed as fractions and their significance in mathematics. Suitable for Grades 7-9.

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