Explain what it means for a number to be irrational

An **irrational number** is a real number that **cannot be expressed as a fraction** of two integers (i.e., it cannot be written in the form \( \frac{a}{b} \), where \( a \) and \( b \) are integers and \( b \neq 0 \)). In decimal form, irrational numbers have the following key properties:

1. Their decimal expansion **does not terminate** (it goes on forever).
2. Their decimal expansion **does not repeat** any fixed pattern.

This makes them different from rational numbers, which either terminate or have repeating decimal patterns. Some well-known examples of irrational numbers include:

- \( \pi \) (pi): The ratio of a circle's circumference to its diameter.
- \( \sqrt{2} \): The square root of 2, which cannot be exactly written as a fraction.
- \( e \): The base of the natural logarithm.

### Properties of Irrational Numbers:
- They are located on the real number line, but cannot be precisely represented by simple fractions.
- Irrational numbers are abundant in mathematics, and they appear in various contexts, such as geometry, algebra, and calculus.

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

Here are 5 related questions to expand on this:

1. How do we prove that a number like \( \sqrt{2} \) is irrational?
2. Can irrational numbers be approximated by rational numbers?
3. What is the difference between algebraic and transcendental irrational numbers?
4. How are irrational numbers distributed on the number line compared to rational numbers?
5. Why is \( \pi \) considered an irrational number?

**Tip:** When working with irrational numbers in practical situations (e.g., calculations), they are often approximated by rational numbers (like using 3.14 for \( \pi \)).

Explore the concept of irrational numbers, which cannot be expressed as a fraction of two integers. This guide covers the properties of irrational numbers, such as non-terminating and non-repeating decimals, with examples like π and √2. Perfect for students in Grades 9-12 studying real numbers.

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