Irrational numbers are real numbers that cannot be expressed as a fraction $\frac{p}{q}$, where $p$ and $q$ are integers and $q \neq 0$. Unlike rational numbers, which can be written in decimal form as terminating or repeating decimals, irrational numbers have non-terminating and non-repeating decimal expansions.

Some common examples of irrational numbers include:

• $\sqrt{2}$ - The square root of 2 is approximately 1.41421356...
• $\pi$ - The ratio of the circumference of a circle to its diameter, approximately 3.14159...
• $e$ - The base of natural logarithms, approximately 2.71828...

To understand irrational numbers better, let's explore some of their properties:

1. Non-terminating and Non-repeating Decimals:

   The decimal expansion of an irrational number continues infinitely without repeating any sequence. For example, the decimal expansion of $\sqrt{2}$ is 1.41421356..., and it does not have any repeating pattern.

2. Density in Real Numbers:

   Between any two rational numbers, there exists at least one irrational number. For example, between 1 and 2, we can find $\sqrt{2} \approx 1.414...$, which is irrational.

3. Operations with Irrational Numbers:

   - The sum or difference of a rational number and an irrational number is always irrational. For instance, $1 + \sqrt{2}$ is irrational.

   - The product or quotient of a non-zero rational number and an irrational number is also irrational. For example, $2 \times \sqrt{3}$ is irrational.

   - However, the sum or difference of two irrational numbers may be rational or irrational. For instance, $\sqrt{2} + (-\sqrt{2}) = 0$, which is rational.

It is essential to understand how to identify irrational numbers. Common methods include:

• Square Roots: Many square roots are irrational, particularly those of non-perfect squares. For example, $\sqrt{3}$ and $\sqrt{5}$ are both irrational.
• Decimal Expansions: If a decimal expansion neither terminates nor repeats, it is likely to be irrational. For instance, $0.1010010001...$ is irrational.

Irrational numbers can also be approximated using rational numbers. For instance, $\pi$ can be approximated by the fraction $\frac{22}{7}$ for practical calculations, but it is essential to remember that $\pi$ itself is not rational.

In geometry, irrational numbers often arise in calculations involving circles, triangles, and other shapes. For instance, the diagonal of a square with a side length of 1 is $\sqrt{2}$, illustrating the presence of irrational numbers in everyday measurements.

Understanding irrational numbers is crucial for building a solid foundation in number theory and for applications in higher mathematics, science, and engineering.