Irrational numbers are real numbers that cannot be expressed as a ratio of two integers and cannot be expressed as a simple repeating decimal.
Unlike rational numbers, the decimal representation of an irrational number neither terminates nor repeats, but goes on forever without repeating.

Examples of irrational numbers:

1. The square root of 2, which is approximately equal to 1.41421356, but has an infinite number of non-repeating decimal places.
2. The mathematical constant π (pi), which represents the ratio of a circle's circumference to its diameter and is approximately equal to 3.14159265.
3. The natural logarithm of 2, denoted as ln(2), is approximately equal to 0.69314718.

These numbers are extremely important in mathematics and science, as they appear frequently in geometry, calculus, physics, and other areas.
They also have important applications in real-world problems, such as in the calculation of distances, areas, and volumes.
There are two main types of irrational numbers:

1. algebraic and
2. transcendental.

Algebraic Irrational Numbers are solutions to algebraic equations with rational coefficients, but cannot be expressed as a ratio of two integers.
Examples

1. √2
2. √5
3. √8

Transcendental Irrational Numbers are not solutions to any algebraic equation with rational coefficients.
Examples

1. π (pi)
2. e (Euler's number)
3. log₂¹⁰

Note: Not all irrational numbers are either algebraic or transcendental, as the classification of some numbers as algebraic or transcendental is still an open question in mathematics.