An irrational number is a number that cannot be expressed as a fraction p/q for any integers p and q. Irrational numbers have decimal expansions that neither terminate nor become periodic. Every transcendental number is irrational.

There is no standard notation for the set of irrational numbers, but the notations Q^_, R-Q, or R\Q, where the bar, minus sign, or backslash indicates the set complement of the rational numbers Q over the reals R, could all be used.

The most famous irrational number is sqrt(2), sometimes called Pythagoras's constant. Legend has it that the Pythagorean philosopher Hippasus used geometric methods to demonstrate the irrationality of sqrt(2) while at sea and, upon notifying his comrades of his great discovery, was immediately thrown overboard by the fanatic Pythagoreans. Other examples include sqrt(3), e, pi, etc. The Erdős-Borwein constant

E = sum_(n=1)^(infty)1/(2^n-1)
(1)
= sum_(n=1)^(infty)(d(n))/(2^n)
(2)
= 1.606695152415291763...

An irrational number is a real number that cannot be expressed as a simple fraction or ratio of two integers.

Irrational numbers are an important concept in mathematics and have various applications in fields such as geometry, calculus, and number theory. They are numbers that cannot be written as a finite or repeating decimal, meaning they have an infinite number of non-repeating decimal places.

Properties of Irrational Numbers:
1. Non-terminating decimal: Irrational numbers have decimal representations that go on forever without repeating. For example, the square root of 2, denoted as √2, is an irrational number with a decimal representation of approximately 1.41421356.

2. Non-repeating decimal: Unlike rational numbers, which have repeating patterns in their decimal representations, irrational numbers do not have any repeating patterns. The decimal representation of an irrational number continues indefinitely without any repeating sequence.

3. Cannot be expressed as a fraction: Irrational numbers cannot be written as a simple fraction or ratio of two integers. This means that they cannot be expressed in the form a/b, where a and b are integers.

4. Existence between rational numbers: Between any two rational numbers, there exists an infinite number of irrational numbers. This property is known as density.

Examples of Irrational Numbers:
1. √2: The square root of 2 is an irrational number because it cannot be expressed as a fraction. Its decimal representation is non-repeating and non-terminating.

2. π (pi): Pi is an irrational number approximately equal to 3.14159. It is the ratio of a circle's circumference to its diameter and has a decimal representation that goes on forever without repeating.

3. √3: The square root of 3 is another example of an irrational number. It cannot be expressed as a fraction and has a decimal representation that does not terminate or repeat.

Applications of Irrational Numbers:
1. Geometry: Irrational numbers play a significant role in geometry, especially in the measurement of lengths, areas, and volumes of geometric figures. For example, the diagonal of a square with sides of length 1 unit is an irrational number (√2 units).

2. Calculus: Irrational numbers are used in calculus to describe and analyze continuous functions. They appear in various mathematical formulas, such as the derivative of trigonometric functions or the area under a curve.

3. Number theory: Irrational numbers are studied extensively in number theory, which deals with properties and relationships of numbers. They provide insights into the nature of numbers and their patterns.

In conclusion, irrational numbers are real numbers that cannot be expressed as fractions and have non-repeating, non-terminating decimal representations. They have various applications in mathematics and are fundamental to many mathematical concepts and theories.