Irrationality

Irrationality is a concept in mathematics that describes a type of number that cannot be expressed as a simple fraction. This is in contrast to rational numbers, which can be expressed as a fraction of two integers. The most famous irrational number is probably the mathematical constant pi, which represents the ratio of a circle's circumference to its diameter.

Definition

An irrational number is any real number that cannot be expressed as a ratio of two integers. This means that it cannot be written in the form a/b, where a and b are integers and b is not zero. The decimal representation of an irrational number never ends or repeats.

Examples

Some examples of irrational numbers include:

• The square root of any number that is not a perfect square (such as √2 or √3)
• The mathematical constants pi (π) and e
• The golden ratio (φ)

Properties

Irrational numbers have several interesting properties. For example, the sum of a rational number and an irrational number is always irrational. The product of a non-zero rational number and an irrational number is also always irrational.

History

The concept of irrationality was first discovered by the ancient Greeks, specifically the Pythagoreans. They discovered that the diagonal of a square is incommensurable with its side, or in other words, the length of the diagonal is an irrational number.

See also

• Rational number
• Real number
• Number theory
• Pythagorean theorem