Irrationality

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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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

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[edit | edit source]

References[edit | edit source]


Irrationality Resources

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