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[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]
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