Commensurable

Commensurable refers to the concept in mathematics where two quantities or numbers share a common measure or can be expressed as a ratio of integers. The term is derived from the Latin commensurabilis, meaning "measurable by a common standard". This concept is fundamental in various branches of mathematics, including number theory, geometry, and algebra.

Definition

Two numbers are commensurable if their ratio is a rational number. In other words, if two numbers \(a\) and \(b\) are commensurable, there exists integers \(p\) and \(q\) (with \(q \neq 0\)) such that:

\[ \frac{a}{b} = \frac{p}{q} \]

Conversely, numbers are incommensurable if their ratio is an irrational number, meaning that no such integers \(p\) and \(q\) exist to express their ratio.

Historical Context

The concept of commensurability dates back to ancient Greek mathematics, where it played a crucial role in the development of theories related to proportions and measurements. The discovery of incommensurable lengths, such as the diagonal of a square relative to its side, was significant in the history of mathematics. This discovery is often attributed to the Pythagoreans, who initially believed that all lengths could be expressed in terms of integer ratios.

Applications

1. 1. 1. Geometry

In geometry, commensurability is important in the study of figures and shapes. For example, the sides of a square are commensurable with each other since their lengths can be described by the same measure. However, as mentioned, the side of a square and its diagonal are incommensurable, a fact that can be proven using the Pythagorean theorem.

1. 1. 1. Number Theory

In number theory, commensurability has implications for the properties of numbers, particularly in the context of Diophantine equations and the approximation of irrational numbers by rational numbers.

1. 1. 1. Algebra

In algebra, particularly in the field of group theory, the concept of commensurability extends to the comparison of subgroups. Two subgroups of a group are commensurable if their intersection has finite index in both of the subgroups.

See Also

• Rational number
• Irrational number
• Pythagorean theorem
• Diophantine equation