Commensurability in general

Generally, two quantities are commensurable if both can be measured in the same units. For example, a distance measured in miles and a quantity of water measured in gallons are incommensurable. A time measured in weeks and a time measured in minutes are commensurable because a week is a constant number of minutes (10080), so that one can convert between the two units by multiplying or dividing by 10080. Measurement is the determination of the size or magnitude of something. ...

Commensurability in mathematics

In mathematics, two nonzero real numbers a and b are said to be commensurable if and only if a/b is a rational number. Euclid, detail from The School of Athens by Raphael. ... Nonzero: The Logic of Human Destiny is a book by Robert Wright originally published in 2000. ... In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... In mathematics, a rational number (or informally fraction) is a ratio or quotient of two integers, usually written as the vulgar fraction a/b, where b is not zero. ...

The usage comes to us from translations of Euclid's Elements, in which two line segments a and b are called commensurable precisely if there is some third segment c that can be laid end-to-end a whole number of times to produce a segment congruent to a, and also, with a different whole number, a segment congruent to b. Euclid did not use any concept of real number, but he used a notion of congruence of line segments, and of one such segment being longer or shorter than another. Euclid Euclid of Alexandria (Greek: ) (ca. ... Euclids Elements (Greek: ) is a mathematical and geometric treatise, consisting of 13 books, written by the Hellenistic mathematician Euclid in Egypt during the early 3rd century BC. It comprises a collection of definitions, postulates (axioms), propositions (theorems) and proofs thereof. ...

That a/b is rational is a necessary and sufficient condition for the existence of some real number c, and integers m and n, such that In logic, the words necessary and sufficient describe relations that hold between propositions or states of affairs, if one is conditional on the other. ... The integers consist of the positive natural numbers (1, 2, 3, …), their negatives (−1, −2, −3, ...) and the number zero. ...

a = mc and b = nc.

Assuming for simplicity that a and b are positive, one can say that a ruler, marked off in units of length c, could be used to measure out both a line segment of length a, and one of length b. That is, there is a common unit of length in terms of which a and b can both be measured; this is the origin of the term. Otherwise the pair a and b are incommensurable. A negative number is a number that is less than zero, such as −3. ... A ruler is an instrument used in geometry and technical drawing to measure short distances and/or to rule straight lines. ... In mathematics, a line segment is a part of a line that is bounded by two end points. ... In general English usage, length (symbols: l, L) is but one particular instance of distance – an objects length is how long the object is – but in the physical sciences and engineering, the word length is in some contexts used synonymously with distance. Height is vertical distance; width (or breadth...

In group theory, a generalisation to pairs of subgroups is obtained, by noticing that in the case given, the subgroups of the real line as additive group, generated respectively by a and by b, intersect in the subgroup generated by dc, where d is the LCM of m and n. This is of finite index, therefore in each of them. This gives rise to a general notion of commensurable subgroups: two subgroups A and B of a group are commensurable when their intersection has finite index in each of them. Sometimes in fact this relation is called commensurate, and to be commensurable requires only to be conjugate to a commensurate subgroup. Group theory is that branch of mathematics concerned with the study of groups. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, the real line is simply the set of real numbers. ... An additive group is a group, and any group can be written as an additive group, so the adjective additive does not describe a class of groups, but rather the notation used to write the group operation. ... In arithmetic and number theory the least common multiple or lowest common multiple (lcm) or smallest common multiple of two integers a and b is the smallest positive integer that is a multiple of both a and b. ... In mathematics, a set is called finite if and only if there is a bijection between the set and some set of the form {1, 2, ..., n} where is a natural number. ... The term intersection can mean: a road junction, where two roads intersect each other, such as a roundabout intersection; in mathematics, the set in which two or more other sets intersect each other; see intersection (set theory); a movie; see Intersection (movie). ... Look up Index in Wiktionary, the free dictionary Index can be defined as: an ordered list, plural indexes a number or variable, plural indices. ...

A relationship can similarly be defined on subspaces of a vector space, in terms of projections that have finite-dimensional kernel and cokernel. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In linear algebra, a projection is a linear transformation P such that P2 = P, i. ... 2-dimensional renderings (ie. ... In abstract algebra, the cokernel of a homomorphism f : X → Y is the quotient of Y by the image of f. ...

In contrast, two subspaces A and B that are given by some moduli space stacks over a Lie algebra are not necessarily commensurable if they are described by infinite dimensional representations. In addition, if the completions of -type modules corresponding to and are not well-defined, then and are also not commensurable. Screenshot (from SSCX Star Warzone). ... In algebraic geometry, the moduli problem is to describe the parameters on which algebraic varieties depend. ... In mathematics, an algebraic stack in algebraic geometry is a concept introduced to generalize algebraic varieties and schemes. ... In mathematics, a Lie algebra is an algebraic structure whose main use lies in studying geometric objects such as Lie groups and differentiable manifolds. ... In mathematical analysis, a metric space M is said to be complete (or Cauchy) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if it doesnt have any holes, if there arent any points missing. For... In abstract algebra, the notion of a module over a ring is the common generalizations of two of the most important notions in algebra, vector space (where we take the ring to be a particular field), and abelian group (where we take the ring to be the ring of integers). ... In mathematics, the term well-defined is used to specify that a certain concept (a function, a property, a relation, etc. ...