Inmathematics, two non-zeroreal numbersa andb are said to becommensurable if their ratio⁠a/b⁠ is arational number; otherwisea andb are calledincommensurable. (Recall that a rational number is one that is equivalent to the ratio of twointegers.) There is a more general notion ofcommensurability in group theory.

For example, the numbers 3 and 2 are commensurable because their ratio,⁠3/2⁠, is a rational number. The numbers${\displaystyle \sqrt{3}}$ and${\displaystyle 2\sqrt{3}}$ are also commensurable because their ratio,$\frac{\sqrt{3}}{2\sqrt{3}}=\frac{1}{2}$, is a rational number. However, the numbers$\sqrt{3}$ and 2 are incommensurable because their ratio,$\frac{\sqrt{3}}{2}$, is anirrational number.

More generally, it is immediate from the definition that ifa andb are any two non-zero rational numbers, thena andb are commensurable; it is also immediate that ifa is any irrational number andb is any non-zero rational number, thena andb are incommensurable. On the other hand, if botha andb are irrational numbers, thena andb may or may not be commensurable.

ThePythagoreans are credited with the proof of the existence ofirrational numbers.^[1][2] When the ratio of thelengths of two line segments is irrational, the line segmentsthemselves (not just their lengths) are also described as being incommensurable.

A separate, more general and circuitous ancient Greekdoctrine of proportionality for geometricmagnitude was developed in Book V of Euclid'sElements in order to allow proofs involving incommensurable lengths, thus avoiding arguments which applied only to a historically restricted definition ofnumber.

Euclid's notion of commensurability is anticipated in passing in the discussion betweenSocrates and the slave boy in Plato's dialogue entitledMeno, in which Socrates uses the boy's own inherent capabilities to solve a complex geometric problem through the Socratic Method. He develops a proof which is, for all intents and purposes, very Euclidean in nature and speaks to the concept of incommensurability.^[3]

The usage primarily comes from translations ofEuclid'sElements, in which two line segmentsa andb are called commensurable precisely if there is some third segmentc that can be laid end-to-end a whole number of times to produce a segment congruent toa, and also, with a different whole number, a segment congruent tob. 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.

That⁠a/b⁠ is rational is anecessary and sufficient condition for the existence of some real numberc, andintegersm andn, such that

a =mc andb =nc.

Assuming for simplicity thata andb arepositive, one can say that aruler, marked off in units of lengthc, could be used to measure out both aline segment of lengtha, and one of lengthb. That is, there is a common unit oflength in terms of whicha andb can both be measured; this is the origin of the term. Otherwise the paira andb areincommensurable.

Ingroup theory, twosubgroups Γ₁ and Γ₂ of a groupG are said to becommensurable if theintersection Γ₁ ∩ Γ₂ is offinite index in both Γ₁ and Γ₂.

Example: Leta andb be nonzero real numbers. Then the subgroup of the real numbersRgenerated bya is commensurable with the subgroup generated byb if and only if the real numbersa andb are commensurable, in the sense thata/b is rational. Thus the group-theoretic notion of commensurability generalizes the concept for real numbers.

There is a similar notion for two groups which are not given as subgroups of the same group. Two groupsG₁ andG₂ are (abstractly)commensurable if there are subgroupsH₁ ⊂G₁ andH₂ ⊂G₂ of finite index such thatH₁ isisomorphic toH₂.

Twopath-connectedtopological spaces are sometimes said to becommensurable if they havehomeomorphic finite-sheetedcovering spaces. Depending on the type of space under consideration, one might want to usehomotopy equivalences ordiffeomorphisms instead of homeomorphisms in the definition. If two spaces are commensurable, then theirfundamental groups are commensurable.

Example: any twoclosed surfaces ofgenus at least 2 are commensurable with each other.

• Real numbers
• Infinite group theory
• Greek mathematics

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