Commensurability (mathematics) | Wikipedia audio article

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00:00:13 1 History of the concept
00:00:27 2 Commensurability in group theory
00:00:41 3 In topology
00:00:55 4 References



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SUMMARY
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In mathematics, two non-zero real numbers a and b are said to be commensurable if their ratio a/b is a rational number; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two integers.) There is a more general notion of commensurability in group theory.
For example, the numbers 3 and 2 are commensurable because their ratio, 3/2, is a rational number. The numbers





3




{\displaystyle {\sqrt {3}}}
and



2


3




{\displaystyle 2{\sqrt {3}}}
are also commensurable because their ratio,






3


2


3





=


1
2




{\textstyle {\frac {\sqrt {3}}{2{\sqrt {3}}}}={\frac {1}{2}}}
, is a rational number. However, the numbers





3




{\textstyle {\sqrt {3}}}
and 2 are incommensurable because their ratio,






3

2




{\textstyle {\frac {\sqrt {3}}{2}}}
, is an irrational number.
More generally, it is immediate from the definition that if a and b are any two non-zero rational numbers, then a and b are commensurable; it is also immediate that if a is any irrational number and b is any non-zero rational number, then a and b are incommensurable. On the other hand, if both a and b are irrational numbers, then a and b may or may not be commensurable.