Commensurability (mathematics)

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, two non-<a href="/facts/Zero/4NBRtKHc">zero</a> <a href="/facts/Real_number/R02gw5Pb">real numbers</a> a and b are said to be commensurable if their ratio ⁠a/b⁠ is a <a href="/facts/Rational_number/VJQmyY05">rational number</a>; otherwise a and b are called incommensurable. (Recall that a rational number is one that is equivalent to the ratio of two <a href="/facts/Integers/PUwwrawx">integers</a>.) There is a more general notion of <a href="/facts/Commensurability_(group_theory)/CzbgSpQz">commensurability in group theory</a>.
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 <a href="/facts/Irrational_number/Psv19TET">irrational number</a>. 
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.

Commensurability (mathematics) open-in-new

Commensurability (mathematics)