Editing Commensurability (mathematics) (section)

{{Short description|1=When two functions have co-rational periods, i.e. n T1 = m T2}}
{{other uses|Commensurability (disambiguation)}}

In [[mathematics]], two non-[[zero]] [[real number]]s ''a'' and ''b'' are said to be '''''commensurable''''' if their ratio ''{{sfrac|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 (group theory)|commensurability in group theory]].

For example, the numbers 3 and 2 are commensurable because their ratio, {{sfrac|3|2}}, is a rational number. The numbers <math>\sqrt{3}</math> and <math>2\sqrt{3}</math> are also commensurable because their ratio, <math display='inline'>\frac{\sqrt{3}}{2\sqrt{3}}=\frac{1}{2}</math>, is a rational number. However, the numbers <math display='inline'>\sqrt{3}</math> and 2 are incommensurable because their ratio, <math display='inline'>\frac{\sqrt{3}}{2}</math>, 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.