 Since "θ" is irrational, the rotation has infinite order in the circle group and the map "T""θ" has no periodic orbits.Alternatively, we can use multiplicative notation for an irrational rotation by introducing the map

The relationship between the additive and multiplicative notations is the group isomorphism

It can be shown that is an isometry.There is a strong distinction in circle rotations that depends on whether is rational or irrational.