A rotation number is an important property of a circle homeomorphism. the denjoy theorem and the poincare. theorem are the well known facts which relate to an irrational rotation number. It is known that an irrational rotation and a circle homeomorphism with an irrational rotation number have many common properties. we review properties of a rotation number of a circle homeomorphism, for example, Arnold tongues and the rotation function. we define a new rotation function and investigate its interesting properties.