 Division is a kind of analysis called cardinal analysis.Multiplication is a kind of synthesis called cardinal synthesis

Classically, the quaternion was viewed as the ratio of two vectors, sometimes called a geometric fraction.If OA and OB represent two vectors drawn from the origin O to two other points A and B, then the geometric fraction was written as

Alternately if the two vectors are represented by α and β the quotient was written as

or

Hamilton asserts