It is well known that the advantage of complex number over real number is that it can express vector, while real number can only express scalar. two real numbers can describe two-dimensional coordinates, and a complex number can describe two-dimensional coordinates; three real numbers can describe three-dimensional coordinates, and four real numbers can describe four-dimensional coordinates, but a quaternion can describe two-dimensional, three-dimensional or four-dimensional coordinates, so a complex or quaternion containing variables can reduce an independent variable when replacing the equation. In this paper, the complex number containing variables is used to describe a circle, a two-dimensional circle, and a quaternion containing variables is used to describe a three-dimensional the surface of the ball and a three-dimensional sphere, using quaternion with variables to describe four-dimensional the surface of the ball and four-dimensional sphere . It is also found that the complex number containing a variable can describe the plane golden spiral and the quaternion containing a variable can describe three-dimensional golden spiral. It is also found that if ab=aC can not deduce b=c when the exponent is a complex number.
Yandong He (Tue,) studied this question.