Substitution equation of complex quaternion with variable can reduce one variable

Abstract

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 a^b=a^C can not deduce b=c when the exponent is a complex number.