Sign Space: A Continuous Generalization of Sign, with Yin–Yang Numbers as a First Instance

AbstractThis paper proposes a new mathematical framework that extends the conventionally dichotomous signs + and - into a continuous topological space called asign space. First we connect the conventional dichotomous sign space {+, −} andextend it into a continuous sign space. We then define a new number system calledthe Yin–Yang numbers on this continuous sign space. Next the Yin Yang axis isintroduced to form the Yin Yang half-plane and geometrically represent Yin Yangnumbers. Then the inverse sign is introduced as an axiom to extend the systemto the Yin Yang plane and then addition define. Subsequently a possible multiplication operation is proposed for the circular sign space and it is proven that thealgebraic structure on the Yin Yang plane is isomorphic to that on the complexplane. However this definition of multiplication has limitations and its naturalaxiomatic derivation is left for future research. This paper presents the circularsign space as a concrete example of a Sign Space. However this structure is onlyone possible example of a Sign Space. In future research, the Sign Space may begeneralized to various topological spaces, such as sphere, torus, and Möbius stripwith new algebraic structures potentially arising.