The Plum-Blossom Wedge Product Method for Multiplication and Division of Multi-digit Integers

TL;DR

This paper introduces the plum-blossom wedge product, a novel mathematical operation that enhances the efficiency of multiplying and dividing large integers, with applications in mental arithmetic and computer hardware design.

Contribution

The paper extends the original plum-blossom product to a new wedge product, providing a more effective method for long integer arithmetic and hardware implementation.

Findings

The plum-blossom wedge product improves multiplication and division efficiency.

It is suitable for mental arithmetic and computer hardware design.

The method maintains good algebraic properties for long integer operations.

Abstract

For two integers $a$ and $b$, the plum-blossom product of $a$ and $b$ is defined as the ones of the usual product of $a$ and $b$ if the ones is less than or equal to $3$, or otherwise as the ones minus $10$. Under this operation, all integers is a commutative semigroup. Since some fine properties of this semigroup, the plum-blossom product method is the efficient method of multiplication and division of long integers, can be used to mental arithmetic and to computer science. In this paper, we improve the theory of plum-blossom products so that the new concept--the plum-blossom wedge product is proposed. The plum-blossom wedge product is also of some good features so that it can be applied to multiplication and division of long integers. The plum-blossom wedge product method of multiplication and division is not only efficient in mental arithmetic, but also could applied to improve the design of multipliers and dividers of computer.