A Multiplication Formula and Its Application

TL;DR

This paper introduces a new vertical multiplication formula for multi-digit integers that improves on existing algorithms by simplifying calculations and enhancing practical application in hardware and mental math.

Contribution

It generalizes the Karatsuba and Offman algorithms with a novel approach using horizontal subtraction, making calculations simpler and more suitable for mental and hardware multiplication.

Findings

The new algorithm reduces the size of involved numbers and their sums.

It is more suitable for mental calculation of large integers.

The formula enables more practical and efficient multiplier designs.

Abstract

This article presents a vertical multiplication formula for calculating the multiplication of any two multi-digit integers, which may be not only used to design the multiplier but also to the mental multiplication. Our algorithm is a generalization of that of Karatsuba and Offman, but it is superior to the latter because the latter is not suitable for oral calculation of multiplication of large integers. These two algorithms are both the vertical multiplication method, but in our algorithm the horizontal subtraction in stead of the horizontal addition used in the Kartsuba algorithm makes the involved numbers are much smaller and their algebraic sum are much smaller too because the positive and negative numbers probably cancel each other out. Because of the above advantages, our vertical multiplication formula allows more practical and efficient multipliers to be designed.