Constructing New Formulae for Finding Squares of Positive Integers by Studying Their Own Patterns

Abstract

Squaring a given number is a very useful and necessary operation in scientific computations. There are some applications of them in real life, most commen usage is Pythagoras theorem. When the number of digits in a specific integer increases, it is very hard to calculate the square using traditional methods. There are some methods developed by mathematicians to find the perfect square for a given number in addition to the use of the definition of squaring a number. In this study, it was mainly focused on constructing the novel formulae for finding perfect squares. To obtain formulae, own patterns of perfect squares were analyzed by using elementary mathematical concepts after categorizing them into the last digit of each perfect square and focusing particularly on the perfect squares ending with five. Ten new formulae could be discovered. These interesting formulae consist of floor and ceiling functions. Furether, an MATHEMATICA program was developed for the aforementioned method using recursive terminology. Finally, for the verification of results obtained from above methods, existing MATHEMATICA commands for finding perfect squares was used. It is also observed that in finding the cube of integer based on last digit has similar patterns as in squaring an integer and this is open for further studies.