*Journal of Applied Sciences & Environmental Management, Vol. 9, No.
3, 2005, pp. 12-13*

**New Performance of Square of Numbers**

^{1}*AKINPELU,
F O;^{2}ADETUNDE, I A; ^{3}OMIDIORA, E O

^{1}Ladoke AkintolaUniversity of
Technology, Department of Pure & Applied Mathematics, Ogbomoso, Oyo Sate.

^{2}University of
Agriculture, Department of Mathematical Sciences, Abeokuta, OgunState.

^{3}Ladoke AkintolaUniversity of
Technology, Department of Computer Science & Engineering, Ogbomoso, OyoState.

*Corresponding author: Email foakinpelu@yahoo.com

**Code Number: ja05051**** **

**ABSTRACT:**

The
new discovery of squaring number can be use in getting a square of any number
be it positive integers or negative integers.@JASEM

The
starting point of Number Theory can be traced back to Piere de Fermat
[1707-1783] (D) Number theory is the field of study which was in the news
recently for the proof of Fermat’s last theorem, and in which Monser Kenku robs
shoulders with the like of Barry Marzu. Indeed a representation theoretic
theorem of Laglands is a vital ingredient in the works of viriles on Fermat’s
last theorem (E). As far back as 200 B.C. China discovered a magic square array
whose entries are taken from a set of consecutive whole numbers – beginning
from 1 – with the property that the numbers in any row, column or diagonal of
the array add up to the same sum. This magic square were subsequently
introduced into India,Japan and later to Europe (F).

Several
investigators have worked on theory of numbers (magic squared) notable among
them were (A, B , C ,F, and G). Early as 400 B.C., Pythagorean investigated
the properties of numbers and Euclid 300 B.C. proved that there are infinitely
many prime (D) one method for finding all prime numbers up to a given n was
devised by Erastospthemes of Cynene about 240.B.C.(D) In 1765 an English man
named John Wilson devised a method for testing whether a number is a prime
(D). In 1896 the French Mathematician Jacques-Hadamard and the Belgian
mathematician Charles Jeandela Valliel poussin-jointly established the prime
number theorem (C ). In this paper, we worked on results of squares of
numbers, which can be of good help to any students at any level.

**METHODOLOGY.**

PERFORMING
THE SQUARE OF A NUMBER M .WE CAN APPLY THE ALGORITHM BELOW.

**ALGORITHM:-**

STEP1
:- LOOK FOR A NUMBER BEFORE M TO BE SQUARE I.E. M-1

STEP2
:- LOOK FOR A NUMBER AFTER M TO BE SQUARE I.E.M+1

STEP3
:- MULTIPLY THE TWO NUMBERS IN STEP1 AND STEP2 TOGETHER

STEP4
:- ADD 1 TO THE RESULT OBTAIN IN STEP3.HENCE THE RESULT.

**PROOF**

A^{2}
= X

FOLLOWING
THE RULES (ALGORITHMS) ABOVE, WE HAVE

A^{2
}= (A-1) X (A+1) + 1= X

A^{2
}=A^{2} +A-A-1+1=X

A^{2}
= A^{2} =X

IT
IS TRUE

EXAMPLE
I,II,…VI.

**EXAMPLE
I:**

PERFORMING
THE SQUARE OF 3.

ALGORITHM

STEP1:
LOOK FOR A NUMBER BEFORE 3 TO BE SQUARE I.E. 2

STEP2:
LOOK FOR A NUMBER AFTER 3 TO BE SQUARE I.E. 4

STEP3:
MULTIPLY THE TWO NUMBERS TOGETHER I.E. 2X4=8

STEP4:
ADD 1 TO THE RESULT OF THE PRODUCT = 8+1 =9

HENCE
3^{2} =9

**EXAMPLE
II**

PERFORMING
THE SQUARE OF 12 = 144

RULE
1: NUMBER BEFORE 12 = 11

RULE
2: NUMBER AFTER 12 = 13

RULE
3: MULTIPLY THE TWO NUMBERS TOGETHER I.E. 11 X 13 =143.

RULE
4: ADD 1 TO THE RESULT OF THE PRODUCT I.E.143 + 1 = 144

**EXAMPLE
III**

PERFORMING
THE SQUARE OF 221 = 48841

RULE
1: NUMBER BEFORE 221 I.E. 220

RULE
2: NUMBER AFTER 221 I.E.222

RULE
3: MULTIPLY THE TWO NUMBERS TOGETHER I.E. 220 X222 = 48840

RULE
4: ADD 1 TO THE RESULT OF THE PRODUCT I.E.48840 + 1 =48841

**EXAMPLE
IV**

PERFORMING
THE SQUARE OF 12500 = 156250000

RULE
1: NUMBER BEFORE 12500 I.E.12499

RULE
2: NUMBER AFTER 12500 I.E.12501

RULE
3: MULTIPLY THE TWO NUMBERS TOGETHER I.E. 12499 X 12501 = 156249999

RULE
4: ADD 1 TO THE RESULT OF THE PRODUCT I.E. 156249999 + 1= 156250000

**EXAMPLE
V**

SQUARE
500500 = 2.5050025 X 10^{11}

NUMBER
BEFORE = 500499

NUMBER
AFTER = 500501

PRODUCT
OF THE TWO NUMBERS + 1 = 500499 X 500501 + 1 =2.505025 X 10^{11}

**EXAMPLE
VI**

FIND
THE SQUARE OF –3

NUMBER
BEFORE –3 I.E. –2

NUMBER
AFTER –3 I.E. –4

PRODUCT
OF THE NUMBER BEFORE AND AFTER = 8

PRODUCT
OF THE NUMBER + 1 = 8 + 1 = 9.

**CONCLUSION.**

Based
on the observations from the examples ( I – VI) and the proof given we
conclude that the method can be used in getting a square of any number be it
positive or negative integers.

Moreover
it is discover that when squaring the natural numbers and find out that their
differences in two places have common differences to be 2 all round and
difference 3 to be 0’s.