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Journal of Applied Sciences & Environmental Management, Vol. 9, No. 3, 2005, pp. 12-13 New Performance of Square of Numbers 1*AKINPELU, F O;2ADETUNDE, I A; 3OMIDIORA, E O 1Ladoke AkintolaUniversity of
Technology, Department of Pure & Applied Mathematics, Ogbomoso, Oyo Sate. *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 A2 = X FOLLOWING THE RULES (ALGORITHMS) ABOVE, WE HAVE A2 = (A-1) X (A+1) + 1= X A2 =A2 +A-A-1+1=X A2 = A2 =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 32 =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 1011 NUMBER BEFORE = 500499 NUMBER AFTER = 500501 PRODUCT OF THE TWO NUMBERS + 1 = 500499 X 500501 + 1 =2.505025 X 1011 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. Squaring of Numbers Diff 1 Diff 2 Diff 3 12 = 1 } 3 22 = 4 } 2 } 5 } 0 32 = 9 } 2 } 7 } 0 42 = 16 } 2 52 = 25 }9 }0 } 2 62 = 36 } 11 }2 } 0 72 = 49 } 13 REFERENCES
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