Difference of Squares of Two Natural Numbers

Difference of Squares of Two Natural NumbersOne of the basic arithmetic operations is convalesceing jogs and inconsistency in the midst of significants of deuce innate represss. Though there are various rules to convey the variety amongst squares of dickens born(p) be, quiesce there are scopes to find simplified and easy approaches. As the grade formed using the passing betwixt squares of deuce congenital deem imitate a subjugate patterns, using progeny patterns may facilitate more easy approach. Also, this epoch has some general properties which are already discussed by some mathematicians in different notations. apart from these, the era has some specific properties like sequence remnant prop, deflection sum airplane propeller, which helps to find the evaluate easily. The sequence like brisk has some relations that assist to form a number pattern.This paper tries to break the general properties, special properties of finding contrast surro unded by the squares of any dickens instinctive poem using number patterns. A rhombus rule relationship among the sequences of numbers formed by considering the difference between squares of the twain inwrought numbers has been defined. A new method to find a2 b2 to a fault has been introduced in some simple cases. This approach will help the secondary tuition lower grade students in identifying and recognizing number patterns and squares of congenital numbers.Mathematical Subject Classifications (2010) 11A25, 11A51, 40C99, 03F50DIFFERENCE BETWEEN SQUARES OF TWO NATURAL NUMBERS RELATIONS, PROPERTIES AND NEW APPROACH penetrationMathematics, a subject of problem solving skills and applications, has wide usage in all the fields. Basic skills of numeric applications in number systems used even in day to day life. Though calculators and computers have greater influences in calculations, still there is a need to find new easy methods of calculations to improve private intell ectual skills.As there has been g rowinging interest, in mathematics education, in teaching and learning, umpteen mathematicians build simple and different methods, rules and relationships in various mathematical field. Though various investigations have made important contributions to mathematics development and education (2), there still room for new research to clarify the common relationship between the numbers and number patterns.In innate numbers, various subsets have been recognized by ancient mathematicians. about are droll numbers, meridian numbers, oblong numbers, triangular numbers and squares. These numbers shall be identified by number patterns. Recognizing number patterns is to a fault an important problem-solving skill. Working with number patterns leads directly to the creation of functions in mathematics a formal description of the relationships among different quantities.One of the basic arithmetic operations is finding squares and difference between squares of ii natural numbers. Already many proofs and relationships were identified and proved in finding difference between squares of dickens natural numbers. We use different methods to find the difference between squares of deuce natural numbers. That is, to find a2 b2. Though, this area of research may be discussed by early mathematicians and researchers in various aspects, still there are many interesting ways to discuss the same in teaching.Teaching number patterns in secondary level education is most important issue as the students develop their analytical and cognitive skills in this stage. Different arithmetic operations and calculations need to be introduced in such way that they help the students in long learning. Easy and simplified approaches will support the students in logical reasoning.This paper tries to identify the general properties, special properties of finding difference between the squares of any two natural numbers using number patterns. Also, this paper trie s to define the rhombus rule relationship between the sequences of numbers formed by the differences of squares of two natural numbers. A new method to find a2 b2 also has been introduced in some simple cases.These may be introduced in secondary school early grades, in front introducing algebraical techniques of finding a2 b2 to develop the knowledge and understanding of number patterns. This will help to recognize and apply number patterns in pull ahead level.Literature check up onTo find the difference between the squares of any two natural numbers, we use different methods. Also, we use various rules to find the square of a natural number. Some properties were also been identified by the researchers and mathematicians.Methods used to find the difference between squares of two natural numbersDirect MethodThe difference between the squares of two natural numbers shall be found out by finding the squares of the numbers directly.Example 252 52 = 625 25 = 600Using algebraic rul eThe algebraic rule a2 b2 = (a b)(a + b) shall be applied to find the difference between the squares of two natural numbers.Example 252 52 = (25 5)(25 + 5) = 20 x 30 = 600Method when a b = 1(2)The difference between the squares of every two square natural numbers is always an odd number, and that it is equal to the sum of these numbers.Example 252 242 = 25 + 24 = 49Methods used to find the square of a natural numberUsing Algebraic MethodThe algebraic rules shall be used to find the square of natural number new(prenominal) than the direct multiplication. In general, (a + b)2, (a b)2 are used to find the squares of a natural number from nearest whole number.Example 992 = ( one C 1)2= 1002 2(100)(1) + 12 = 10000 200 + 1= 9801Square of a number using previous number(8)The following rule may be applied to find the square of a number using previous number.(n + 1)2 = n2 + n + (n+1)Example 312 = 302 + 30 + 31 = 900 + 30 + 31 = 961The Gilbreth Method of finding square(9)The Gilbr eth method uses binomial theorem to find the square of a natural number. The rule isn2 = 100(n 25) + (50 n)2Example 992 = 100(99 25) + (50 99)2= 7400 + 2401 = 9801Other than the higher up mentioned methods various methods are used based on the knowledge and requirements.Properties of differences between squares of the natural numbers2.3.1. The difference between squares of any two consecutive natural numbers is always odd.To prove this property, permit us consider two consecutive natural numbers, opine 25 and 26Now allow us find 262 252262 252 = (26 + 25)(26 25) Using algebraic rule= 51 x 1 = 51, an odd number2.3.2. The difference between squares of any two election natural numbers is always even.To prove this property, let us consider two ersatz natural numbers, say 125 and 127Now let us find 1272 12521272 1252 = (127 + 125)(127 125) Using algebraic rule= 252 x 2 = 504, an even numberSome other properties were also identified and discussed by various mathematicians and researchers.Number Patterns and Difference Between the Squares of Two Natural Numbers Discussions and FindingsSome of the properties stated above shall be proved by using number pattern. Number patterns are interesting area of arithmetic that stimulates the logical reasoning. They shall be applied in various notations to identify the sequences and relations between the numbers.3.1. Sample bow for the difference between squares of two natural numbersTo find the properties and relations that are satisfied by the sequences formed by the differences between the squares of two natural numbers, let us form a number pattern. For discussion purposes, let us consider first 10 natural numbers 1, 2, 3 10.Now, let us find the difference between two consecutive natural numbers.That is, 22 12 = 3 32 22 = 5 and so on. past the sequence will be as follows 3, 5, 7, 9, 11, 13, 15, 17 and 19.The sequence is a set of odd numbers starting from 3.i.e., Difference 1 x x is an odd number greater t han or equal to 3, x NIn the same way, let us form the sequence for the difference between squares of two alternative natural numbers.That is, 32 12 = 8, 42 22 = 12, and so on.Then the sequence will be 8, 12, 16, 20, 24, 28, 32 and 36 thusly the sequence is a set of even numbers and multiples of 4 starting from 8.i.e., Difference 2 x x is an multiple of 4 greater than or equal to 8, x NBy proceeding this way, the sequences for other differences shall be formed. permit us represent the sequences in a table for discussion purposes.In Table 1, N is the natural number.S is the square of the corresponding natural number.D1 represents the difference between the squares of two consecutive natural numbers. That is, the difference between the numbers is 1.D2 represents the difference between the squares of two alternate natural numbers. That is, the difference between the numbers is 2.D3 represents the difference between the squares of 4th and 1st number. That is, the difference between the numbers is 3, and so on.3.2. Relationship between the row elements of each columnNow, let us discuss the relationship between the elements of rows and columns of the table.From the above table,Column D1 shows that the difference between squares of two consecutive numbers is odd.Column D2 shows that the difference between squares of two alternate numbers is even.The other columns show that the difference between the squares of two numbers is either odd or even.From the above findings, the following properties shall be defined for the difference between squares of any two natural numbers.3.3. General Properties of the difference between squares of two natural numbersThe difference between squares of any two consecutive natural numbers is always odd.Proof Column D1 proves this property.This may also be tested randomly for big numbers.Let us consider two digit consecutive natural numbers, say 96 and 97.Now, 972 962 = 9409 9216= 493, an odd numberLet us consider three digit consecu tive natural numbers, say 757 and 758.Thus, 7582 7572 = 574564 573049= 1515, an odd numberThis property may also be further tested for big numbers and proved. For example, let us consider five digit two consecutive natural numbers, say 15887 and 15888.Then, 158882 158872 = 252428544 252396769= 31775, an odd numberApart from these, the property shall also be easily derived by the natural numbers properties. As the difference between two consecutive numbers is 1, the natural number property The sum of odd and even natural numbers is always odd, shall be applied to prove this property.The difference between squares of any two alternative natural numbers is always even.Proof Column D2 proves this property.This may also be verified for big numbers by considering different digit natural numbers as discussed above.Apart from this, as the difference between two alternate natural numbers is 2, the natural numbers property A natural number said to be even if it is a multiple of two shall also be used for proving the stated property.The difference between squares of any two natural numbers is either odd or even, depending upon the difference between the numbers.Proof The other columns of Table 1 prove this property.In Table 1, as D3 represents the sequence formed by the difference between two natural numbers whose difference is 3, an odd number, the sequence is also odd. Thus, the property may be proved by testing the other Columns D4, D5, Also, the addition, subtraction and multiplication properties of natural numbers prove this property.Example112 62Here the difference (11 6 = 5) is odd.So, the essence will be odd.i.e. 112 62 = 121 36 = 85, an odd number122 82Here the difference (12 8 = 4) is even.So, the result will be even.i.e. 122 82 = gross 64 = 80, an even number3.4. Special Properties of the difference between squares of the two natural numbersTable 1 also facilitates to find some special properties stated below.Sequence Difference PropertyTable 1 sh ows that the sequences formed are following a number pattern with a common property between them. Let us consider the number sequences of each column.Let us consider the first column D1 elements. D1 3, 5, 7, 9, 11 As D1 represents the difference between the squares of two consecutive natural numbers, let us say, a and b with a b, the difference between them will be 1.That is a b = 1Let us consider the difference between the elements in the sequence.The difference between the numbers in the sequence is 2.Thus the difference between the elements of the sequence shall be expressed as, 2 x 1. Thus, Difference = 2(a b)Now, let us consider the second column D2 elements. D2 8, 12, 16, 26, As D2 represents the difference between the squares of two alternative natural numbers, the difference between the natural numbers, say a and b is always 2. That is a b = 2If we consider the difference between the elements in the sequence, the difference is 4.Thus, the difference between the elemen ts in the sequence shall be expressed as 2 x 2.That is, difference = 2 (a b)In the same way, D3 15, 21, 27, 33, D3 represents the difference between squares of the 4th and 1st numbers, difference is 3. That is a b = 3The difference between the numbers in the sequence is 6.Thus, difference = 2 x 3 = 2(a b)All other columns also show that the difference between the numbers in the corresponding sequence is 2 (a b)Thus, this may be infer as following propertyThe difference between elements of the number sequence, formed by the difference between any two natural numbers, is equal to two times of the difference between those corresponding natural numbers.Difference Sum PropertyFrom Table 1, we shall also identify another relationship between the elements of the sequence formed.Let us consider the columns from table 1 other than D1.Consider D2 8, 12, 16, 20, This sequence shall be formed by adding two numbers of Column D1.i.e. 8 = 3 + 512 = 5 + 716 = 7 + 920 = 9 + 11And so on.Thus, if the difference between the natural numbers taken is 2, therefore the number sequence of the difference between the two natural numbers shall be formed by adding 2 natural numbers.Consider D3 15, 21, 27, This sequence shall be formed by adding three numbers from Column D1.i.e. 15 = 3 + 5 + 721 = 5 + 7 + 927 = 7 + 9 + 11And so on.Thus, if the difference between the natural numbers taken is 3, then the number sequence of the difference between the two natural numbers shall be formed by adding 3 natural numbers.This may also be verified with respect to the other columns.Table 2 shows the above relationship between the differences of the squares of the natural numbers.Now the above relation shall be generalized asIf a b = k 1, then a2 b2 shall be written as the sum of k natural numbersAs Column D1 elements are odd natural numbers, this property may be defined asIf a b = k 1, then a2 b2 shall be written as the sum of k odd natural numbersAs these odd numbers are consecutive, th e property may be further precisely defined asIf a b = k 1, then a2 b2 shall be written as the sum of k consecutive odd natural numbers3.5. New Method to find the difference between squares of two natural numbersUsing the above difference sum property, the difference between squares of two natural numbers shall be found as follows.The property shows that, a2 b2 is equal to sum of k consecutive odd numbers. Now, the principal idea is to find those k consecutive odd numbers.Let us consider two natural numbers, say 7 and 10.The difference between them 10 7 = 3Thus, 102 72 = sum of three consecutive odd numbers.102 72 = 100 49 = 51Now, 51 = Sum of 3 consecutive odd numbersi.e., 51 = 15 + 17 + 19Let we try to find these 3 numbers with respect to either the first number, let us say, a or the second number, say, b.Assume, for bAs general form for odd numbers is either (2n + 1) or (2n 1), as b 15 = 2(7) + 1 = 2b + 117 = 2(7) + 3 = 2b + 319 = 2(7) + 5 = 2b + 5Thus, 102 72 shall be written as the sum of 3 consecutive odd numbers starting from 15.i.e. starting from 2b + 1This idea may also be applied for higher digit numbers. Let us consider two 3 digit numbers, ci and 105. Let us find 1052 1012Here the difference is 4. Thus 1052 1012 shall be written as the sum of 4 consecutive odd numbers.The numbers shall be found as followsHere b = 101The first odd number = 2b + 1 = 2(101) + 1 = 203Thus, the 4 consecutive odd numbers are 203, 205, 207, 209So,1052 1012 = 203 + 205 + 207 + 209 = 824This shall be verified for any number of digits. Let us consider two 6 digit numbers 100519, 100521. Let us find 1005212 1005192Here the difference is 2. Thus 1005212 1005192 shall be written as the sum of two odd numbers.Applying the same idea,The first odd number = 2(100519) + 1 = 201039Thus the 2 consecutive odd numbers are 201039, 2010411005212 1005192 = 201039 + 201041 = 402080The above result shall be verified by using other methods.For example 1052 10121052 1012 = 11025 10201 = 824 (Using Direct Method)1052 1012 = (105 + 101) (105 101) = 206 x 4 = 824 (Using Algebraic Rule)Thus, this idea shall be generalized as followsa2 b2shall be found by adding the (a b) consecutive odd numbers starting from 2b + 1This shall also be found using the first term a. As a b, let us consider (2n 1) form of odd numbers.From Table 1, 102 62 = 13 + 15 + 17 + 19 = 64Here, 2a 1 = 2(10) 1 = 192a 3 = 2(10) 3 = 172a 5 = 2(10) 5 = 152a 7 = 2(10) 7 = 13Thus, as the difference between the numbers is 4, 102 62 shall be written as the sum of four consecutive odd numbers in reverse order starting from 2a 1.Thus proceeding, this may be generalized as,a2 b2shall be found by adding the (a b) consecutive odd numbers starting from 2a 1 in reverse orderFinding the first number of each columnLet us check the number pattern followed by the first numbers of each column. From Table 1, the first numbers of each column are 3, 8, 15, 24 Let us find the difference bet ween elements of this sequence.The difference between two consecutive terms of this sequence is 5, 7, 9 i.e. D2 D1 = 8 3 = 5 D3 D2 = 7 D4 D3 = 9 and so on.As D2 represents the difference between two alternate natural numbers, (say a and b) which implies that the difference between a and b is 2.Now, 5 = 2 (2) +1i.e. 2 times of the difference between the numbers + 1In the same idea, D3 D2 = 15 8 = 7As D3 represents the difference between squares of the 4th and 1st natural numbers, (say a and b) which implies that the difference between a and b is 3.Thus, 7 = 2(3) + 1This also shows that the difference shall be found by= 2 times of the difference between the numbers + 1Thus,The first term of the each column shall be found by adding the previous column first term with 2 times of the difference between the numbers + 1Finding the elements row wiseThe elements of the table shall also be formed in row wise.If we check the elements of each row, we can find that they follow a number pa ttern sequence with some property.Let us consider the elements of row when N = 5 20, 40, 60, 8020 = 2 x 5 x 2Here, 5 represent the row natural number.2 represent the difference between the elements using which the column is formed.Thus Row element = 2 x N x differenceIn the same way, 40 = 2 x 5 x 4= 2 x N x differenceThus, the elements shall be formed by the ruleRow Element = 2 x N x differenceThis shall be applied for middle rows also.For example, let us consider the row between 5 6The elements in this intermediate row are 11, 33, 55, 77, 99Here N is the mid value of 5 6. i.e. N = 5.5Let us consider the elements and apply the above stated rule.11 = 2 x N x difference= 2 x 5.5 x 1In the same way other elements shall also be formed.Thus the elements of the table shall be formed in row wise using the stated rule.baseball field Rule RelationLet us consider the elements in D2, D3 and D4.Consider the elements in the rhombus drawn, 24, 33, 39 and 4824 + 48 = 7233 + 39 = 72Thus the sums of the elements in the opposite corners are equal.The other column elements also prove the same.Thus, Rhombus Rule RelationSum of the elements the same row of the sequence of alternative columns is equal to the sum of the two elements in the intermediate columnApplication of the Properties in Finding the Square of a numberThe square of a natural number shall be found by various methods. Here is one of the suggested methods.This method uses nearest 10s and 100s to find the square of a number.This method is also based on the algebraic formula a2 b2 = (a b)(a + b)If a b, b2 = a2 (a2 b2)If b a, b2 = a2 + (b2 a2)Example Square of 32As we need to find 322, let us assume b = 32.The nearest multiple of 10 is 30. Let a = 30Here b a. b2 = a2 + (b2 a2)322 = 302 + (322 302)Using the Difference Sum Property,322 = 900 + 61 + 63 = 1024Example 2 Square of 9972Let b = 997Nearest multiple 10 is 1000. Let a = 1000Here a b, so b2 = a2 (a2 b2)9972 = 10002 (10002 9972)Using Difference Su m Property,9972 = 1000000 (1995 + 1997 + 1999)= 994009ConclusionThough this method shall be applied to find the difference between squares of any two natural numbers, if the difference is big, it will be cumbersome. Thus, this method shall be used for finding the difference between squares of any two natural numbers where the difference is manageable. The properties shall be used for easy calculation.This properties and approach shall be introduced in secondary school lower grade levels, to make the students to identify the number patterns. This approach will surely help the students to understand the properties of squares, difference and natural numbers. The new approach will surely help the students in create their reasoning skills.LimitationsAs number systems, number patterns and arithmetic operations have wide applications in various fields, the above properties, rules and relations shall be further canvas intensively based on the requirements. Thus, new properties and relati ons shall be identified and discussed with respect to other nations.