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NUMERICAL AND GEOMETRIC SQUARE (General Equation Of Pythagoras Theorem) What Pythagoras said some 2500 years ago, and what generation to come understood from it did not exactly match. A part of his theorem that a square area can be divided into two square areas. Was thoroughly dealt with; while the rest was ignored, or left unattended. Pythagoras, however, did not only clothe the theorem in words, but demonstrated it practically. What he did was cut a square area of arbitrary size and touched its adjacent corners to mutually perpendicular lines, now known as x-axis and y-axis of rectangular co-ordinate system. The line segments between origin (point of intersection of mutually perpendicular lines) and points were corners touched were sides of two squares whose combined area was equal to the original area, as shown below. That is,  z2  =  x2   +  y2.

Area of z2 is equal to the sum of areas x2   and  y2, i.e. z2  =  x2   +  y2. But this is not all what we can infer from Pythagoras Theorem. Area z2, in fact, resolves into two square areas,  x2  and  y2, as do a vector  ̿Ṝ which resolves into its components  Ṝx and  Ṝy. Different values of θ, which changes orientation of z2, gives different components x2  and y2 for the same.

What our mathematicians now fully understand is that the magnitude of z2 equal to combine magnitude of  x2 and  y2. But, what they completely ignore is that z2 actually resolves into two component * x2 and y2. This resolution is analogous to the resolution of a vector into its rectangular component. In this sense, z2 (or any area ) is a vector quantity. Thus same z2, by touching at different points, could have different components x2 and y2 because of side way orientation.
 * these components, in fact, lie in orthogonal planes. But, for simplicity, we have shown them lying in xy-planes, here in above figure and all other figures to come.

The term side-way-orientation is new. It is extensively used in the paper to represent angle θ of the figure above. Parallel-orientation, on the other hand, means constant value of θ. This second of orientation (Parallel-orientation) is existed when scalar number n2 is multiplied to both sides of equation z2 = x2  +  y2 for particular values of x2, y2, and z2 ; i.e.(nz) 2 = (nx) 2 + (ny) 2.

Man and numbers lived together for thousands and thousands of years, but, historically, there were three periods when he really appreciated the value of numbers. First of them was Pythagorean period, some six hundred years before Christ, when Pythagoras expressed their numbers (x, y and z) in terms of the sides of a right-angle triangle and give a relation: z2 = x2 +  y2    ………………(1) were x, y were two sides and z was hypotenuse.

Since x, y and z represented three lines; they would not necessarily be natural numbers. Pythagoras, himself, provided formulas for finding three natural numbers (called Pythagorean triplets), which would satisfy equation, —(1), as		z = m2  + 1 x = m2  -  1 y = 2m where m  could be any natural number. Though these formulas produced infinitely many Pythagorean triplets, one corresponding to each value of m, they would not give all. It was left to Diophantine period*, around 250 A.D., when right equations were discovered Which would give all possible Pythagorean triplets. These formulas can be written as: z  = a2   +   b2   ………….(2) x  = a2   -    b2   ………….(3) y  = 2ab         …………  ..(4) where a and b are arbitrary natural numbers. A complete solution of these formulas, sometimes called Diophantine problem, can be found in the 10th book of Euclid. But a comprehensive derivation of general formulas are given on next page. One can not only find the above formulas, but also an infinite numbers like them.
 * This period belongs to Diophantus, a Greek mathematician. He was the first to   introduce    symbolism into Greek algebra.

Numbers do not mean anything unless we attribute some physical concepts to them. A numerical square, for example, can be equal to a numerical cube, such as: (8)2 = (4)3 But, in geometry, a square and cube are two separate entities. An extreme care must therefore be taken while dealing with numbers alone. It was Pythagoras who first established a truly remarkable relation between numbers and geometry, and then developed formulas on the basis of numbers to relate geometric figures. Thus one Pythagorean triplet represents a particular relation among geometric figures, and formulas yielding all triplets relate one set of figures to all other sets. In other words, one triplet is like a point on a continuous process. If all triplets represent ‘mile-stones’ of the continuous process, then this justifies the validity of relation at every other point; i.e.  z2 =x2 + y2, where x2 and y2 are any integers, and z2 is a resulting value of the square. [Note that it would be more conceptual if each member of a triplet were represented by its square value than its linear value].

The triplets I have discussed so far tell a story of only one kind of sequence, but there could be infinite many other sequences. For example, 3 and 5 generate the following square: (3)2 + (5)2 + (3)(5) = (7)2 This triplet, (7; 3,5), could be a member of a sequence generated by the equation, x2 + y2 + xy = p2 …………………………………(5). Some other triplets of this equation are, (p; x,y)=(13; 7,8)=( 13; -15,7)=(273; 225,33) etc. Similarly, the equation  x2 + y2 + 3xy = p2 ……………..(6) Could produce triplets like (p; x,y) = (11;3,7),= (11; -3,16), = (79; 16, 57).

Now question arises, is there a General equation with General formulas for triplets from which all the above forms could be deduced? The answer is yes. The following derivation gives the desired solutions. Suppose the general equation is of the form;

x2 + y2 + ℓxy = p2  ……(7) (General Equation Of Pythagoras Theorem)

where, for simplicity, ℓ is any integer (i.e. ℓ = 0, ±1 , ±2 , ±3……). Pythagoras equation (1) could simply be derived by substituting ℓ = 0. Other equation like eq.(5) and eq.(6) can be obtained by putting ℓ = 1 and ℓ = 3 respectively.

In order to obtain a general formulas for triplets we proceed as follows: Let a, b ϵ Z {a and b belong to Z, which is a set of all integers}, then P can have particular values of a and b. an equation which produces all the possible integers for p is:

P = a2 + b2 + ℓab Squaring both sides, (p) 2 = (a2 + b2 + ℓab ) 2 And rearranging the right-hand-side,

p2 = (a2- b2) 2 + (2ab + ℓb2) 2 + ℓ(a2-b2)(2ab + ℓb2). But, from the equation – (7)

P2 = x2 + y2 + ℓxy Therefore, P = a2 + b2 + ℓab …………(8) x = a2 – b2 ………………...(9) y = 2ab + ℓb2 ……………..(10)

The triplet (p; x,y) = [(a2+b2+ℓab); (a2-b2), (2ab + ℓb2)]

Is a general triplet from which all the formulas for different equations can be obtained for particular values of ℓ.

I have so far observed only one aspect of the square p, which is given by the general equation-(7). There could be other forms such as: P2 = x2 + y2 + z2 The quadiplets for this equation can similarly be derived as:

P = a2 + b2 + c2                  {where a,b,c ϵ Z}	x = a2 +  b2 – c2	y = 2ac z = 2bc

The pentiplets of the equation, P2 = x2 + y2 + z2 + t2

P = a2 + b2 + c2 + d2  { where a,b,c,d ϵ Z } x  = a2 + b2 +c2 – d2	y  = 2ad y = 2bd t  = 2cd Similarly one can add as many squares as he likes to get a resultant square p, and can also find natural numbers satisfying the equation.

P2 = x12 +x22 + x32 +……….+ xn2 (Another General Equation Of                                     Pythagoras Theorem for n=2 n=2 ) Were P = a12 + a22 + a32 + ……..……..+an2 x1= (a12 + a22 + a32+…….+an-12) – an2 x2 = 2a1an x3 = 2a2an …	…	…		…	xn = 2an-1an Where (a1,a2,a3,……,an ϵ Z)

I started with the Pythagoras theorem and continued on to find a resultant square of a combination of areas. But my scope of work remained limited to numbers. Naturally, one may like to see the geometric interpretation of study the Pythagoras relation, z2 = x2 + y2 Since this relation holds for arbitrary values of the areas x2 and y2, I can, therefore, take any square along x-axes and any other square along y-axis. The resulting area z2 will have a position in geometric space as shown  in the figure (0.1). In other words., z2 can be decomposed into its component area x2 and y2, or conversely x2 and y2 can be combined to give resultant z2. AN IMPORTANT MESSAGE WE GET FROM THE ABOVE DESCRIPTION IS THAT THE MAGNITUDE OF Z2 IS AS IMPORTANT AS ITS ORIENTATION ALONG CO-ORDINATE AXIS. Hence same magnitude of z2 can have different orientations in geometric space with different component squares x2 and y2. So area is a vector quantity with two directions: one perpendicular to it and the other side-way orientation. Geometric effects of multiplying n2 (when n is a rational number), to both sides of equation z2 =x2 + y2,    i.e       (nz) 2 = (nx) 2 + (ny) 2

If, now, we increase z2 by n2 [n is a rational number] times; that is, (nz) 2, then the corresponding increase in components x2 and y2 will be (nx) 2 and (ny) 2. This means that z2 and (nz) 2 are in parallel orientation (θ remains constant); and the increase in z2 distributes itself with the same ratio (n2) along both x-axis and y-axis, see figure (2.1). Similarly, the figure (2.2) is a geometric illustration of equation:

Z2 = x2 + Y2   and  p2 = x2 + y2 + xy, where

(For z2 = x2 + Y2 )                                  (For p2 = x2 + y2 + xy ) z = a2 + b2					p= a2 + b2 + ab x = a2 - b2 					x = a2 – b2 Y = 2ab 					y = 2ab +b2

For same values of a and b, z2 and p2 cannot be in parallel-orientation. They cannot even be made parallel for any integer values of a  and  b, because of the irrationality of. . 		Term √( y2 + xy)        in  p2 = [ x2 + (√ y2 + xy)2] For the same values of a and b the x-component is same for both the equation; that is x = a2 – b2. The expression ‘xy’ of x2 +y2 +xy would, therefore, distributes its area only along y-axis, as shown by dotted lines. The geometric location of z2 and p2 will be as shown in figure (2.2). the area p2, by Pythagoras relation, will be 			P2= x2 + ( √ (y2 + xy )) 2 Since y2 + xy  can never be a rational number, Therefore, for a rational m, the term x2 + ( √ (y2 + xy )) 2  cannot be equated to any (mz) 2 for any values of a and b. hence p2 and z2 cannot be in parallel-orientation.

Naeem Ahmad 423-AI Gulbarg III Ph# 00923009488180			   Lahore (Pakistan) email= naeema_3@hotmail.com www.anewmaths.com