Fun with mathematics

Creative thought-

From   R ^2   to   R ^2 – relation

          2  1           2  2

The Pythagorean triples are having its own importance in

recreational mathematics. It is a source of generating

higher order like power numeral relations,particularly

with squares. In fact there is a close relationship between 

   R ^2   and   R ^2 –relations

2   1            2   2

In   R ^2 . a square is equated with a sum of two squares  and in 

    2   1

 R ^2  , a sum of two squares is equated with a sum of two other

2  2     

squares. By using a Pythagorean triple or two sets of triples,

one can generate     R^2

        2  2

 With one Pythagorean triple (x,y,z) we can generate

(x+z)^2 + (y+z)^2 = (z+x+y)^2 + (y-x)^2

For example (3,4,5) yields,

8^2 + 9^2 = 12^2 + 1^2

It is curious to  know that a single R^2 relation can generate
                                                                     2  1

a series of R^2 relations . For example the above numeral

relation can be written with numbers existing and number

added

(3+5)^2 + (4+5)^2 = (5+7)^2 + (0+1)^2

If all the existing numbers only or if  all the added numbers

only are multiplied with a number n , the equality of the relation

is not affected.

 (3n+5)^2 + (4n+5)^2 = (5n+7)^2 + 1^2

or

(3+5n)^2 +(4+5n)^2 = (5+7n)^2 + n^2

Again if all the existing numbers are multiplied with m and

all added numbers with n, the equality of the relation n is

still preserved.

(3m+5n)^2 + (4m+5n)^2 = (5m+7n)^2 + n^2  

Fun with Mathematics

Pythagorean triples with one same number

All the numbers in the Pythagorean triple cannot be odd,

because Sum of two odd squares will be even and cannot

be an odd square.Knowing the odd-even nature of the two

numbers of the Pythagorean triple, the odd-even nature of

the other number can be predicted.The permitted and

forbidden set of triples in terms of its odd-even

characteristics are given below                                  

                 permitted                                forbidden

        x        y        z                            x      y       z

      even  even  even                     even  even  odd

      even  odd    odd                      even  odd   even

      odd   odd   even                      odd   odd    odd

There are many puzzles associated with Pythagorean

triples. One among them is to find out two or more set

of triples with one identical number in them. This

identical number may be either the smallest or the

biggest of the set.

Factorisation technique is advantageous to generate such

sets.Any number (N), composite or prime can be expressed

as the product of its two factors (x,y). For primes it will the

1 and the prime number itself.

                                  N = x y

The  number N can be expressed in term of its factors as

N = [(x+y)/2]^2 – [(x-y)/2]^2

If N happens to be a square number N = n^2, then

n^2 + [(x-y)/2]^2 = [ (x+y)/2]^2

By using different pairs of two even or two odd factors for n^2

Pythagorean triples with one identical number n can be formed.

For example,

64 = 4x16 = 2x 32

which give (8,6,10) and (8,15,17)

From

 81 = 3x27 = 1x81 we have

(9,12,15) and (9,40,41)

If the square number has many pair of such factors

(both even or  both odd) ,then we can get three or

 even more sets with an identical number in them. For example,

144 = 2x72 =4x36=6x24=8x18

It gives,(12,35,37),(12,16,20),(12,9,15),(12,5,17)

Another way of getting such set of triples with identical bigger

number is linear combination of numbers in the given

Pythagorean triples(x,y,z) or with another

Pythagorean triples (a,b,c)

(ax+by)^2 + (ay-bx)^2 = (a^2+b^2)(x^2+y^2)= (cz)^2

For example

      x   y   z       a      b      c 

      3   4   5       7    24     25  gives 117^2 + 44^2 = 125^2

      3   4   5      24    7      25  gives 100^2 + 75^2 = 125^2

      3   4   5      20   15     25  gives 120^2 + 35^2 + 125^2

                                                                         (reducible)

It is very useful to generate double or triple coincidence in

numeral relations where sum of two squares is equated

with sum of two other squares. It is a different puzzle

which we will see in the next blog.

Fun with mathematics

Pythagoras theorem with different statement

Pythagoras theorem states that if square of a number is equal

to sum of squares of two other numbers ,then these three

numbers represent the three sides of a right-angled triangle.

Conversely, if (x,y,z) denotes the length of opposite side,

adjacent side and hypotenuse of a right-angled triangle

respectively, then

                           x^2 + y^2 = z^2

Where z>x and z>y but x+y >z.(3,4,5) and (5,12,13)  are the

two well-known examples for Pythagorean triples.

                          3^2 + 4^2 = 5^2 ;

                          5^2 + 12^2 = 13^2

 Since x^2 represents a square of a side x, we usually consider

 squares in each side of a right-angled triangle  to show that

the area of the square drawn on the hypotenuse is equal to the

sum of the areas of the squares drawn on other two sides.

It is curious to know that the Pythagoras theorem holds good

for many geometrical figures such as rectangle, circle,

semi-circle, equilateral triangle,

isosceles triangle etc.,

Squares need one variable- the side length, which is taken

as the length of each side of the given right-angled triangle.

But rectangle or isosceles triangle requires two variables,

the length and breadth  or base  and height.  The length of the

side of the triangle is taken as one of the variables for the

rectangles/isosceles triangles to be drawn on each side of the

triangle. The other variable required is derived from the other

sides of the given right-angled triangle . For example, the

length of the rectangle or the height of the isosceles triangle

drawn on the hypotenuse  is equal to the sum of the length

of the other two sides of the triangle  and for  other sides 

 it is the length of the hypotenuse.  It can be shown that the

areas of the rectangle/isosceles triangle drawn on hypotenuse

is equal to the sum of areas of the rectangles/isosceles

triangles drawn on other two sides.

               For rectangle :     xz + yz = z(x+y)

              For isosceles triangle :  ½ xz + ½ yz = ½ z(x+y)

                                                           Or xz + yz = z(x+y)

 If the length of the rectangle is taken as its breadth which is

 equal to the length of the side of the triangle over which it is

 drawn, it becomes a square. It holds good even in the case

of isosceles triangle. If the height of the isosceles triangle

is equal to the length of the side of the given right-angled triangle,

their areas make the very same Pythagoras expression

x^2+y^2= z^2.

If semi-circles/circles are drawn by keeping the sides of the

given right-angled triangle as its diameter, the n the area of

 the semi-circle/circle drawn on hypotenuse will be equal to

the sum of the areas of the semi-circles/circles drawn on

other two sides.