Tuesday, November 23, 2010

creative thoughts-16

creative thought-16

Difference of two numbers equals to its product

The difference between two numbers may be equal to its product.

Mathematically it is expressed as

a-b = d and ab = d

hence a – b = ab, where a > b .The dependency between these two variables becomes

a = b/(1-b) ; b = a/(a+l)

It shows that for ‘a’ to be positive, ’b’ must be less than unity but

greater than zero ,that is ‘b’ must be a fraction. ‘a’ may be a whole

number or a fraction ,but ‘b’ will always be a fraction. If ‘b’ is given integral value, ‘a’
becomes negative. Thus there is no solution with whole integral value for both ‘a’and ‘b’.

When ‘a’ takes a whole integral value n , ‘b’ becomes a fraction

n/(n+1) and when ‘a’ takes a fractional value x/y, ‘b’ becomes

x/(x+y).Few typical solutions are given in Table.1.

a is an integer        a is a frac tion

  a      b                     a       b

  1     ½                   1/3      ¼

  2    2/3                 2/5      2/7

  3    ¾                  3/8      3/11

…   …                  …         ….

 n   n/(n+1)          N/n     N/(N+n)

The pair of numbers whose difference d and product are equal can be related to d.
By eliminating one of the dependent variables (either ‘a’ or ‘b’) ,one can obtain
a quadratic equation

(axa) – d a – d = 0 . The positive root of the equation is

a = [d + √ (dxd) + 4d ]/2

which gives,

b = a – d = [ -d + √ (dxd) +4d]/2

The positive values of the pair of numbers under the given condition for a given d
are shown in Table.2.

d         a or b

6       √ 15 ± 3

8        √ 24 ± 4

10      √ 35 ± 5

….        …….

2n      √ n(n+2) ± n

Under the given condition ,the fractional pair of numbers may be with numerator or
the denominator identical. When the numerators are same, the pair of numbers is
assumed as x/b and x/a so that

(x/b) – (x/a) = (x/b).(x/a)

It gives an additional condition x = a-b. Thus the pair becomes [(a-b)/b, (a-b)/a].
The pair of numbers whose denominators are same can be derived directly from
the pair of numbers whose numerators are identical. By multiplying both the numerator
and the denominator of a fractional pair by the denominator of the other same . e.g.,

a   b     a-b/b      a-b/a       a(a-b)/ab     b(a-b)/ab

7   3      4/3        4/7          28/21           12/21

3   5     2/3        2/5          10/15            6/15

7  11    4/7       4/11         44/77           28/77

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