class Rational

This file adds generate_unit_fractions method to the Rational class. It generates a sum of unit fractions that the denominators are all different. For example,

3/4 = 1/2 + 1/6 + 1/12

The sum can have two or more patterns.

3/4 = 1/2 + 1/6 + 1/12 = 1/2 + 1/4

But the method generates only one pattern because it uses the following algorithm.

a/b is given.
find the solution (x, y) of the equation ax + by = 1
a/b = (-y)/x + 1/bx, and x < b

The second term of the right hand side is a unit fraction. The denominator of the first term is less than b So, if you do the same to (-y)/x, the denominator of the first term goes smaller and smaller. Finally, you can get the sum of unit fractions.

Public Instance Methods

generate_unit_fractions() click to toggle source

Returns an array of unit fractions. If self is not a proper fraction, the first element of the array is an integer.

# File lib/math_programs/unit_fractions.rb, line 30
def generate_unit_fractions
  i = self.floor
  a = (self-i).numerator
  b = (self-i).denominator
  s = []
  while a > 1 do
    axby1 = Axby1.new(a, b)
    x, y = axby1.solve
    return nil unless x
    s << Rational(1,b*x)
    a, b = -y, x
  end
  s << Rational(a,b)
  s << i unless i == 0
  return s.reverse
end