如何将有理数格式化为小数？

给定一个任意大（或小）Rational具有有限十进制表示形式的数字，例如：

r = Rational(1, 2**15)
#=> (1/32768)

如何以字符串形式获取其完整的十进制值？

上述数字的预期输出是：

"0.000030517578125"

to_f显然不起作用：

r.to_f
#=> 3.0517578125e-05

And sprintf要求我指定位数：

sprintf('%.30f', r)
#=> "0.000030517578125000000000000000"

Most ten-year-olds know how to do that: use long division!¹

Code

def finite_long_division(n,d)
  return nil if d.zero?
  sign = n*d >= 0 ? '' : '-'
  n, d = n.abs, d.abs
  pwr =
  case n <=> d
  when 1 then power(n,d)
  when 0 then 0
  else        -power(d,n)-1
  end            
  n *= 10**(-pwr) if pwr < 0
  d *= 10**(pwr)  if pwr >= 0
  s = ld(n,d)
  t = s.size == 1 ? '0' : s[1..-1]
  "%s%s.%s x 10^%d" % [sign, s[0], t, pwr]
end

def power(n, d)
  # n > d
  ns = n.to_s
  ds = d.to_s
  pwr = ns.size - ds.size - 1
  pwr += 1 if ns[0, ds.size].to_i >= ds.to_i
  pwr
end

def ld(n,d)
  s = ''
  loop do # .with_object('') do |s|
    m,n = n.divmod(d)
    s << m.to_s
    return s if n.zero?
    n *= 10
  end
end

Examples²

finite_long_division(1, 2**15)
  #=> "3.0517578125 x 10^-5"
finite_long_division(-1, 2**15)
  #=> "-3.0517578125 x 10^-5"
finite_long_division(-1, -2**15)
  #=> "3.0517578125 x 10^-5"

finite_long_division(143, 16777216)
  #=> "8.523464202880859375 x 10^-6"
143/16777216.0
  #=> 8.52346420288086e-06 

finite_long_division(8671,
  803469022129495137770981046170581301261101496891396417650688)
  #=> "1.079195309486679194852923588206549145803161531099624\
  #      804222395643336829571798416196370119711226461255452\
  #      67714596064934085006825625896453857421875 x 10^-56"      

Recall that every rational number has either a decimal representation or contains an infinitely-repeating sequence of digits (e.g., 1/3 #=> 0.33333..., 3227/555 #=> 5.8144144144... and 1/9967 #=> 0.00010033109260559848...³). This method would therefore never terminate if the rational was of the repeating sequence variety. Since one generally doesn't know in advance which type a rational number is, it might be useful to modify the method to first determine if the rational number has a finite decimal representation. It is known that a rational number n/d which cannot be reduced (by removing common factors) has this property if and only if d is divisible by 2 or by 5 and is not divisible by any other prime number.⁴ We could easily construct a method to determine if an already-reduced rational number has that property.

require 'prime'

def decimal_representation?(n, d)
  primes = Prime.prime_division(d).map(&:first)
  (primes & [2,5]).any? && (primes - [2, 5]).empty?
end

^{1 At least that was true when I was a kid.}

^{2 See here http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/fractions.html#section3.1.1 for a partial list of rational numbers that have finite decimal representations}.

^{3 This rational number's repeating sequence contains 9,966 digits.}

^{4 Reference http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fractions/fractions.html#section3.1.1}.