Class: Numerals::Numeral

Inherits:
Object
  • Object
show all
Includes:
ModalSupport::BracketConstructor, ModalSupport::StateEquivalent
Defined in:
lib/numerals/numeral.rb

Overview

A Numeral represents a numeric value as a sequence of digits (possibly repeating) in some numeric base.

A numeral can have a special value (infinity or not-a-number).

A non-special numeral is defined by:

  • radix (the base)

  • digits (a Digits object)

  • sign (+1/-1)

  • point: the position of the fractional point; 0 would place it before the first digit, 1 before the second, etc.

  • repeat: the digits starting at this position repeat indefinitely

A Numeral is equivalent to a Rational number; a quotient of integers can be converted to a Numeral in any base and back to a quotient without altering its value (although the fraction might be simplified).

By default a Numeral represents an exact quantity (rational number). A numeral can also represent an approximate value with a specific precision: the number of significant digits (numeral.digits.size), which can include significant trailing zeros. Approximate numerals are never repeating.

Exact numerals are always repeating, but, when the repeating digits are just zeros, the repeating? method returns false.

Instance Attribute Summary collapse

Class Method Summary collapse

Instance Method Summary collapse

Constructor Details

#initialize(*args) ⇒ Numeral

Special numerals may be constructed with the symbols :nan, :infinity, :negative_infinity, :positive_infinity (or with :infinity and the :sign option which should be either +1 or -1)

Examples:

Numeral[:nan]
Numeral[:infinity, sign: -1]

For non-special numerals, the first argument may be a Digits object or an Array of digits, and the remaining parameters (:base, :sign, :point and :repeat) are passed as options.

Examples:

Numeral[1,2,3, base: 10, point: 1] # 1.23
Numeral[1,2,3,4, point: 1, repeat: 2] # 1.234343434...

The :normalize option can be used to specify the kind of normalization to be applied to the numeral:

  • :exact, the default, produces a normalized :exact number where no trailing zeros are kept and there is always a repeat point (which may just repeat trailing zeros)

  • :approximate produces a non-repeating numeral with a fixed number of digits (where trailing zeros are significant)

  • false or nil will not normalize the result, mantaining the digits and repeat values passed.



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# File 'lib/numerals/numeral.rb', line 83

def initialize(*args)
  if Hash === args.last
    options = args.pop
  else
    options = {}
  end
  options = { normalize: :exact }.merge(options)
  normalize = options.delete(:normalize)
  @point  = nil
  @repeat = nil
  @sign   = nil
  @radix  = options[:base] || options[:radix] || 10
  if args.size == 1 && Symbol === args.first
    @special = args.first
    case @special
    when :positive_infinity
      @special = :inf
      @sign = +1
    when :negative_infinity
      @special = :inf
      @sign = -1
    when :infinity
      @special = :inf
    end
  elsif args.size == 1 && Digits === args.first
    @digits = args.first
    @radix = @digits.radix || @radix
  elsif args.size == 1 && Array === args.first
    @digits = Digits[args.first, base: @radix]
  else
    if args.any? { |v| Symbol === v }
      @digits = Digits[base: @radix]
      args.each do |v|
        case v
        when :point
          @point = @digits.size
        when :repeat
          @repeat = @digits.size
        else # when Integer
          @digits.push v
        end
      end
    elsif args.size > 0
      @digits = Digits[args, base: @radix]
    end
  end
  if options[:value]
    @digits = Digits[value: options[:value], base: @radix]
  end
  @sign    ||= options[:sign] || +1
  @special ||= options[:special]
  unless @special
    @point   ||= options[:point]  || @digits.size
    @repeat  ||= options[:repeat] || @digits.size
  end
  case normalize
  when :exact
    normalize! Numeral.exact_normalization
  when :approximate
    normalize! Numeral.approximate_normalization
  when Hash
    normalize! normalize
  end
end

Instance Attribute Details

#digits ⇒ Object

Returns the value of attribute digits.



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# File 'lib/numerals/numeral.rb', line 148

def digits
  @digits
end

#point ⇒ Object

Returns the value of attribute point.



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# File 'lib/numerals/numeral.rb', line 148

def point
  @point
end

#radix ⇒ Object

Returns the value of attribute radix.



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# File 'lib/numerals/numeral.rb', line 148

def radix
  @radix
end

#repeat ⇒ Object

Returns the value of attribute repeat.



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# File 'lib/numerals/numeral.rb', line 148

def repeat
  @repeat
end

#sign ⇒ Object

Returns the value of attribute sign.



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# File 'lib/numerals/numeral.rb', line 148

def sign
  @sign
end

#special ⇒ Object

Returns the value of attribute special.



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# File 'lib/numerals/numeral.rb', line 148

def special
  @special
end

Class Method Details

.approximate_normalization ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 222

def self.approximate_normalization
  { remove_extra_reps: false, remove_trailing_zeros: false, remove_leading_zeros: true, force_repeat: false }
end

.exact_normalization ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 226

def self.exact_normalization
  { remove_extra_reps: true, remove_trailing_zeros: true, remove_leading_zeros: true, force_repeat: true }
end

.from_coefficient_scale(coefficient, scale, options = {}) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 510

def self.from_coefficient_scale(coefficient, scale, options={})
  radix = options[:base] || options[:radix] || 10
  if coefficient < 0
    sign = -1
    coefficient = -coefficient
  else
    sign = +1
  end
  digits = Digits[base: radix]
  digits.value = coefficient
  point = scale + digits.size
  normalization = options[:normalize] || :exact
  normalization = :approximate if options[:approximate]
  Numeral[digits, base: radix, point: point, sign: sign, normalize: normalization]
end

.from_quotient(*args) ⇒ Object

Create a Numeral from a quotient (Rational number). The quotient can be passed as an Array [numerator, denomnator]; to allow fractions with a zero denominator (representing indefinite or infinite numbers).



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# File 'lib/numerals/numeral.rb', line 401

def self.from_quotient(*args)
  r = args.shift
  if Integer === args.first
    r = [r, args.shift]
  end
  options = args.shift || {}
  raise "Invalid number of arguments" unless args.empty?
  max_d = options.delete(:maximum_number_of_digits) || Numeral.maximum_number_of_digits
  if Rational === r
    x, y = r.numerator, r.denominator
  else
    x, y = r
  end
  return integer(x, options) if (x == 0 && y != 0) || y == 1

  radix = options[:base] || options[:radix] || 10

  xy_sign = x == 0 ? 0 : x < 0 ? -1 : +1
  xy_sign = -xy_sign if y < 0
  x = x.abs
  y = y.abs

  digits = Digits[base: radix]
  repeat = nil
  special = nil

  if y == 0
    if x == 0
      special = :nan
    else
      special = :inf
    end
  end

  return Numeral[special, sign: xy_sign] if special

  point = 1
  k = {}
  i = 0

  while (z = y*radix) < x
    y = z
    point += 1
  end

  while x > 0 && (max_d <= 0 || i < max_d)
    break if repeat = k[x]
    k[x] = i
    d, x = x.divmod(y)
    x *= radix
    digits.push d
    i += 1
  end

  while digits.size > 1 && digits.first == 0
    digits.shift
    repeat -= 1 if repeat
    point -= 1
  end

  Numeral[digits, sign: xy_sign, repeat: repeat, point: point]
end

.indeterminate ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 375

def self.indeterminate
  nan
end

.infinity(sign = +1) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 367

def self.infinity(sign=+1)
  Numeral[:inf, sign: sign]
end

.integer(x, options = {}) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 379

def self.integer(x, options={})
  base = options[:base] || options[:radix] || 10
  if x == 0
    # we also could conventionally keep 0 either as Digits[[], ...]
    digits = Digits[0, base: base]
    sign = +1
  else
    if x < 0
      sign = -1
      x = -x
    else
      sign = +1
    end
    digits = Digits[value: x, base: base]
  end
  Numeral[digits, sign: sign]
end

.maximum_number_of_digits ⇒ Object

Return the maximum number of digits that Numeral objects can handle.



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# File 'lib/numerals/numeral.rb', line 50

def self.maximum_number_of_digits
  @maximum_number_of_digits
end

.maximum_number_of_digits=(n) ⇒ Object

Change the maximum number of digits that Numeral objects can handle.



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# File 'lib/numerals/numeral.rb', line 44

def self.maximum_number_of_digits=(n)
  @maximum_number_of_digits = [n, 2048].max
end

.nan ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 371

def self.nan
  Numeral[:nan]
end

.negative_infinity ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 363

def self.negative_infinity
  Numeral[:inf, sign: -1]
end

.positive_infinity ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 359

def self.positive_infinity
  Numeral[:inf, sign: +1]
end

.zero(options = {}) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 355

def self.zero(options={})
  integer 0, options
end

Instance Method Details

#-@ ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 347

def -@
  negated
end

#approximate(number_of_digits = nil) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 645

def approximate(number_of_digits = nil)
  dup.approximate! number_of_digits
end

#approximate!(number_of_digits = nil) ⇒ Object

Expand to the specified number of digits, then truncate and remove repetitions. If no number of digits is given, then it will be converted to approximate numeral only if it is not repeating.



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# File 'lib/numerals/numeral.rb', line 632

def approximate!(number_of_digits = nil)
  if number_of_digits.nil?
    if exact? && !repeating?
      @repeat = nil
    end
  else
    expand! number_of_digits
    @digits.truncate! number_of_digits
    @repeat = nil
  end
  self
end

#approximate? ⇒ Boolean

An approximate Numeral has limited precision (number of significant digits). In an approximate Numeral, trailing zeros are significant.

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 605

def approximate?
  !exact?
end

#base ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 150

def base
  @radix
end

#base=(b) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 154

def base=(b)
  @radix = b
end

#digit_value_at(i) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 207

def digit_value_at(i)
  if i < 0
    0
  elsif i < @digits.size
    @digits[i]
  elsif @repeat.nil? || @repeat >= @digits.size
    0
  else
    repeated_length = @digits.size - @repeat
    i = (i - @repeat) % repeated_length
    i += @repeat
    i < 0 ? 0 : @digits[i]
  end
end

#dup ⇒ Object

Deep copy



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# File 'lib/numerals/numeral.rb', line 332

def dup
  duped = super
  duped.digits = duped.digits.dup
  duped
end

#exact ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 653

def exact
  dup.exact!
end

#exact! ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 649

def exact!
  normalize! Numeral.exact_normalization
end

#exact? ⇒ Boolean

An exact Numeral represents exactly a rational number. It always has a repeat position, although the repeated digits may all be zero.

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 599

def exact?
  !!@repeat
end

#expand(minimum_number_of_digits) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 623

def expand(minimum_number_of_digits)
  dup.expand! minimum_number_of_digits
end

#expand!(minimum_number_of_digits) ⇒ Object

Make sure the numeral has at least the given number of digits; This may denormalize the number.



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# File 'lib/numerals/numeral.rb', line 611

def expand!(minimum_number_of_digits)
  if @repeat
    while @digits.size < minimum_number_of_digits
      @digits.push @digits[@repeat] || 0
      @repeat += 1
    end
  else
    @digits.push 0 while @digits.size < minimum_number_of_digits
  end
  self
end

#indeterminate? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 170

def indeterminate?
  nan?
end

#infinite? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 174

def infinite?
  @special == :inf
end

#inspect ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 592

def inspect
  to_s
end

#nan? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 166

def nan?
  @special == :nan
end

#negate! ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 338

def negate!
  @sign = -@sign
  self
end

#negated ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 343

def negated
  dup.negate!
end

#negative_infinite? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 182

def negative_infinite?
  @special == :inf && @sign == -1
end

#nonrepeating? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 199

def nonrepeating?
  !special && !repeating?
end

#normalize!(options = {}) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 230

def normalize!(options = {})
  if @special
    if @special == :nan
      @sign = nil
    end
    @point = @repeat = nil
  else

    defaults = { remove_extra_reps: true, remove_trailing_zeros: true }
    options = defaults.merge(options)
    remove_trailing_zeros = options[:remove_trailing_zeros]
    remove_extra_reps = options[:remove_extra_reps]
    remove_leading_zeros = options[:remove_extra_reps]
    force_repeat = options[:force_repeat]

    # Remove unneeded repetitions
    if @repeat && remove_extra_reps
      rep_length = @digits.size - @repeat
      if rep_length > 0 && @digits.size >= 2*rep_length
        while @repeat > rep_length && @digits[@repeat, rep_length] == @digits[@repeat-rep_length, rep_length]
          @repeat -= rep_length
          @digits.replace @digits[0...-rep_length]
        end
      end
      # remove unneeded partial repetitions
      if rep_length > 0 && @digits.size > rep_length
        removed = 0
        while @repeat > 0 && @digits[@repeat-1] == @digits[@repeat-1+rep_length]
          @repeat -= 1
          removed += 1
        end
        @digits.replace @digits[0...-removed] if removed > 0
      end

    end

    # Replace 'nines' repetition 0.999... -> 1
    if @repeat && @repeat == @digits.size-1 && @digits[@repeat] == (@radix-1)
      @digits.pop
      @repeat = nil

      i = @digits.size - 1
      carry = 1
      while carry > 0 && i >= 0
        @digits[i] += carry
        carry = 0
        if @digits[i] > @radix
          carry = 1
          @digits[i] = 0
          @digits.pop if i == @digits.size
        end
        i -= 1
      end
      if carry > 0
        digits.unshift carry
        @point += 1
      end
    end

    # Remove zeros repetitions
    if remove_trailing_zeros
      if @repeat && @repeat >= @digits.size
        @repeat = @digits.size
      end
      if @repeat && @repeat >= 0
        unless @digits[@repeat..-1].any? { |x| x != 0 }
          @digits.replace @digits[0...@repeat]
          @repeat = nil
        end
      end
    end

    if force_repeat
      @repeat ||= @digits.size
    else
      @repeat = nil if @repeat && @repeat >= @digits.size
    end

    # Remove leading zeros
    if remove_leading_zeros
      # if all digits are zero, we consider all to be trailing zeros
      unless !remove_trailing_zeros && @digits.zero?
        while @digits.first == 0
          @digits.shift
          @repeat -= 1 if @repeat
          @point -= 1
        end
      end
    end

    # Remove trailing zeros
    if remove_trailing_zeros && !repeating?
      while @digits.last == 0
        @digits.pop
        @repeat -= 1 if @repeat
      end
    end
  end
  self
end

#normalized(options = {}) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 351

def normalized(options={})
  dup.normalize! options
end

#parameters ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 550

def parameters
  if special?
    params = { special: @special }
    params.merge! sign: @sign if @special == :inf
  else
    params = {
      digits: @digits,
      sign:   @sign,
      point:  @point
    }
    params.merge! repeat: @repeat if @repeat
    if approximate?
      params.merge! normalize: :approximate
    end
  end
  params
end

#positive_infinite? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 178

def positive_infinite?
  @special == :inf && @sign == +1
end

#repeating? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 195

def repeating?
  !special? && @repeat && @repeat < @digits.size
end

#repeating_position ⇒ Object

unlike the repeat attribute, this is nevel nil



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# File 'lib/numerals/numeral.rb', line 191

def repeating_position
  @repeat || @digits.size
end

#scale ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 158

def scale
  @point - @digits.size
end

#scale=(s) ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 203

def scale=(s)
  @point = s + @digits.size
end

#special? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 162

def special?
  !!@special
end

#split ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 526

def split
  if @special || (@repeat && @repeat < @digits.size)
    raise NumeralError, "Numeral cannot be represented as sign, coefficient, scale"
  end
  [@sign, @digits.value, scale]
end

#to_base(other_base) ⇒ Object

Convert a Numeral to a different base



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# File 'lib/numerals/numeral.rb', line 541

def to_base(other_base)
  if other_base == @radix
    dup
  else
    normalization = exact? ? :exact : :approximate
    Numeral.from_quotient to_quotient, base: other_base, normalize: normalization
  end
end

#to_quotient ⇒ Object

Return a quotient (Rational) that represents the exact value of the numeral. The quotient is returned as an Array, so that fractions with a zero denominator can be handled (representing indefinite or infinite numbers).



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# File 'lib/numerals/numeral.rb', line 467

def to_quotient
  if @special
    y = 0
    case @special
    when :nan
      x = 0
    when :inf
      x = @sign
    end
    return [x, y]
  end

  n = @digits.size
  a = 0
  b = a

  repeat = @repeat
  repeat = nil if repeat && repeat >= n

  for i in 0...n
    a *= @radix
    a += @digits[i]
    if repeat && i < repeat
      b *= @radix
      b += @digits[i]
    end
  end

  x = a
  x -= b if repeat

  y = @radix**(n - @point)
  y -= @radix**(repeat - @point) if repeat

  d = Numerals.gcd(x, y)
  x /= d
  y /= d

  x = -x if @sign < 0

  [x.to_i, y.to_i]
end

#to_s ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 568

def to_s
  case @special
  when :nan
    'Numeral[:nan]'
  when :inf
    if @sign < 0
      'Numeral[:inf, sign: -1]'
    else
      'Numeral[:inf]'
    end
  else
    args = ''
    if @digits.size > 0
      args = @digits.digits_array.to_s.unwrap('[]')
      args << ', '
    end
    params = parameters
    params.delete :digits
    params.merge! base: @radix
    args << params.to_s.unwrap('{}')
    "Numeral[#{args}]"
  end
end

#to_value_scale ⇒ Object 



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# File 'lib/numerals/numeral.rb', line 533

def to_value_scale
  if @special || (@repeat && @repeat < @digits.size)
    raise NumeralError, "Numeral cannot be represented as value, scale"
  end
  [@digits.value*@sign, scale]
end

#zero? ⇒ Boolean

Returns:

  • (Boolean) 


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# File 'lib/numerals/numeral.rb', line 186

def zero?
  !special? && @digits.zero?
end