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REPRESENTATIONS @tomstuart, RubyConf AU 2019 https://codon.com/representations-count COUNT

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This is a fairy tale.

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$ irb >> 5 * 2 => 10 >> 5 - 2 => 3 >> 2 - 5 ImpossibleNumberError (0 is the smallest number)

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class SignedNumber def self.positive(number) end def self.negative(number) end def ==(other) end def +(other) end def -(other) end def *(other) end def <(other) end end

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$ irb >> require 'signed_number' => true >> def pos(number) SignedNumber.positive(number) end => :pos >> def neg(number) SignedNumber.negative(number) end => :neg

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>> pos(2) + pos(5) == pos(7) => true >> pos(2) - pos(5) == neg(3) => true >> (pos(9) - pos(3)) * (neg(4) + neg(3)) == neg(42) => true >> pos(2) * neg(3) < pos(4) + neg(9) => true

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Take a sec to think.

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class SignedNumber def self.positive(number) end def self.negative(number) end def ==(other) end def +(other) end def -(other) end def *(other) end def <(other) end end

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3 -

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3 sign (+ or –) size -

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class SignedNumber def self.positive(number) end def self.negative(number) end # … end protected attr_reader :sign, :size def initialize(sign, size) @sign, @size = sign, size end new(:negative, number) new(:positive, number)

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==

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sign == other.sign && size == other.size class SignedNumber # … def ==(other) end # … end

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>> pos(1) == pos(1) => true >> neg(2) == neg(2) => true >> pos(1) == pos(2) => false >> pos(1) == neg(1) => false ✔ ✔ ✔ ✔

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+

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SignedNumber.new(sign, size + other.size) end # … end class SignedNumber # … def +(other)

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>> pos(3) + pos(1) == pos(4) => true >> neg(3) + neg(1) == neg(4) => true >> pos(3) + neg(1) == pos(2) => false >> pos(3) + neg(1) => # ✔ ✔ ✘

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else SignedNumber.new(sign, size - other.size) class SignedNumber # … def +(other) if sign == other.sign class SignedNumber # … def +(other) SignedNumber.new(sign, size + other.size) end # … end end

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>> pos(3) + neg(1) == pos(2) => true >> neg(3) + pos(1) == neg(2) => true >> pos(1) + neg(3) == neg(2) ImpossibleNumberError (0 is the smallest number) ✔ ✔

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else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size ) end end end # … end class SignedNumber # … def +(other) if sign == other.sign SignedNumber.new(sign, size + other.size) else SignedNumber.new(sign, size - other.size) if size >= other.size

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>> pos(1) + neg(3) == neg(2) => true >> neg(1) + pos(3) == pos(2) => true >> pos(3) + neg(3) == neg(3) + pos(3) => false >> pos(3) + neg(3) => # >> neg(3) + pos(3) => # >> pos(0) == neg(0) => false ✔ ✘ ✘ ✔

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class SignedNumber # … def ==(other) end # … end ( || size.zero?) && size == other.size sign == other.sign

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>> pos(3) + neg(3) == neg(3) + pos(3) => true ✔

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class SignedNumber # … def ==(other) (sign == other.sign || size.zero?) && size == other.size end def +(other) if sign == other.sign SignedNumber.new(sign, size + other.size) else if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size )a end end end def -(other) if sign == other.sign if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size )b end else SignedNumber.new(sign, size + other.size) end end def *(other) if sign == other.sign SignedNumber.new(:positive, size * other.size) else SignedNumber.new(:negative, size * other.size) end end def <(other) if sign == other.sign if sign == :positive size < other.size else other.size < size end else sign == :negative && !(size.zero? && other.size.zero?) end end # … end

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class SignedNumber # … def ==(other) (sign == other.sign || size.zero?) && size == other.size end def +(other) if sign == other.sign SignedNumber.new(sign, size + other.size) else if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size )a end end end def -(other)

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)a end end end def -(other) if sign == other.sign if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size )b end else SignedNumber.new(sign, size + other.size) end end def *(other) if sign == other.sign SignedNumber.new(:positive, size * other.size)

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other.size - size )b end else SignedNumber.new(sign, size + other.size) end end def *(other) if sign == other.sign SignedNumber.new(:positive, size * other.size) else SignedNumber.new(:negative, size * other.size) end end def <(other) if sign == other.sign if sign == :positive size < other.size else other.size < size

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SignedNumber.new(:positive, size * other.size) else SignedNumber.new(:negative, size * other.size) end end def <(other) if sign == other.sign if sign == :positive size < other.size else other.size < size end else sign == :negative && !(size.zero? && other.size.zero?) end end # … end

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class SignedNumber # … def ==(other) (sign == other.sign || size.zero?) && size == other.size end def +(other) if sign == other.sign SignedNumber.new(sign, size + other.size) else if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size )a end end end def -(other) if sign == other.sign if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size )b end else SignedNumber.new(sign, size + other.size) end end def *(other) if sign == other.sign SignedNumber.new(:positive, size * other.size) else SignedNumber.new(:negative, size * other.size) end end def <(other) if sign == other.sign if sign == :positive size < other.size else other.size < size end else sign == :negative && !(size.zero? && other.size.zero?) end end # … end

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class SignedNumber # … def ==(other) (sign == other.sign || size.zero?) && size == other.size end def +(other) if sign == other.sign SignedNumber.new(sign, size + other.size) else if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size ) end end end def -(other) if sign == other.sign if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size ) end else SignedNumber.new(sign, size + other.size) end end def *(other) if sign == other.sign SignedNumber.new(:positive, size * other.size) else SignedNumber.new(:negative, size * other.size) end end def <(other) if sign == other.sign if sign == :positive size < other.size else other.size < size end else sign == :negative && !(size.zero? && other.size.zero?) end end # … end

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This was too hard.

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Can we do better?

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class SignedNumber def self.positive(number) end def self.negative(number) end def ==(other) end def +(other) end def -(other) end def *(other) end def <(other) end end

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3 -

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$3 -

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Your balance: -$3 Your balance: $7 deposit $10

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Your balance: -$3 Your balance: $97 deposit $100

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Your balance: $3 Your balance: $13 deposit $10

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+2 +2 +3

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-2 -2 -3

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This has useful properties.

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Regularity.

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-2 :negative, 2 :negative, 1 :negative, 0 :positive, 0 :positive, 1 :positive, 2 -1 0 +1 +2

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-2 -1 0 +1 +2 … … … … …

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Visual intuition.

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+2

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-2

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+3 +2 +

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+5

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Compositionality.

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+3 +2 -

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+1

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protected attr_reader :left, :right def initialize(left, right) @left, @right = left, right end end class SignedNumber def self.positive(number) end def self.negative(number) end # … new(number, 0) new(0, number)

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==

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+2 +3 = ?

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+3 +2 = ?

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= ? +2 +2

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+2 = +2 +2 = +3 +3 = +2

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+2 = +2 +2 = +3 +3 = +2 left right right left + +

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left + other.right == other.left + right end # … end class SignedNumber # … def ==(other)

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✔

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<

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+2 < +2 +2 < +3 +3 < +2

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+2 < +2 +2 < +3 +3 < +2 left right right left + +

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other.left + right < left + other.right end # … end class SignedNumber # … def <(other)

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✔

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+

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+3 +2 +

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left right left right + +

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SignedNumber.new(left + other.left, right + other.right) end # … end class SignedNumber # … def +(other)

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✔

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-

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+2 +3 -

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left right right left + +

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SignedNumber.new(left + other.right, other.left + right) end # … end class SignedNumber # … def -(other)

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✔

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*

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× +4 +5

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=

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- =

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- = +

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- + = -

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- + = - 48 16 6 18 20

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= 54 34 20 -

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- =

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× = +4 +5 +20

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SignedNumber.new( left * other.right + other.left * right, left * other.left + right * other.right ) end # … end class SignedNumber # … def *(other)

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✔

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class SignedNumber # … def ==(other) left + other.right == other.left + right end def +(other) SignedNumber.new(left + other.left, right + other.right) end def -(other) SignedNumber.new(left + other.right, other.left + right) end def *(other) SignedNumber.new( left * other.right + other.left * right, left * other.left + right * other.right ) end def <(other) other.left + right < left + other.right end # … end

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class SignedNumber # … def ==(other) left + other.right == other.left + right end def +(other) SignedNumber.new(left + other.left, right + other.right) end def -(other) SignedNumber.new(left + other.right, other.left + right) end def *(other) SignedNumber.new( left * other.right + other.left * right, left * other.left + right * other.right ) end def <(other) other.left + right < left + other.right end # … end

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class SignedNumber # … def ==(other) left + other.right == other.left + right end def +(other) SignedNumber.new(left + other.left, right + other.right) end def -(other) SignedNumber.new(left + other.right, other.left + right) end def *(other) SignedNumber.new( left * other.right + other.left * right, left * other.left + right * other.right ) end def <(other) other.left + right < left + other.right end # … end class SignedNumber # … def ==(other) (sign == other.sign || size.zero?) && size == other.size end def +(other) if sign == other.sign SignedNumber.new(sign, size + other.size) else if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size ) end end end def -(other) if sign == other.sign if size >= other.size SignedNumber.new(sign, size - other.size) else SignedNumber.new( sign == :positive ? :negative : :positive, other.size - size ) end else SignedNumber.new(sign, size + other.size) end end def *(other) if sign == other.sign SignedNumber.new(:positive, size * other.size) else SignedNumber.new(:negative, size * other.size) end end def <(other) if sign == other.sign if sign == :positive size < other.size else other.size < size end else sign == :negative && !(size.zero? && other.size.zero?) end end # … end

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This is a fairy tale.

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Regularity. Visual intuition. Compositionality.

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Representations count, so choose wisely.

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tomstuart/negative-numbers

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@tomstuart https://codon.com/representations-count THANKS!