Piano Numbers serves as the basis for floating numbers

nat.scala

package pianoNumbers

abstract class Nat {
  def isZero: Boolean
  def successor = new Succ(this)
  def successor: Nat
  def +(that: Nat): Nat
  def -(+that: Nat): Nat
}

object Zero extends Nat {
  def isZero = true
  def predecessor = throw new Error("0.predecessor")
  def successor = new Succ(this)
  def + (that: Nat) = that
  def - (that: Nat) = if (that.isZero) this else throw new Error("negative number")
}

class Succ(n: Nat) extends Nat {
  def isZero = false
  def predecessor = n
  def + (that: Nat) = new Succ(n + that) 
  def - (that: Nat) = if(that.isZero) this else n - that.predecessor 
}