Scala で分数を扱う ref: http://qiita.com/akaneko3/items/e23f2b2f7bd8f4940cec

Main.scala

package jp.mydns.akanekodou.scala

import math.Rational

object Main extends App {
  // ここから
  implicit val RationalIsFractional = new Fractional[Rational] {
    // compare の実装
    def compare(x : Rational, y : Rational) : Int =
      (x.numerator * y.denominator) compare (x.denominator * y.numerator)

    // 四則演算
    def plus(x : Rational, y : Rational) : Rational =
      Rational(x.numerator * y.denominator + x.denominator * y.numerator ,x.denominator * y.denominator)
    def minus(x : Rational, y : Rational) : Rational =
      Rational(x.numerator * y.denominator - x.denominator * y.numerator ,x.denominator * y.denominator)
    def times(x : Rational, y : Rational) : Rational =
      Rational(x.numerator * y.numerator, x.denominator * y.denominator)
    def div(x : Rational, y : Rational) : Rational =
      Rational(x.numerator * y.denominator, x.denominator * y.numerator)

    // 正負反転
    def negate(x : Rational) : Rational = Rational(- x.numerator, x.denominator)

    // 型変換
    def fromInt(x : Int) : Rational = Rational(x)
    def toInt(x : Rational) : Int = (x.numerator / x.denominator).toInt
    def toLong(x : Rational) : Long = (x.numerator / x.denominator).toLong
    def toFloat(x : Rational) : Float =
      x.numerator.toFloat / x.denominator.toFloat
    def toDouble(x : Rational) : Double =
      x.numerator.toDouble / x.denominator.toDouble
  }

  import scala.math.Fractional.Implicits._
  import scala.math.Ordering.Implicits._
  // ここまでを Main の外に出したい !

  val a = Rational(2, 12)
  val b = a.copy(n = 3)
  val c = a + b
  println(a, b, a + b, a == b, a < b)
}

NewRational.scala

case class NewRational(n : BigInt, d : BigInt = 1) extends Fractional[NewRational] {
  require(d != 0) // 分母は 0 でないことを要求する
  private val g : BigInt = n gcd d
  val numerator : BigInt = n * d.signum / g // 分子
  val denominator : BigInt = d * d.signum / g // 分母(分母は必ず正)

  def compare(x : NewRational, y : NewRational) : Int =
    (x.numerator * y.denominator) compare (x.denominator * y.numerator)

  def plus(x : NewRational, y : NewRational) : NewRational =
    NewRational(x.numerator * y.denominator + x.denominator * y.numerator ,x.denominator * y.denominator)
  def minus(x : NewRational, y : NewRational) : NewRational =
    NewRational(x.numerator * y.denominator - x.denominator * y.numerator ,x.denominator * y.denominator)
  def times(x : NewRational, y : NewRational) : NewRational =
    NewRational(x.numerator * y.numerator, x.denominator * y.denominator)
  def div(x : NewRational, y : NewRational) : NewRational =
    NewRational(x.numerator * y.denominator, x.denominator * y.numerator)
  def negate(x : NewRational) : NewRational = NewRational(- x.numerator, x.denominator)

  def fromInt(x : Int) : NewRational = NewRational(x)
  def toInt(x : NewRational) : Int = (x.numerator / x.denominator).toInt
  def toLong(x : NewRational) : Long = (x.numerator / x.denominator).toLong
  def toFloat(x : NewRational) : Float =
    x.numerator.toFloat / x.denominator.toFloat
  def toDouble(x : NewRational) : Double =
    x.numerator.toDouble / x.denominator.toDouble
}

Rational.scala

package jp.mydns.akanekodou.scala.math

// 分数クラス(分母はデフォルトで 1)
case class Rational(n : BigInt, d : BigInt = 1) {
  require(d != 0) // 分母は 0 でないことを要求する
  private val g : BigInt = n gcd d
  val numerator : BigInt = n * d.signum / g
  val denominator : BigInt = d * d.signum / g

  // toString の override
  override def toString() : String =
    s"${numerator}" + (if (denominator != 1) s"/${denominator}" else "")
}