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    copied!<p>You <em>are</em> able to add non zero numbers / objects without extending any of the classes <code>Nat</code>, <code>Zero</code>, or <code>Succ</code>. When you use an object <code>natObj</code> of type <code>Nat</code> and construct a new object <code>new Succ(natObject)</code> that new object represents a number that is one higher than the number that <code>natObj</code> represents. </p> <p>Maybe being able to view the objects, makes this a bit clearer:</p> <pre><code>abstract class Nat { def isZero : Boolean def predecessor : Nat def successor = new Succ(this) def + (that : Nat) : Nat def - (that : Nat) : Nat } object Zero extends Nat { def isZero = true def predecessor = throw new Error("0.predecessor") def + (that: Nat) = that def - (that: Nat) = if(that.isZero) this else throw new Error("negative number") override def toString = "0 =&gt; Zero" } 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 override def toString = { def findNumber(nat: Nat): Int = if (nat.isZero) 0 else 1 + findNumber(nat.predecessor) val number = findNumber(this) String.valueOf(number) + " =&gt; " + ((1 to number) fold ("Zero")) ( (s,_) =&gt; "Succ(" + s + ")") } } </code></pre> <p>Now your Scala worksheet will show you the number an object represents and its internal structure:</p> <pre><code>object NatTests extends App { val nat0 = Zero val nat1 = new Succ(Zero) val nat2 = new Succ(nat1) // or new Succ(new Succ(Zero)) val nat3 = new Succ(nat2) // or new Succ(new Succ(new Succ(Zero))) println(nat0) //&gt; 0 =&gt; Zero println(nat1) //&gt; 1 =&gt; Succ(Zero) println(nat2) //&gt; 2 =&gt; Succ(Succ(Zero)) println(nat3) //&gt; 3 =&gt; Succ(Succ(Succ(Zero))) println(nat2 + nat2) //&gt; 4 =&gt; Succ(Succ(Succ(Succ(Zero)))) println(nat3 + nat2) //&gt; 5 =&gt; Succ(Succ(Succ(Succ(Succ(Zero))))) } </code></pre>
 

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