Added simple explanation of recursion

Author: Alienpruts

chap-05/recursion-explained.php

@@ -0,0 +1,70 @@
+<?php
+/**
+ * Created by PhpStorm.
+ * User: bert
+ * Date: 4/10/18
+ * Time: 14:14
+ */
+
+/**
+ * Recursion is a way to solve larger problems by dividing them into smaller problems. Recursive functions MUST call
+ * themselves to break down the problems. For example :
+ *
+ * Mathematical factorials :
+ *
+ *  5! = 5 * 4 * 3 * 2 * 1
+ *  4! = 4 * 3 * 2 * 1
+ *  3! = 3 * 2 * 1
+ *  2! = 2 * 1
+ *  1! = 1
+ *
+ * The above can be rewritten :
+ *
+ *  5! = 5 * 4!
+ *  4! = 4 * 3!
+ *  3! = 3 * 2!
+ *  2! = 2 * 1!
+ *  1! = 5 * 0!
+ *  0! = 1
+ *
+ * Mathematically this boils down do :
+ *
+ *  n! = n * (n-1)!
+ *
+ * Functionally it goes like this for 3! = 3 * 2! :
+ *
+ *  Step 1 : 3! = 3 * 2!
+ *  Step 2 : 2! = 2 * 1!
+ *  Step 3 : 1! = 1 * 0!
+ *  Step 4 : 0! = 1             <== This is the actual end of recursion, where we start to get results in.
+ *  Step 5 : 1! = 1 * 1 = 1
+ *  Step 6 : 2! = 2 * 1 = 2
+ *  Step 7 : 3! = 3 * 2 = 6
+ *
+ * Recursive functions have these three properties which should be met :
+ *
+ *  1) Should be called on a smaller subproblem of the larger problem
+ *  2) Should have a base case which ends the recursion.
+ *  3) Should not be any cycle : should not make a recursive call to the same problem (endless loop)
+ *
+ */
+
+
+$n = 10;
+
+echo "The factor of {$n} is : " .factorial($n) . PHP_EOL;
+
+echo "Be sure to check the docblock of this file for more explanation!" . PHP_EOL;
+
+function factorial(int $n) : int
+{
+
+    // This is the base case which ends recursion.
+    if ($n == 0){
+        return 1;
+    }
+
+    // This is the recursive call. Notice how we lower n to eventually reach the base case.
+    return $n * factorial($n - 1);
+}
+