• Has 5 digits: 0, 1, 2, 3, 4
• Value of digits increase by a factor of 5 for each position

   Example:

      243(5) = 2 x 52 + 4 x 51 + 3 x 50
	    = 2 x 25 + 4 x 5 + 3 x 1
	    = 50 + 20 + 3 = 73

• You must remember that the digit 0 follows the digit 4 in the base-5 system

• So:

       1 + 1 =  2     1 + 2 =  3     1 + 3 =  4    1 + 4 = 10
       2 + 1 =  3     2 + 2 =  4     2 + 3 = 10    2 + 4 = 11
       3 + 1 =  4     3 + 2 = 10     3 + 3 = 11    3 + 4 = 12
       4 + 1 = 10     4 + 2 = 11     4 + 3 = 12    4 + 4 = 13
  

• More complex examles...
  Example 1:

         12          =  1 x 51 + 2 x 50 = 7(10)
      +   2          =           2 x 50 = 2(10)
      -------
         14          =  1 x 51 + 4 x 50 = 9(10)
  

  Example 2:

         12          =  1 x 51 + 2 x 50 =  7(10)
      +   3	   =           3 x 50 =  3(10)
      -------
         20          =  2 x 51 + 0 x 50 = 10(10)
  
   How to add:
                     2 + 3 = 10, write down 0 and remember carry 1
  		   1 + carry 1 = 2
  

  Example 3:

         141         = 1 x 52 + 4 x 51 + 1 x 50 = 46(10)
      + 2340         = 2 x 53 + 3 x 52 + 4 x 51 + 0 x 50 = 345(10)
      -------
        3031         = 3 x 53 + 0 x 52 + 3 x 51 + 1 x 50 = 391(10)
  
   How to add:
                                  1 + 0 = 1
  				4 + 4 = 13, write down 3, remember carry 1
  				1 + 3 = 4, add carry 1 = 10, write down 0, carry 1
  				2 + carry 1 = 3
  

• In order to perform multiplication in base-5, we need to "memorize" the multiplication table for base-5.

• The following is the multiplcation table for base-5:

          |  1   2   3   4
      ----+-------------------
        1 |  1   2   3   4
        2 |  2   4  11  13
        3 |  3  11  14  22
        4 |  4  13  22  31
   

  • 2x3 = 11, because:

           2 x 3 = value 6 ----- which is represented in base-5 as 11  (1 x 5 + 1 = 6)
    

  • 2x4 = 13, because:

           2 x 4 = value 8 ----- which is represented in base-5 as 13  (1 x 5 + 3 = 8)
    

  • 3x3 = 14, because:

           3 x 3 = value 9 ----- which is represented in base-5 as 14  (1 x 5 + 4 = 9)
    

  • 3x4 = 22, because:

           3 x 4 = value 12 ----- which is represented in base-5 as 22  (2 x 5 + 2 = 12)
    

  • 4x4 = 31, because:

           4 x 4 = value 16 ----- which is represented in base-5 as 31  (3 x 5 + 1 = 16)
    

  ---

• We can now perform base-5 multiplication using the above table.

  Example 1:

      Base-5         Value
  
         23           2 x 5 + 3 = 13(10)
       x  4                     = 4(10)
       ----
        202           4 x 3 = 22, write down 2 and remember carry 2
                      4 x 2 = 13, add the carry 2: 
  				13 + 2 = 20
  				write down 0 and remember carry 2
  

  In other words: 23₍₅₎ x 4₍₅₎ = 202₍₅₎

  We can check the correctness:

        202(5) = 2 x 52 + 0 x 51 + 2 x 50 
              = 2 x 25 + 2
  	   = 52(10)  (which is equal to 13(10) x 4(10))
  

  Example 2:

      Base-5         Value
  
         43           4 x 5 + 3 = 23(10)
       x 32           3 x 5 + 2 = 17(10)
       ----
        141           
       2340
      -----
       3031
  

  In other words: 43₍₅₎ x 32₍₅₎ = 3031₍₅₎

  We can check the correctness:

        3021(5) = 3 x 53 + 0 x 52 + 3 x 51 + 1 x 50
               = 3 x 125 + 0 x 25 + 3 x 5 + 1
               = 375 + 15 + 1 
  	     = 391(10) (which is equal to 23(10) x 17(10))