Important property of all positional representation systems
 

Property of any positional number system:

  • The arithmetic opertions (+, - , x, /) are independent from the positional number system used to represent the values
 

I will illustrate this fact by performing some addition and multiplication in the base 5 number system

The Base 5 number system

The Base 5 positional number system:

  • Has 5 digits: 0, 1, 2, 3, 4

  • The value of digits are weighted by the factor 5pos if the digit is in the position pos

Example of a Base-5 number: 

   What decimal value is represented by:  243(5) ?     

   Answer:

      243(5) = 2 x 52 + 4 x 51 + 3 x 50
	     =    50  +   20   +    3  =  73(10)

Addition table of the Base-5 number system
 

Addition table for the Base-5 number system: 

      + |  1 |  2 |  3 |  4 
    ----+----+----+----+----       
      1 |  2 |  3 |  4 | 10 
    ----+----+----+----+----
      2 |  3 |  4 | 10 | 11 
    ----+----+----+----+----
      3 |  4 | 10 | 11 | 12 
    ----+----+----+----+----
      4 | 10 | 11 | 12 | 13

Adding Base-5 numbers works exactly like adding decimal numbers

The only difference is:

The carry happens when the total (sum) ≥ 5

Base-5 addition - Example
 

Example: 

      Base-5         Decimal equivalent      

        324                  89
      + 243                + 73
     -------              ------
       1122                 162
 

(I will do the addition step-by-step in class)

Multiplication table of the Base-5 number system
 

Multiplication table for the Base-5 number system: 

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

Multiplying Base-5 numbers works exactly like adding decimal numbers

The only difference is:

The carry happens when the total (sum) ≥ 5

Base-5 multiplication - Example

Example: 

      Base-5         Decimal equivalent      

        324                  89
      x 243                x 73
     -------              ------
       2032                 267
      24110                6230
     120300              -------
   --------                6497
     201442

   202042(5) = 2*55 + 0*54 + 1*53 + 4*52 + 4*51 + 2*50   
             = 2*3125 + 0 + 1*125 + 4*25 + 4*5 + 2
             = 6250   +     125   + 100  +  20 + 2   = 6497(10)
 

(I will do the addition step-by-step in class)