Binary Values and Number Systems

2 Positional Notation

The base of a number system specifies the number of digits used in the system. The digits always begin with 0 and continue through one less than the base. For example,

  • There are 2 digits in base 2: 0 and 1.
  • There are 8 digits in base 8: 0 through 7.
  • There are 10 digits in base 10: 0 through 9.

The base also determines what the positions of digits mean. When you add 1 to the last digit in the number system, you have a carry to the digit position to the left.

Numbers are written using positional notation:

  • The rightmost digit represents its value multiplied by the base to the zeroth power (b0).
  • The digit to the left of that one represents its value multiplied by the base to the first power (b1).
  • The next digit represents its value multiplied by the base to the second power (b2).
  • The next digit represents its value multiplied by the base to the third power (b3),

and so on. You are so familiar with positional notation that you probably do not think about it:

2 * 103 = 2 * 1000 = 2000
+ 0 * 102 = 0 * 100 = 0
+ 1 * 101 = 1 * 10 = 10
+ 9 * 100 = 9 * 1 = 9
        ---------
2019

In the previous calculation, we assumed that the number base is 10. This is a logical assumption because our number system is base 10. Other bases, such as base 2 (binary), are particularly important in computer processing.