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 (b
^{0}). - The digit to the left of that one represents its value multiplied by the base to the first power (b
^{1}). - The next digit represents its value multiplied by the base to the second power (b
^{2}). - The next digit represents its value multiplied by the base to the third power (b
^{3}),

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

2 * 10^{3} |
= | 2 * 1000 | = | 2000 |

+ 0 * 10^{2} |
= | 0 * 100 | = | 0 |

+ 1 * 10^{1} |
= | 1 * 10 | = | 10 |

+ 9 * 10^{0} |
= | 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.