# Numbering Systems Overview

A numbering system is a consistent method of representing numerical values, which is closely related to methods of counting and performing mathematical operations.

We are most familiar with the base-ten numbering system, also known as decimal or denary. Using this system, each digit in a number is allowed to take one of ten possible values in the range 0–9. Moving from right to left, each digit is worth ten times as much as the digit immediately to its right. The value or 'weight' of each digit is commonly known as the place value.

Fractional numbers may also be represented using this system. The digit immediately to the right of the decimal point is worth one tenth of a unit (0.1) and each subsequent digit moving to the right is worth one tenth of the digit immediately to its left.

## Structure of a 'Base-N' Number

The principles used to construct a base-ten number may be generalised to allow numbers to be represented using a wide range of number bases. The rules for a base-N number, where N is an integer larger than one, would be: -

• each digit can vary in the range 0 to N-1,
• moving to the left, each digit is worth N times as much as the digit immediately to its right.

Numbering systems commonly used with electronics and microprocessor-based systems include binary (base-2), octal (base-8), and hexadecimal (base-16). Binary is the numbering system used internally by digital computers, while octal and hexadecimal are primarily used as compact representations of binary values, due in part to the ease of conversion between the two.