Number Systems

For every mathematics to be implemented, a full-fledged number system is needed. A computer performs a variety of tasks that include Text, Image, Audio or Video Processing. All these tasks ultimately are processed in terms of the number systems. The best feasible number system for a computer is Binary Number System.Following is a list of the most common number systems used so far:

• Decimal Number System
• Binary Number System
• Octal Number System
• Hexadecimal Number System

Every valid number system possesses a well defined RANGE of discrete values and a BASE which is normally the total count of values defined in the range. The following table best describes the common number systems:

NOTE: Since every data being processed by the computer gets converted first to Binary Number System, computer needs different techniques to convert different number systems into Binary Number System. Following are the principal techniques:

1. Binary Coded Octal


2. Binary Coded Hexadecimal


3. Binary Coded Decimal

Binary Coded Octal:

Binary Coded Octal is a 3-digit binary representation of a single Octal digit. It is based on an analogy that every Octal digit can be represented by only 3-digit binary number. Basically the analogy is: 2³ always equals to 8 which is the base of Octal Number System.

The adjacent table shows the count of Octal Number System in terms of 3-digit binary numbers:

Binary Coded HexaDecimal:

Very similar to BCO, Binary Coded Hexadecimal is a 4-digit binary representation of a single Hexadecimal digit. It is based on the analogy that every Hexadecimal digit can be represented by only a 4-digit binary number. Basically the analogy is: 2⁴ always equals to 16 which is the base of Hexadecimal Number System.

The adjacent table shows the count of Hexadecimal Number System in terms of 4-digit binary numbers:

Binary Coded Decimal:

Binary Coded Decimal is also a 4-digit binary representation of a single Decimal digit. Since the base of Decimal Number System is 10 which can never be represented by 3-digit binary representation. If we use 4-digits we'll be wasting 6 different representations. But ultimately, better is to choose 4-digits.

The adjacent table shows the count of Decimal Number System in terms of 4-digit binary numbers:

NOTE: Always keep in mind that converted binary is equal to coded binary but only for Octal and Hexadecimal numbers. For Decimal Numbers converted binary is different from coded binary.