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Octal and Hexadecimal Number Systems

Since binary representation of a large number forms an inconveniently long binary number, it is more convenient to combine groups of 4 bits into octal (base 8) or hexadecimal (base 16) digits respectively. Both representations are much shorter and more convenient for use in computers.

The octal number systems has a base of  8, uses eight symbols 0 through 7 and positional values which are ascending powers of 8.

For example:
(236)8 = 2 x 8^2 + 3 x 8^1 + 6 x 8^
            = 2 x 64 + 3 x 8 + 6 x 1
            = (158)10  

Hexadecimal number system has a base of 16 and employ 16 distinct digits (0 – 9 and A – F) as symbols. The positional values are increasing powers of 16 starting from 16^0 at the least significant digit.

For example,
(27A)16 = 2 x 16^2 + 7 x 16^1 + A x 16^0
              = 2 x 256 + 7 x 16  + 10 x 1
                = 512 + 112 + 10
                = (634)10

The rules of conversions from and to decimal system are similar to binary number system. All these number systems have a set of rules for arithmetic operations.

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