What is octal system of numeration?

Answer 1

The octal (base-8) system of notation uses eight symbols, 0, 1, 2, 3, 4, 5, 6 and 7, rather than the ten symbols we use in decimal. We denote an octal number by prefixing it with "oct:" or by postfixing 8 to differentiate it from decimal notation (formally prefixed with "dec:" or postfixed with 10).

In decimal, each position represents an increasing power of 10, with the units position on the right. In octal, each position is an increasing power of 8. Thus 108 is decimal 1x81 which is 810, while 1008 is decimal 1x82, which is 6410.

Octal is typically used as shorthand method of denoting binary numbers (base-2) where each octal digit represents a group of 3 binary digits (bits), because 3 bits have 8 different combinations, including 000. Thus:

0002 = 08

0012 = 18

0102 = 28

0112 = 38

1002 = 48

1012 = 58

1102 = 68

1112 = 78

Thus a longer binary number such 11001110 can be split into 3 groups of three (with an implied leading zero):

(0)11 001 110

Translating each of these groups into octal we get 3, 1 and 6, which can be combined to form 3168. In decimal this means (3x82) + (1x81) + (6x80) which is 20610.

Thus, 110011102 = 3168 = 20610.

Hexadecimal, base-16, allows us to do the same thing, except the bits are grouped in groups of 4 rather than 3. This is easier to work with because a byte is 8 bits, thus a byte can be represented in 2 hexadecimal digits (where each digit represent a nybble, or half a byte). The symbols used in hexadecimal are: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E and F.

110011102 is therefore CE16, which means (12x161) + (14x160), which as we already know is 20610.

Therefore, 110011102 = 3168 = 20610 = CE16.