Octal Number System

As the value of a decimal number grows, the number of bits (0/1) in its binary representation grows with it. Sometimes a binary number becomes so long that it is difficult to manage — imagine reading or copying a 16-bit or 32-bit string without a mistake. The octal number system was devised precisely for a compact representation of binary numbers.

The base-8 system

• Octal is called the base-8 system: it has a total of eight digits, 0–7.
• Positional values are expressed in powers of 8.
• Examples of octal numbers: (237.05)8, (13)8, (617.24)8.

Writing the base as a subscript

The base value is what distinguishes a number in one system from the same-looking number in another. It is written as a subscript of the number:

(70)8   →  70 read as an OCTAL number
(70)10  →  70 read as a DECIMAL number   (a completely different value!)

Without the subscript, "70" is ambiguous — always carry the base along.

Why octal compresses binary so neatly: 3 bits per octal digit

Since 8 = 2^3, three binary digits are sufficient to represent any octal digit. Every octal digit stands for exactly one 3-bit pattern, so a long binary string can be read off three bits at a time as a much shorter octal string.

The decimal and binary equivalents of the 8 octal digits:

| Octal digit | Decimal value | 3-bit binary | …