How was the octal number system invented?

Answer 1

It was invented just like any other base system that uses place value columns:

Each place value column is eight times bigger than the place value column to its right.

There is no reason to use the base 10 system other than we normally have 2 thumbs and 8 fingers giving 10 digits on our hands. In fact the Babylonians used sexagesimal (base 60) numbers; their legacy can be seen in the degrees in a circle and in our measurement of time: the colon used to separate the digits of a 24 hour time in the format "hh:mm:ss" are separators of sexagesimal digits that are represented by the base 10 numbers 0-59 - times are just the number of seconds since midnight expressed as a sexagesimal number.

Some cultures have used a base 20 number system.

When it comes to digital computers, they tend to use binary numbers as each location of memory can store one of two states: 0 or 1; these are binary-digits or bits (for short).

The bits are then grouped together:

4 bits make 1 nybble

2 nybbles = 8 bits = 1 byte.

We're used to decimal numbers so 1 byte can store the binary numbers 0 - 1111 1111 = 0-255 (in decimal). Two bytes allow storage of 0 - 1111 1111 1111 1111 = 0-65535 (in decimal). This conversion from binary to decimal is not very easy; to overcome this Binary Coded Decimal (BCD) can be used. BCD only uses the binary patterns of decimal numbers, ie 0-9 = 0000-1001, thus a byte can now store two decimal digits 0-99 (but this is very wasteful).

A much more useful representation of numbers in a digital computer is to use numbers with a base that is a power of 2 - that way every binary pattern is used.

One such system is hexadecimal or base 16. In this case each digit of a number is a number in the range 0-15 in decimal; to make each digit occupy a single character, the letters a-f (in either lower case or upper case a A-F) are used for 10-15, so that each hexadecimal digit is 0-f. 15 (0xf) is represented in binary as 1111 - 4 bits or 1 nybble. This means that each hexadecimal digit is exactly half a byte, and so two hexadecimal digits represent a byte exactly; this makes writing numbers extremely efficient and easy to convert to binary.

This ability to easily convert hexadecimal numbers to binary is useful for computer programmers as they will often use blocks of bits as flags. For example file permissions under *nix are a grouping of 3 bits: Read/Write/eXecute for different users. These permissions can be written easily into hexadecimal (for example 0x1ed) but it is not very obvious what the permissions are; however as there are three in each group it is better to use base 2³ = 8 numbers, or octal numbers. In octal each digit can contain the digits 0-7 (which avoids the need to create symbols for digits after the decimal digit 9). In the previous example of permissions 0x1ed (hexadecimal) = 0755 (octal) which is easily converted into the permissions rwxr-xr-x and the user know instantly what it means.

Octal was not invented specifically as a number system; it just exists as one as a consequence of the invention of the place value system. However, the grouping of bits in digital computers makes it useful for some applications there.