Computers use a binary or base-2 counting system for reasons
that are beyond this post. Since binary numbers are large to write,
programmers adopted larger base systems that could directly represent the bits,
namely octal (base-8,3 bits) and hexadecimal (base-16, 4 bits) with hexadecimal more commonly used. Two
hexadecimal digits represent a byte and other computer words all have a number of bits that can be divided evenly into bytes. For example, 32-bit address is 4 bytes or 8 hexadecimal digits. To write these
extra digits, A-B-C-D-E-F is used, representing decimal the numbers
10-11-12-13-14-15 and to make the distinction clear numbers are often prefixed with 0x or #.
Decimal numbers, or base-10 is what humans mostly use and while it seems logical when looking at our hands, unlike computers, using this system is arbitrary. Humans can use any base system (see below). The result is
that computers have to convert the base of numbers that are displayed to the
users and also convert numbers that are entered by the user. While this is not difficult, this extra step itself can create issues that need not be created.
One of these issues is rounding error.
In both systems 1/3 cannot be represented exactly, as the best that can be done in base-10 is
0.33333... and hexadecimal (and
convertible to octal and binary), the number is 0x0.5555... There are also numbers that can
be represented in decimal but not hexadecimal. For example, 0.1 is 0x109999....
While people easily understand the issues with base-10 figures, such as
seeing a graph with three 33% and knowing they really add up to 100%, issues with
this conversion are often not noticed or create odd numbers such as 0.9999 when people expect to see 1.
Another issue is that many things in computers are limited by the
binary system but with base-10 the limitation seems arbitrary. Take the RGB colour values used by many programs, including photo editors and CSS. The user can only enter 0
to 255. There seems no rational reason
that they cannot change the third digit 255 to 256 unless the user realises that it
really is a two digit hexadecimal number. Then they would understand clearly that 0xFF + 0x1 =
0x100, a three digit number too big to fit is the two digit space.
A high point is that the cost of learning is minimal because people already have the tools to use hexadecimal numbers. For example, to calculate 0x4D + 0xC5 using
elementary school techniques:
- Add 0xD + 0x5 to get 0x12
- Carry the 0x1
- Add 0x1 + 0x4 + 0xC and you get 0x11
- Carry the 0x1 again and since it is the only number the first digit becomes 0x1
- The answer is 0x112
While there is a necessity for new symbols and vocabulary, people also no longer using Roman Numerals (base-1 with compression?) and have also switched to the Metric system. As computers are now the primary tool used in mathematical calculations, it seem only prudent that humans consider using the native numbering system of computing when doing math themselves.
