Why Count in Tens? Choosing How Many Digits to Have

A positional number system represents a quantity by a string of digits in which each position carries a value equal to the base raised to a power, so in the familiar system the positions count ones, tens, hundreds and so on, and the digit in each position says how many of that unit are present. The base determines how many distinct digits are needed, which is always equal to the base itself, and it determines how quickly the values grow across positions. Nothing about the quantity changes when the base changes, only the notation, so the same number of objects is written differently and behaves identically. The zero is essential to the arrangement, since without a symbol for an empty position the notation is ambiguous, which is why positional systems appeared late and spread slowly.

Several have been adopted seriously and traces of most survive:

• •Ten, almost certainly because of fingers, and now near universal
• •Twenty, using fingers and toes, surviving in French number words and in the English score
• •Sixty, used in Mesopotamia and surviving in minutes, seconds and degrees
• •Twelve, favoured for its divisibility and surviving in dozens, inches and old currency
• •Two, used by computers because a circuit is either on or off
• •Eight and sixteen, used in computing as compact ways of writing binary

The choice is a matter of physical reliability rather than mathematical elegance. A circuit representing a digit must distinguish reliably between the possible values despite noise, component variation and temperature, and distinguishing two states, meaning roughly high and roughly low voltage, is enormously easier and more robust than distinguishing ten. Machines using ten states were built in the early computing era and were abandoned for exactly that reason. Two states also map directly onto logic, where a proposition is true or false, which lets arithmetic and logical operations use the same circuitry. The cost is that numbers need many more digits, which is why programmers write binary values in groups using base sixteen, where each digit corresponds exactly to four binary digits and the conversion is mechanical.

Not every way of writing numbers works by position, and the alternatives clarify what the positional idea supplies. Roman numerals are additive with a subtractive rule, so each symbol carries a fixed value wherever it appears, which makes reading straightforward and arithmetic awkward, since there is no column structure to add down. Tally systems group marks in fives and are excellent for counting and useless for calculation. Egyptian numerals used distinct symbols for each power of ten and repeated them. Chinese numerals combine a digit with a symbol naming its place, which is positional in effect while writing the place out. What positional notation adds is that arithmetic becomes a mechanical procedure on digits, which is why algorithms for multiplication and division could be taught to clerks and later to machines.

A fraction written in a given base terminates only if its denominator divides a power of that base, which explains a familiar irritation. In base ten the prime factors available are two and five, so halves, quarters, fifths, eighths and tenths terminate while thirds and sevenths repeat forever. In base two only the factor two is available, so a fifth, which is perfectly tidy written as a decimal, repeats endlessly in binary and cannot be stored exactly in a computer, which is the origin of the small errors that appear when money is handled as ordinary floating point numbers. Base twelve has factors two and three, so thirds and quarters both terminate, which is the argument its advocates make and which explains why twelve was favoured for measures divided in practice.

The base sets how many digits exist and what each position is worth, while the quantity itself is unchanged, so only notation differs. Ten is a choice about fingers, and twelve, twenty and sixty all survive in minutes, dozens and scores. Computers use two because distinguishing two voltage levels is far more reliable than ten, at the cost of longer numbers.