The digit
Definition
A digit is a symbol used to write numbers. The decimal system has exactly ten of them: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
A digit is not the quantity: it is the sign that records it.
Digit or number
| Digit | Number | |
|---|---|---|
| Nature | A symbol | A quantity |
| How many | Ten, in base ten | Infinitely many |
| Examples | 0, 7 | 7, 128, −3, π |
The number 128 is written with three digits: 1, 2 and 8. The number 7 is written with a single digit, which bears the same name — hence the common confusion between the two words.
A useful analogy
Digits are to numbers what letters are to words: a finite alphabet that records an infinite vocabulary.
It has its limit, as every analogy does. In a word, the place of a letter gives it no value; in a number, the place of a digit fixes one, and precisely.
Positional notation
A digit is worth something according to where it is written:
205=2×102+0×101+5×100
The digit 2 is worth two in 512 and two hundred in 205. Only its position changed.
Zero plays a role of its own here: it adds no quantity, but it holds a rank. Without it, 205 and 25 would be written the same way.
Not every numeral system works this way. Roman numerals are additive and subtractive, but not positional: X is worth ten in XVI as in XIV, whatever its rank.
Changing base
The number of available digits defines the base. A base b is written with exactly b digits, from 0 to b−1.
| Base | Name | Digits | Thirteen is written |
|---|---|---|---|
| 2 | binary | 0, 1 | 1101 |
| 8 | octal | 0 to 7 | 15 |
| 10 | decimal | 0 to 9 | 13 |
| 16 | hexadecimal | 0 to 9, A to F | D |
In base sixteen, sixteen symbols are needed: the ten usual digits do not suffice, so letters are borrowed. This is the sign that "digit" names a role and not a shape — A is a digit in base sixteen.
Summary
The digit is the symbol, the number is the quantity; and it is the digit's position that gives it its weight.