Zero (mathematics)

Zero is the name for a character, the digit 0, and for the number "zero". If used as a digit, 0 is often also called null, nil, nought.

In addition to that, in mathematics, a zero of a function is any value of the argument for which the function takes the value 0.

The number zero

In the context of counting, zero is the smallest cardinality, the number of elements in the empty set. Whether zero is considered as a natural number is a matter of convention — often it is, but sometimes not. However, in modern mathematics, it is convenient and usual to consider zero as a natural number because it is the smallest cardinal number.
Similarly, zero is also the smallest ordinal number (the order type of the empty set).

In the context of the number system (used for measurements), zero is an integer, a rational number (the fraction 0/1), a real number (with the decimal expansion 0.000...), and a complex number (0+0i).
Zero is the neutral (or identity) element for the addition,

The character "0"

In the ASCII 8-bit character set, the character "0" has code 48 (hexadecimal 30x).

The digit "0"

The digit 0 is used in the decimal representation of numbers (and in other positional systems, like the binary and the hexadecimal system). It has its origin in Asia, was brought to Europe by the Arabs, and then adopted in Europe.

The numeral 0

As an one-digit numeral, 0 denotes the number zero in all positional numeral systems which use the Arabic digits.

The symbol "0"

In mathematics, the character 0 is also used as a symbol, usually for objects which are, in some way, related or analogous to the number zero. Examples for this usage are

the neutral element in a vector space (the zero vector), or,

more generally, the neutral element in additively written groups (the zero element),

functions which take the constant value 0, etc.

Sometimes, the symbol 0 is also used to denote the empty set.