0 in the context of "Negative number"

An integer is the number zero (0), a positive natural number (1, 2, 3, ...), or the negation of a positive natural number (−1, −2, −3, ...). The negations or additive inverses of the positive natural numbers are referred to as negative integers. The set of all integers is often denoted by the boldface Z or blackboard bold ${\displaystyle \mathbb {Z} }$.

The set of natural numbers ${\displaystyle \mathbb {N} }$ is a subset of ${\displaystyle \mathbb {Z} }$, which in turn is a subset of the set of all rational numbers ${\displaystyle \mathbb {Q} }$, itself a subset of the real numbers ⁠${\displaystyle \mathbb {R} }$⁠. Like the set of natural numbers, the set of integers ${\displaystyle \mathbb {Z} }$ is countably infinite. An integer may be regarded as a real number that can be written without a fractional component. For example, 21, 4, 0, and −2048 are integers, while 9.75, ⁠5+1/2⁠, 5/4, and the square root of 2 are not.

In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set of the natural numbers is commonly denoted by a bold N or a blackboard bold ⁠${\displaystyle \mathbb {N} }$⁠.

The natural numbers are used for counting, and for labeling the result of a count, like "there are seven days in a week", in which case they are called cardinal numbers. They are also used to label places in an ordered series, like "the third day of the month", in which case they are called ordinal numbers. Natural numbers may also be used to label, like the jersey numbers of a sports team; in this case, they have no specific mathematical properties and are called nominal numbers.

In mathematics, an infinitesimal number is a non-zero quantity that is closer to 0 than any non-zero real number is. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the "infinitieth" item in a sequence.

Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities; the augmentations are the reciprocals of one another.

In mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. The exponential of a variable ⁠${\displaystyle x}$⁠ is denoted ⁠${\displaystyle \exp x}$⁠ or ⁠${\displaystyle e^{x}}$⁠, with the two notations used interchangeably. It is called exponential because its argument can be seen as an exponent to which a constant number e ≈ 2.718, the base, is raised. There are several other definitions of the exponential function, which are all equivalent although being of very different nature.

The exponential function converts sums to products: it maps the additive identity 0 to the multiplicative identity 1, and the exponential of a sum is equal to the product of separate exponentials, ⁠${\displaystyle \exp(x+y)=\exp x\cdot \exp y}$⁠. Its inverse function, the natural logarithm, ⁠${\displaystyle \ln }$⁠ or ⁠${\displaystyle \log }$⁠, converts products to sums: ⁠${\displaystyle \ln(x\cdot y)=\ln x+\ln y}$⁠.

In mathematics, the empty set or void set is the unique set having no elements; its size or cardinality (count of elements in a set) is zero. Some axiomatic set theories ensure that the empty set exists by including an axiom of empty set, while in other theories, its existence can be deduced. Many possible properties of sets are vacuously true for the empty set.

Any set other than the empty set is called non-empty.

In mathematics, −1 (negative one or minus one) is the additive inverse of 1, that is, the number that when added to 1 gives the additive identity element, 0. It is the negative integer greater than negative two (−2) and less than 0.