0 in the context of Binary 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, a negative number is the opposite of a positive real number. Equivalently, a negative number is a real number that is less than zero. Negative numbers are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset. If a quantity, such as the charge on an electron, may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common-sense idea of an opposite is reflected in arithmetic. For example, −‍(−3) = 3 because the opposite of an opposite is the original value.

Negative numbers are usually written with a minus sign in front. For example, −3 represents a negative quantity with a magnitude of three, and is pronounced and read as "minus three" or "negative three". Conversely, a number that is greater than zero is called positive; zero is usually (but not always) thought of as neither positive nor negative. The positivity of a number may be emphasized by placing a plus sign before it, e.g. +3. In general, the negativity or positivity of a number is referred to as its sign.

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.

In mathematics, the additive inverse of an element x, denoted −x, is the element that when added to x, yields the additive identity. This additive identity is often the number 0 (zero), but it can also refer to a more generalized zero element.

In elementary mathematics, the additive inverse is often referred to as the opposite number, or the negative of a number. The unary operation of arithmetic negation is closely related to subtraction and is important in solving algebraic equations. Not all sets where addition is defined have an additive inverse, such as the natural numbers.

In mathematics, the sign of a real number is its property of being either positive, negative, or 0. Depending on local conventions, zero may be considered as having its own unique sign, having no sign, or having both positive and negative sign. In some contexts, it makes sense to distinguish between a positive and a negative zero.

In mathematics and physics, the phrase "change of sign" is associated with exchanging an object for its additive inverse (multiplication with −1, negation), an operation which is not restricted to real numbers. It applies among other objects to vectors, matrices, and complex numbers, which are not prescribed to be only either positive, negative, or zero.

The unary numeral system is the simplest numeral system to represent natural numbers: to represent a number N, a symbol representing 1 is repeated N times.

In the unary system, the number 0 (zero) is represented by the empty string, that is, the absence of a symbol. Numbers 1, 2, 3, 4, 5, 6, ... are represented in unary as 1, 11, 111, 1111, 11111, 111111, ...

An imaginary number is the product of a real number and the imaginary unit i, which is defined by its property i = −1. The square of an imaginary number bi is −b. For example, 5i is an imaginary number, and its square is −25. The number zero is considered to be both real and imaginary.

Originally coined in the 17th century by René Descartes as a derogatory term and regarded as fictitious or useless, the concept gained wide acceptance following the work of Leonhard Euler in the 18th century, and Augustin-Louis Cauchy and Carl Friedrich Gauss in the early 19th century.