0 instance of additive identity
Mathematics
0 is the integer immediately preceding 1. Zero is an even number because it is divisible by 2 with no remainder. 0 is neither positive nor negative, or both positive and negative; cf. Signed zero. Many definitions include 0 as a natural number, in which case it is the only natural number that is not positive. Zero is a number which quantifies a count or an amount of null size. In most cultures, 0 was identified before the idea of negative things (i.e., quantities less than zero) was accepted.
As a value or a number, zero is not the same as the digit zero, used in numeral systems with positional notation. Successive positions of digits have higher weights, so the digit zero is used inside a numeral to skip a position and give appropriate weights to the preceding and following digits. A zero digit is not always necessary in a positional number system (e.g., the number 02). In some instances, a leading zero may be used to distinguish a number.Elementary algebra
The number 0 is the smallest non-negative integer. The natural number following 0 is 1 and no natural number precedes 0. The number 0 may or may not be considered a natural number, but it is an integer, and hence a rational number and a real number (as well as an algebraic number and a complex number).
The number 0 is neither positive nor negative, and is usually displayed as the central number in a number line. It is neither a prime number nor a composite number. It cannot be prime because it has an infinite number of factors, and cannot be composite because it cannot be expressed as a product of prime numbers (as 0 must always be one of the factors). Zero is, however, even (i.e., a multiple of 2, as well as being a multiple of any other integer, rational, or real number).
The following are some basic (elementary) rules for dealing with the number 0. These rules apply for any real or complex number x, unless otherwise stated.Addition: x + 0 = 0 + x = x. That is, 0 is an identity element (or neutral element) with respect to addition.
Subtraction: x − 0 = x and 0 − x = −x.
Multiplication: x · 0 = 0 · x = 0.
Division: 0/x = 0, for nonzero x. But x/0 is undefined, because 0 has no multiplicative inverse (no real number multiplied by 0 produces 1), a consequence of the previous rule.
Exponentiation: x0 = x/x = 1, except that the case x = 0 is considered undefined in some contexts. For all positive real x, 0x = 0.The expression 0/0, which may be obtained in an attempt to determine the limit of an expression of the form f(x)/g(x) as a result of applying the lim operator independently to both operands of the fraction, is a so-called "indeterminate form". That does not mean that the limit sought is necessarily undefined; rather, it means that the limit of f(x)/g(x), if it exists, must be found by another method, such as l'Hôpital's rule.
The sum of 0 numbers (the empty sum) is 0, and the product of 0 numbers (the empty product) is 1. The factorial 0! evaluates to 1, as a special case of the empty product.Related mathematical terms
A zero of a function f is a point x in the domain of the function such that f(x) = 0. When there are finitely many zeros, these are called the roots of the function. This is related to zeros of a holomorphic function.
The zero function (or zero map) on a domain D is the constant function with 0 as its only possible output value, i.e., the function f defined by f(x) = 0 for all x in D. The zero function is the only function that is both even and odd. A particular zero function is a zero morphism in category theory; e.g., a zero map is the identity in the additive group of functions. The determinant on non-invertible square matrices is a zero map.
Several branches of mathematics have zero elements, which generalize either the property 0 + x = x, or the property 0 · x = 0, or both.
type of example of manifestation of representation of