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Negative numbers: Real numbers that are less than zero. Because zero itself has no sign, neither the positive numbers nor the negative numbers include zero. When zero is a possibility, the following terms are often used: Non-negative numbers: Real numbers that are greater than or equal to zero. Thus a non-negative number is either zero or positive.
Some authors use for non-zero integers, while others use it for non-negative integers, or for {–1,1} (the group of units of ). Additionally, Z p {\displaystyle \mathbb {Z} _{p}} is used to denote either the set of integers modulo p (i.e., the set of congruence classes of integers), or the set of p -adic integers .
The numbers d i are non-negative integers less than β. This is also known as a β-expansion, a notion introduced by Rényi (1957) and first studied in detail by Parry (1960). Every real number has at least one (possibly infinite) β-expansion. The set of all β-expansions that have a finite representation is a subset of the ring Z[β, β −1].
Alternatively, since the natural numbers naturally form a subset of the integers (often denoted ), they may be referred to as the positive, or the non-negative integers, respectively. [50] To be unambiguous about whether 0 is included or not, sometimes a superscript " ∗ {\displaystyle *} " or "+" is added in the former case, and a subscript ...
In mathematics, the factorial of a non-negative integer, denoted by !, is the product of all positive integers less than or equal to . The factorial of also equals the product of with the next smaller factorial: ! = () = ()! For example, ! =! = =
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.
By the definition of non-prime numbers, has factors ,, where , are integers greater than one and less than . Since a , b < n {\displaystyle a,b<n} , they are not in C {\displaystyle C} as n {\displaystyle n} is the smallest element of C {\displaystyle C} .
This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, respectively. All 5-smooth numbers are of the form 2 a × 3 b × 5 c, where a, b and c are non-negative integers.