Search results
Results from the WOW.Com Content Network
Notably, is the first uncountable cardinal number that can be demonstrated within Zermelo–Fraenkel set theory not to be equal to the cardinality of the set of all real numbers: For any natural number , we can consistently assume that =, and moreover it is possible to assume that is as least as large as any cardinal number we like.
Positive numbers: Real numbers that are greater than zero. 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 ...
A number is positive if it is greater than zero. A number is negative if it is less than zero. A number is non-negative if it is greater than or equal to zero. A number is non-positive if it is less than or equal to zero. When 0 is said to be both positive and negative, [citation needed] modified phrases are used to refer to the sign of a number:
In a vector space, the additive inverse −v (often called the opposite vector of v) has the same magnitude as v and but the opposite direction. [11] In modular arithmetic, the modular additive inverse of x is the number a such that a + x ≡ 0 (mod n) and always exists. For example, the inverse of 3 modulo 11 is 8, as 3 + 8 ≡ 0 (mod 11). [12]
An integer is positive if it is greater than zero, and negative if it is less than zero. Zero is defined as neither negative nor positive. The ordering of integers is compatible with the algebraic operations in the following way: If a < b and c < d, then a + c < b + d; If a < b and 0 < c, then ac < bc
The real numbers are more numerous than the natural numbers. Moreover, R {\displaystyle \mathbb {R} } has the same number of elements as the power set of N {\displaystyle \mathbb {N} } . Symbolically, if the cardinality of N {\displaystyle \mathbb {N} } is denoted as ℵ 0 {\displaystyle \aleph _{0}} , the cardinality of the continuum is
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 " infinity - eth " item in a sequence .
On the other hand, every real number greater than or equal to zero is certainly an upper bound on this set. Hence, 0 {\displaystyle 0} is the least upper bound of the negative reals, so the supremum is 0.