In mathematics, the surreal number system is an arithmetic continuum containing the real numbers as well as infinite and infinitesimal numbers, respectively larger or smaller in absolute value than any positive real number. The surreals share many properties with the reals, including a total order ≤ and the usual arithmetic operations (addition, subtraction, multiplication, and division); as such, they form an ordered field. (Strictly speaking, the surreals are not a set, but a proper class.) If formulated in Von Neumann–Bernays–Gödel set theory, the surreal numbers are the largest possible ordered...

In mathematics and computer science, the floor and ceiling functions map a real number to the largest previous or the smallest following integer, respectively. More precisely, floor(x) = ⌊ x ⌋ {\displaystyle \lfloor x\rfloor } is the largest integer less than or equal to x and ceiling(x) = ⌈ x ⌉ {\displaystyle \lceil x\rceil } is the smallest integer greater than or equal to x. == Notation == Carl Friedrich Gauss introduced the square bracket notation [ ...