Signed zero

Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, and −0, +0 and 0 are three ways of writing the same number. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero), regarded as equal by the numerical comparison operations but with possible different behaviors in particular operations. This occurs in the sign–magnitude and ones' complement signed number representations for integers, and in most floating-point number representations. The number 0 is typically expressed as +0. The IEEE 754 standard for floating-point arithmetic (presently used by most computers and programming languages that support floating-point numbers) requires both +0 and −0. Its arithmetic can be considered to model a variant of the extended real number line (the real numbers including additional numbers +∞ and −∞). In IEEE 754 arithmetic, ⁠1/−0⁠ = −∞ and ⁠1/+0⁠ = +∞; the divisions ⁠±0/±0⁠ and ⁠±∞/±∞⁠ return "Not a Number" (NaN). Negatively signed zero echoes the mathematical analysis concept of approaching 0 from below as a one-sided limit, which may be denoted by x → 0−, x → 0−, or x → ↑0. The notation "−0" may be used informally to denote a negative number that has been rounded to zero. The concept of negative zero also has some theoretical applications in statistical mechanics and other disciplines. It is claimed that the inclusion of signed zero in IEEE 754 makes it much easier to achieve numerical accuracy in some critical problems, in particular when computing with complex elementary functions. On the other hand, the concept of signed zero runs contrary to the usual assumption made in mathematics that negative zero is the same value as zero. Representations that allow negative zero can be a...