|
Signed zero is zero with an associated sign. In ordinary arithmetic, the number 0 does not have a sign, so that −0, +0 and 0 are identical. 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 and magnitude'' and ''ones' complement'' signed number representations for integers, and in most floating-point number representations. The number 0 is usually encoded as +0, but can be represented by either +0 or −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. Real arithmetic with signed zeros can be considered a variant of the extended real number line such that 1/−0 = −∞ and 1/+0 = +∞; division is only undefined for ±0/±0 and ±∞/±∞. 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 small 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,〔William Kahan, "Branch Cuts for Complex Elementary Functions, or Much Ado About Nothing's Sign Bit", in (''The State of the Art in Numerical Analysis'' ) (eds. Iserles and Powell), Clarendon Press, Oxford, 1987.〕 in particular when computing with complex elementary functions.〔William Kahan, (Derivatives in the Complex z-plane ), p 10.〕 On the other hand, the concept of signed zero runs contrary to the general assumption made in most mathematical fields (and in most mathematics courses) that negative zero is the same thing as zero. Representations that allow negative zero can be a source of errors in programs, as software developers do not realize (or may forget) that, while the two zero representations behave as equal under numeric comparisons, they yield different results in some operations. ==Representations== The widely used two's complement encoding does not allow a negative zero. In a 1+7-bit sign-and-magnitude representation for integers, negative zero is represented by the bit string 1000 0000. In an 8-bit one's complement representation, negative zero is represented by the bit string 1111 1111. In all three encodings, positive zero is represented by . In IEEE 754 binary floating point numbers, zero values are represented by the biased exponent and significand both being zero. Negative zero has the sign bit set to one. One may obtain negative zero as the result of certain computations, for instance as the result of arithmetic underflow on a negative number, or −1.0, or simply as −0.0.In IEEE 754 decimal floating point encoding, a negative zero is represented by an exponent being any valid exponent in the range for the encoding, the true significand being zero, and the sign bit being one. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Signed zero」の詳細全文を読む スポンサード リンク
|