Signed zero


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 equivalent. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 and +0, 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 usually encoded as +0, but can still be represented by +0, −0, or 0.
The IEEE 754 standard for floating-point arithmetic requires both +0 and −0. Real arithmetic with signed zeros can be considered a variant of the extended real number line such that = −∞ and = +∞; division is undefined only for 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 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 source of errors in programs, if software developers do not take into account that while the two zero representations behave as equal under numeric comparisons, they yield different results in some operations.

Representations

Binary integer formats can use various encodings. In the widely used two's complement encoding, zero is unsigned. 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 ones' complement representation, negative zero is represented by the bit string 1111 1111. In all these three encodings, positive or unsigned zero is represented by 0000 0000. However, the latter two encodings are uncommon for integer formats. The most common formats with a signed zero are floating-point formats, described below.
Image:IEEE 754 Single Negative Zero.svg|thumb|right|400px|Negative zero by IEEE 754 representation in binary32
In IEEE 754 binary floating-point formats, 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 × 0.0, or simply as −0.0.
In IEEE 754 decimal floating-point formats, a negative zero is represented by an exponent being any valid exponent in the range for the format, the true significand being zero, and the sign bit being one.

Properties and handling

The IEEE 754 floating-point standard specifies the behavior of positive zero and negative zero under various operations. The outcome may depend on the current IEEE rounding mode settings.

Notation

In systems that include both signed and unsigned zeros, the notation and is sometimes used for signed zeros.

Arithmetic

Addition and multiplication are commutative, but there are some special rules that have to be followed, which mean the usual mathematical rules for algebraic simplification may not apply. The sign below shows the obtained floating-point results.
The usual rule for signs is always followed when multiplying or dividing:
There are special rules for adding or subtracting signed zero:
Because of negative zero, the expressions and, for floating-point variables x and y, cannot be replaced by. However can be replaced by x with rounding to nearest.
Some other special rules:
Division of a non-zero number by zero sets the divide by zero flag, and an operation producing a NaN sets the invalid operation flag. An exception handler is called if enabled for the corresponding flag.

Comparisons

According to the IEEE 754 standard, negative zero and positive zero should compare as equal with the usual comparison operators, like the operators of C and Java. In those languages, special programming tricks may be needed to distinguish the two values:
  • Type punning the number to an integer type, so as to look at the sign bit in the bit pattern;
  • using the ISO C copysign function to copy the sign of the zero to some non-zero number;
  • using the ISO C signbit macro that returns whether the sign bit of a number is set;
  • taking the reciprocal of the zero to obtain either = +∞ or = −∞.
Note: Casting to integral type will not always work, especially on two's complement systems.
However, some programming languages may provide alternative comparison operators that do distinguish the two zeros. This is the case, for example, of the method in Java's Double wrapper class.

In rounded values

Informally, one may use the notation "−0" for a negative value that was rounded to zero. This notation may be useful when a negative sign is significant; for example, when tabulating Celsius temperatures, where a negative sign means below freezing.

In statistical mechanics

In statistical mechanics, one sometimes uses negative temperatures to describe systems with population inversion, which can be considered to have a temperature greater than positive infinity, because the coefficient of energy in the population distribution function is. In this context, a temperature of −0 is a temperature larger than any other negative temperature, corresponding to the maximum conceivable extent of population inversion, the opposite extreme to +0.