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Signed zero

Signed zero is zero with an associated sign. In ordinary arithmetic, −0 = +0 = 0. However, in computing, some number representations allow for the existence of two zeros, often denoted by −0 (negative zero) and +0 (positive zero). This occurs in some 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. The zeroes can be considered as a variant of the extended real number line such that 1/−0 = −∞ and 1/+0 = +∞, division by zero is only undefined for ±0/±0. Negatively signed zero echoes the mathematical analysis concept of approaching 0 from below as a one-sided limit, which may be denoted by x â†’ 0−, x â†’ 0−, or x â†’ â†‘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, in particular when computing with complex elementary functions. 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 are different bit patterns and 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 both encodings, positive zero is represented by 0000 0000. 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*0.0, or simply as −0.0. In IEEE 754 and in IBM's General Decimal Arithmetic encoding specification, a floating-point representation that uses decimal arithmetic, 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. 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. Arithmetic Multiplication and division follow their usual rules for combining signs: \frac{-0}{ \left| x \right| } = -0\,\! (for x different from 0) (-0) \cdot (-0) = +0\,\! \left| x \right| \cdot (-0) = -0\,\! Addition and subtraction are handled specially if the values could cancel: x + (\pm 0) = x\,\! (for x different from 0) (-0) + (-0) = (-0) - (+0) = -0\,\! (+0) + (+0) = (+0) - (-0) = +0\,\! x - x = x + (-x) = +0\,\! (for any finite x, −0 when rounding toward negative) Because of negative zero (and only because of it), the statements z = -(x - y) and z = (-x) - (-y), for floating-point variables x, y, and z, cannot be optimized to z = y - x. Some other special rules: \sqrt{-0} = -0\,\! \frac{-0}{-\infty} = +0\,\! (follows the sign rule for division) \frac{\infty}{-0} = -\infty\,\! (follows the sign rule for division) {\pm 0} \times {\pm \infty} = \mbox{NaN}\,\! (Not a Number or interrupt for indeterminate form) \frac{\pm 0}{\pm 0} = \mbox{NaN}\,\! Comparisons According to the IEEE 754 standard, negative zero and positive zero should compare as equal with the usual (numerical) comparison operators, like the == operators of C and Java. In those languages, special programming tricks may be needed to distinguish the two values: casting the number to an integer type, so as to compare the bit patterns; using the IEEE 754 copysign() function to copy the sign of the zero to some non-zero number; taking the reciprocal of the zero to obtain either 1/(+0) = +∞ or 1/(−0) = âˆ’∞. However, some programming languages may provide alternative comparison operators that do distinguish the two zeros. This is the case, for example, of the equals method in Java's Double class. [http://java.sun.com/javase/6/docs/api/java/lang/Double.html#equals%28java.lang.Object%29] Scientific uses 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, 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 −1/Temperature. In this context, a temperature of −0 is a (theoretical) temperature larger than any other negative temperature, corresponding to the (theoretical) maximum conceivable extent of population inversion, the opposite extreme to +0.

From Yahoo Answers

Question:what does an equal sign with a slash through it mean?

Answers:It means "is not equal to". The slash is used to negate other signs as well, for instance a < sign with a slash through it means "is not less than". Generally any sign with a slash through it means the opposite of the same sign without the slash. In text forums it can be written as /= or =/ but is also often written in these forms, which derive from computer programming languages: != and <>.

Question:I'm a little confused as to when to use the approximately equals sign. (Looks like the equal sign but has two squiggles instead of two straight lines.) Do you use it when you round? Example: 6.56986 is approximately equaled to 6.57? Because the answer isn't exact? Some clarity would be appreciated. Thank you.

Answers:see the Pearson's chi square test

Question:i'm taking physics and i noticed that there are equations that have an equality sign that has 3 lines instead of 2.. could someone explain this to me please? if it means "identical w/"... what's the difference btw having 2 and 3 lines?

Answers:The equal sign with three lines means: "Exactly equal to" or "Identical to" The equal sign where the top one is curvey means: "Approximately equal to"

Question:what does it mean in an equation when instead of the usual equals sign with 2 lines, it's 3 horizontal lines? does it still just mean equals or something else?


From Youtube

Stop Sign Extravaganza :This is how you stop at a stop sign: stop smoothly; stop before the stop line, if there's one; if there's no stop line, stop before entering the crosswalk; if there's no crosswalk, stop before entering the intersection. Pedestrians have right-of-way at stop signs, meaning they have the right to cross the street before motorized vehicles. You have to stop, even on a right turn, going straight or turning left. Do use a turn signal at stop signs, so other drivers know what you're going to do. If it's a four-way or multi-way stop, and two or more drivers somehow stop at the same time, the driver on the right (counterclockwise) goes first. If you're facing each other and stop at the same time, straight or right-turning traffic trumps left-turning traffic. If you do stop first, go; don't just sit there and look. This causes drivers to creep, then stop. Just because no one's around, doesn't mean you can roll through the stop sign. I've done this before and have gotten two tickets because of it. That's why I always stop now. I do a "stop and go," not a "stop and pray." Stop and go means you stop, with the car ceasing movement, and then go. Stop and pray means you stop, sit and sit and sit. Thanks for reading and watching. Be well!