title: Probability theory

text: Consider an experiment that can produce a number of outcomes. The set of all outcomes is called the "sample space" of the experiment. The "power set" of the sample space (or equivalently, the event space) is formed by considering all different collections of possible results. For example, rolling an honest die produces one of six possible results. One collection of possible results corresponds to getting an odd number. Thus, the subset {1,3,5} is an element of the power set of the sample space of die rolls. These collections are called "events". In this case, {1,3,5} is the event that the die falls on some odd number. If the results that actually occur fall in a given event, that event is said to have occurred.


title: Probability theory

text: Probability is a way of assigning every "event" a value between zero and one, with the requirement that the event made up of all possible results (in our example, the event {1,2,3,4,5,6}) be assigned a value of one. To qualify as a probability distribution, the assignment of values must satisfy the requirement that if you look at a collection of mutually exclusive events (events that contain no common results, e.g., the events {1,6}, {3}, and {2,4} are all mutually exclusive), the probability that any of these events occurs is given by the sum of the probabilities of the events.


title: Probability theory

text: The probability that any one of the events {1,6}, {3}, or {2,4} will occur is 5/6. This is the same as saying that the probability of event {1,2,3,4,6} is 5/6. This event encompasses the possibility of any number except five being rolled. The mutually exclusive event {5} has a probability of 1/6, and the event {1,2,3,4,5,6} has a probability of 1, that is, absolute certainty.


title: Probability theory

text: When doing calculations using the outcomes of an experiment, it is necessary that all those elementary events have a number assigned to them. This is done using a random variable. A random variable is a function that assigns to each elementary event in the sample space a real number. This function is usually denoted by a capital letter. In the case of a die, the assignment of a number to a certain elementary events can be done using the identity function. This does not always work. For example, when flipping a coin the two possible outcomes are "heads" and "tails". In this example, the random variable "X" could assign to the outcome "heads" the number "0" (formula_1) and to the outcome "tails" the number "1" (formula_2).


title: Probability theory

text: Initially the probability of an event to occur was defined as the number of cases favorable for the event, over the number of total outcomes possible in an equiprobable sample space: see Classical definition of probability.


title: Probability theory

text: For example, if the event is "occurrence of an even number when a die is rolled", the probability is given by formula_3, since 3 faces out of the 6 have even numbers and each face has the same probability of appearing.


title: Probability theory

text: The modern definition starts with a finite or countable set called the sample space, which relates to the set of all "possible outcomes" in classical sense, denoted by formula_4. It is then assumed that for each element formula_5, an intrinsic "probability" value formula_6 is attached, which satisfies the following properties:


title: Probability theory

text: That is, the probability function "f"("x") lies between zero and one for every value of "x" in the sample space "Ω", and the sum of "f"("x") over all values "x" in the sample space "Ω" is equal to 1. An is defined as any subset formula_9 of the sample space formula_10. The of the event formula_9 is defined as


title: Probability theory

text: The function formula_6 mapping a point in the sample space to the "probability" value is called a abbreviated as . The modern definition does not try to answer how probability mass functions are obtained; instead, it builds a theory that assumes their existence.


title: Probability theory

text: The classical definition breaks down when confronted with the continuous case. See Bertrand's paradox.


title: Probability theory

text: If the sample space of a random variable "X" is the set of real numbers (formula_14) or a subset thereof, then a function called the (or ) formula_15 exists, defined by formula_16. That is, "F"("x") returns the probability that "X" will be less than or equal to "x".


title: Probability theory

text: If formula_15 is absolutely continuous, i.e., its derivative exists and integrating the derivative gives us the cdf back again, then the random variable "X" is said to have a or or simply formula_21


title: Probability theory

text: Whereas the "pdf" exists only for continuous random variables, the "cdf" exists for all random variables (including discrete random variables) that take values in formula_26


title: Probability theory

text: These concepts can be generalized for multidimensional cases on formula_27 and other continuous sample spaces.


title: Probability theory

text: The "raison d'être" of the measure-theoretic treatment of probability is that it unifies the discrete and the continuous cases, and makes the difference a question of which measure is used. Furthermore, it covers distributions that are neither discrete nor continuous nor mixtures of the two.


title: Probability theory

text: An example of such distributions could be a mix of discrete and continuous distributions—for example, a random variable that is 0 with probability 1/2, and takes a random value from a normal distribution with probability 1/2. It can still be studied to some extent by considering it to have a pdf of formula_28, where formula_29 is the Dirac delta function.


title: Probability theory

text: Other distributions may not even be a mix, for example, the Cantor distribution has no positive probability for any single point, neither does it have a density. The modern approach to probability theory solves these problems using measure theory to define the probability space:


title: Probability theory

text: Given any set formula_10 (also called ) and a σ-algebra formula_31 on it, a measure formula_32 defined on formula_31 is called a if formula_34


title: Probability theory

text: If formula_31 is the Borel σ-algebra on the set of real numbers, then there is a unique probability measure on formula_31 for any cdf, and vice versa. The measure corresponding to a cdf is said to be by the cdf. This measure coincides with the pmf for discrete variables and pdf for continuous variables, making the measure-theoretic approach free of fallacies.


title: Probability theory

text: Along with providing better understanding and unification of discrete and continuous probabilities, measure-theoretic treatment also allows us to work on probabilities outside formula_27, as in the theory of stochastic processes. For example, to study Brownian motion, probability is defined on a space of functions.


title: Probability theory

text: When it's convenient to work with a dominating measure, the Radon-Nikodym theorem is used to define a density as the Radon-Nikodym derivative of the probability distribution of interest with respect to this dominating measure. Discrete densities are usually defined as this derivative with respect to a counting measure over the set of all possible outcomes. Densities for absolutely continuous distributions are usually defined as this derivative with respect to the Lebesgue measure. If a theorem can be proved in this general setting, it holds for both discrete and continuous distributions as well as others; separate proofs are not required for discrete and continuous distributions.


title: Probability theory

text: Certain random variables occur very often in probability theory because they well describe many natural or physical processes. Their distributions, therefore, have gained "special importance" in probability theory. Some fundamental "discrete distributions" are the discrete uniform, Bernoulli, binomial, negative binomial, Poisson and geometric distributions. Important "continuous distributions" include the continuous uniform, normal, exponential, gamma and beta distributions.


title: Probability theory

text: In probability theory, there are several notions of convergence for random variables. They are listed below in the order of strength, i.e., any subsequent notion of convergence in the list implies convergence according to all of the preceding notions.


title: Probability theory

text: As the names indicate, weak convergence is weaker than strong convergence. In fact, strong convergence implies convergence in probability, and convergence in probability implies weak convergence. The reverse statements are not always true.


title: Probability theory

text: Common intuition suggests that if a fair coin is tossed many times, then "roughly" half of the time it will turn up "heads", and the other half it will turn up "tails". Furthermore, the more often the coin is tossed, the more likely it should be that the ratio of the number of "heads" to the number of "tails" will approach unity. Modern probability theory provides a formal version of this intuitive idea, known as the . This law is remarkable because it is not assumed in the foundations of probability theory, but instead emerges from these foundations as a theorem. Since it links theoretically derived probabilities to their actual frequency of occurrence in the real world, the law of large numbers is considered as a pillar in the history of statistical theory and has had widespread influence.


title: Probability theory

text: identically distributed random variables formula_59 converges towards their common expectation formula_60, provided that the expectation of formula_61 is finite.


title: Probability theory

text: It is in the different forms of convergence of random variables that separates the "weak" and the "strong" law of large numbers


title: Probability theory

text: It follows from the LLN that if an event of probability "p" is observed repeatedly during independent experiments, the ratio of the observed frequency of that event to the total number of repetitions converges towards "p".


title: Probability theory

text: For example, if formula_66 are independent Bernoulli random variables taking values 1 with probability "p" and 0 with probability 1-"p", then formula_67 for all "i", so that formula_68 converges to "p" almost surely.


title: Probability theory

text: The central limit theorem (CLT) explains the ubiquitous occurrence of the normal distribution in nature, and this theorem, according to David Williams, "is one of the great results of mathematics."


title: Probability theory

text: The theorem states that the average of many independent and identically distributed random variables with finite variance tends towards a normal distribution "irrespective" of the distribution followed by the original random variables. Formally, let formula_50 be independent random variables with mean formula_60 and variance formula_71 Then the sequence of random variables


title: Probability theory

text: For some classes of random variables, the classic central limit theorem works rather fast, as illustrated in the Berry–Esseen theorem. For example, the distributions with finite first, second, and third moment from the exponential family; on the other hand, for some random variables of the heavy tail and fat tail variety, it works very slowly or may not work at all: in such cases one may use the Generalized Central Limit Theorem (GCLT).


title: Time signature

text: The time signature (also known as meter signature, metre signature, or measure signature) is a notational convention used in Western musical notation to specify how many beats (pulses) are contained in each measure (bar), and which note value is equivalent to a beat.


title: Time signature

text: In a music score, the time signature appears at the beginning as a time symbol or stacked numerals, such as or (read "common time" or "four-four time", respectively), immediately following the key signature (or immediately following the clef symbol if the key signature is empty). A mid-score time signature, usually immediately following a barline, indicates a change of meter.


title: Time signature

text: There are various types of time signatures, depending on whether the music follows regular (or symmetrical) beat patterns, including simple (e.g., and ), and compound (e.g., and ); or involves shifting beat patterns, including complex (e.g., or ), mixed (e.g., & or & ), additive (e.g., ), fractional (e.g., ), and irrational meters (e.g., or ).


title: Time signature

text: For instance, means two quarter-note (crotchet) beats per bar, while means three eighth-notes (quavers) per bar, which are beats at slower tempos (but at faster tempos, becomes compound time, with one beat per bar). The most common simple time signatures are , , and .


title: Time signature

text: In compound meter, subdivisions (which are what the upper number represents in these meters) of the beat are in three equal parts, so that a dotted note (half again longer than a regular note) becomes the beat. The upper numeral of compound time signatures is commonly 6, 9, or 12 (multiples of 3 in each beat). The lower number is most commonly an 8 (an eighth-note or quaver): as in or .


title: Time signature

text: In the examples below, bold denotes a more-stressed beat, and "italics" denotes a less-stressed beat.


title: Time signature

text: "Simple": is a simple triple meter time signature that represents three quarter notes (crotchets). It is felt as


title: Time signature

text: "Compound": In principle, comprises not three groups of two eighth notes (quavers) but two groups of three eighth-note (quaver) subdivisions. It is felt as


title: Time signature

text: Time signatures indicating "two" beats per bar (whether in simple or compound meter) are called "duple meter", while those with "three" beats to the bar are "triple meter". Terms such as "quadruple" (4), "quintuple" (5), and so on, are also occasionally used.


title: Time signature

text: While changing the bottom number and keeping the top number fixed only formally changes notation, without changing meaning – , , , and are all three beats to a meter, just noted with eighth notes, quarter notes, half notes, or whole notes – these conventionally imply different performance and different tempi. Conventionally, larger numbers in the bottom correspond to faster tempi and smaller numbers correspond to slower tempi. This convention is known as "tempo giusto", and means that the tempo of each note remains in a narrower, "normal" range. For illustration, a quarter note might correspond to 60–120 bpm, a half note to 30–60 bpm, a whole note to 15–30 bpm, and an eighth note to 120–240 bpm; these are not strict, but show an example of "normal" ranges.


title: Time signature

text: This convention dates to the Baroque era, when tempo changes were indicated by changing time signature during the piece, rather than by using a single time signature and changing tempo marking. For example, while , , , and have the same beat pattern, they would conventionally be used for increasingly slow music. A 20th century example is "O Fortuna" (1935–1936) by Carl Orff, which begins slowly in , and then speeds up and changes to .


title: Time signature

text: To the ear, a bar may seem like one singular beat. For example, a fast waltz, notated in time, may be described as being "one in a bar". Correspondingly, at slow tempos, the beat indicated by the time signature could in actual performance be divided into smaller units.


title: Time signature

text: On a formal mathematical level, the time signatures of, e.g., and are interchangeable. In a sense, "all" simple triple time signatures, such as , , , etc.—and all compound duple times, such as , and so on, are equivalent. A piece in can be easily rewritten in , simply by halving the length of the notes.


title: Time signature

text: Other time signature rewritings are possible: most commonly a simple time signature with triplets translates into a compound meter.


title: Time signature

text: Though formally interchangeable, for a composer or performing musician, by convention, different time signatures often have different connotations. First, a smaller note value in the beat unit implies a more complex notation, which can affect ease of performance. Second, beaming affects the choice of actual beat divisions. It is, for example, more natural to use the quarter note/crotchet as a beat unit in or than the eighth note/quaver in or . Third, time signatures are traditionally associated with different music styles—it would seem strange to notate a conventional rock song in or , rather than .


title: Time signature

text: Signatures that do not fit the usual duple or triple categories are called "complex", "asymmetric", "irregular", "unusual", or "odd"—though these are broad terms, and usually a more specific description is appropriate. The term "odd meter", however, sometimes describes time signatures in which the upper number is simply odd rather than even, including and .


title: Time signature

text: The irregular meters (not fitting duple or triple categories) are common in some non-Western music, but rarely appeared in formal written Western music until the 19th century. Early anomalous examples appeared in Spain between 1516 and 1520, but the Delphic Hymns to Apollo (one by Athenaeus is entirely in quintuple meter, the other by Limenius predominantly so), carved on the exterior walls of the Athenian Treasury at Delphi in 128 BC, are in the relatively common cretic meter, with five beats to a foot.


title: Time signature

text: The third movement of Frédéric Chopin's Piano Sonata No. 1 (1828) is an early, but by no means the earliest, example of time in solo piano music. Anton Reicha's Fugue No. 20 from his "Thirty-six Fugues", published in 1803, is also for piano and is in . The waltz-like second movement of Tchaikovsky's "Pathétique" Symphony (shown below), often described as a "limping waltz", is a notable example of time in orchestral music.


title: Time signature

text: In the Western popular music tradition, unusual time signatures occur as well, with progressive rock in particular making frequent use of them. The use of shifting meters in The Beatles' "Strawberry Fields Forever" and the use of quintuple meter in their "Within You, Without You" are well-known examples, as is Radiohead's "Paranoid Android" (includes ).


title: Time signature

text: Paul Desmond's jazz composition "Take Five", in time, was one of a number of irregular-meter compositions that The Dave Brubeck Quartet played. They played other compositions in ("Eleven Four"), ("Unsquare Dance"), and ("Blue Rondo à la Turk"), expressed as . This last is an example of a work in a signature that, despite appearing merely compound triple, is actually more complex. Brubeck's title refers to the characteristic "aksak" meter of the Turkish "karşılama" dance.


title: Time signature

text: However, such time signatures are only unusual in most Western music. Traditional music of the Balkans uses such meters extensively. Bulgarian dances, for example, include forms with 5, 7, 9, 11, 13, 15, 22, 25 and other numbers of beats per measure. These rhythms are notated as "additive rhythms" based on simple units, usually 2, 3 and 4 beats, though the notation fails to describe the metric "time bending" taking place, or compound meters. See Additive meters below.


title: Time signature

text: While time signatures usually express a regular pattern of beat stresses continuing through a piece (or at least a section), sometimes composers place a different time signature at the beginning of each bar, resulting in music with an extremely irregular rhythmic feel. In this case, the time signatures are an aid to the performers and not "necessarily" an indication of meter. The Promenade from Modest Mussorgsky's "Pictures at an Exhibition" (1874) is a good example. The opening measures are shown below:


title: Time signature

text: Igor Stravinsky's "The Rite of Spring" (1913) is famous for its "savage" rhythms. Five measures from "Sacrificial Dance" are shown below:


title: Time signature

text: In such cases, a convention that some composers follow (e.g., Olivier Messiaen, in his "La Nativité du Seigneur" and "Quatuor pour la fin du temps") is to simply omit the time signature. Charles Ives's "Concord Sonata" has measure bars for select passages, but the majority of the work is unbarred.


title: Time signature

text: Some pieces have no time signature, as there is no discernible meter. This is sometimes known as "free time". Sometimes one is provided (usually ) so that the performer finds the piece easier to read, and simply has "free time" written as a direction. Sometimes the word "FREE" is written downwards on the staff to indicate the piece is in free time. Erik Satie wrote many compositions that are ostensibly in free time but actually follow an unstated and unchanging simple time signature. Later composers used this device more effectively, writing music almost devoid of a discernibly regular pulse.


title: Time signature

text: If two time signatures alternate repeatedly, sometimes the two signatures are placed together at the beginning of the piece or section, as shown below:


title: Time signature

text: To indicate more complex patterns of stresses, such as additive rhythms, more complex time signatures can be used. Additive meters have a pattern of beats that subdivide into smaller, irregular groups. Such meters are sometimes called "imperfect", in contrast to "perfect meters", in which the bar is first divided into equal units.


title: Time signature

text: For example, the time signature means that there are 8 quaver beats in the bar, divided as the first of a group of three eighth notes (quavers) that are stressed, then the first of a group of two, then first of a group of three again. The stress pattern is usually counted as


title: Time signature

text: This kind of time signature is commonly used to notate folk and non-Western types of music. In classical music, Béla Bartók and Olivier Messiaen have used such time signatures in their works. The first movement of Maurice Ravel's Piano Trio in A Minor is written in , in which the beats are likewise subdivided into to reflect Basque dance rhythms.


title: Time signature

text: Romanian musicologist Constantin Brăiloiu had a special interest in compound time signatures, developed while studying the traditional music of certain regions in his country. While investigating the origins of such unusual meters, he learned that they were even more characteristic of the traditional music of neighboring peoples (e.g., the Bulgarians). He suggested that such timings can be regarded as compounds of simple two-beat and three-beat meters, where an accent falls on every first beat, even though, for example in Bulgarian music, beat lengths of 1, 2, 3, 4 are used in the metric description. In addition, when focused only on stressed beats, simple time signatures can count as beats in a slower, compound time. However, there are two different-length beats in this resulting compound time, a one half-again longer than the short beat (or conversely, the short beat is the value of the long). This type of meter is called "aksak" (the Turkish word for "limping"), "impeded", "jolting", or "shaking", and is described as an "irregular bichronic rhythm". A certain amount of confusion for Western musicians is inevitable, since a measure they would likely regard as , for example, is a three-beat measure in "aksak", with one long and two short beats (with subdivisions of , , or ).


title: Time signature

text: Folk music may make use of metric time bends, so that the proportions of the performed metric beat time lengths differ from the exact proportions indicated by the metric. Depending on playing style of the same meter, the time bend can vary from non-existent to considerable; in the latter case, some musicologists may want to assign a different meter. For example, the Bulgarian tune "Eleno Mome" is written in one of three forms: (1) , (2) , or (3) , but an actual performance (e.g., "Eleno Mome") may be closer to . The Macedonian meter is even more complicated, with heavier time bends, and use of quadruples on the threes. The metric beat time proportions may vary with the speed that the tune is played. The Swedish Boda Polska (Polska from the parish Boda) has a typical elongated second beat.


title: Time signature

text: In Western classical music, metric time bend is used in the performance of the Viennese waltz. Most Western music uses metric ratios of 2:1, 3:1, or 4:1 (two-, three- or four-beat time signatures)—in other words, integer ratios that make all beats equal in time length. So, relative to that, 3:2 and 4:3 ratios correspond to very distinctive metric rhythm profiles. Complex accentuation occurs in Western music, but as syncopation rather than as part of the metric accentuation.


title: Time signature

text: Brăiloiu borrowed a term from Turkish medieval music theory: "aksak". Such compound time signatures fall under the "aksak rhythm" category that he introduced along with a couple more that should describe the rhythm figures in traditional music. The term Brăiloiu revived had moderate success worldwide, but in Eastern Europe it is still frequently used. However, aksak rhythm figures occur not only in a few European countries, but on all continents, featuring various combinations of the two and three sequences. The longest are in Bulgaria. The shortest aksak rhythm figures follow the five-beat timing, comprising a two and a three (or three and two).


title: Time signature

text: A method to create meters of lengths of any length has been published in the Journal of Anaphoria Music Theory and Xenharmonikon 16 using both those based on the Horograms of Erv Wilson and Viggo Brun's algorithm written by Kraig Grady.


title: Time signature

text: Irrational time signatures (rarely, "non-dyadic time signatures") are used for so-called "irrational bar lengths", that have a denominator that is not a power of two (1, 2, 4, 8, 16, 32, etc.). These are based on beats expressed in terms of fractions of full beats in the prevailing tempo—for example or . For example, where implies a bar construction of four quarter-parts of a whole note (i.e., four quarter notes), implies a bar construction of four third-parts of it. These signatures are of utility only when juxtaposed with other signatures with varying denominators; a piece written entirely in , say, could be more legibly written out in .


title: Time signature

text: According to Brian Ferneyhough, metric modulation is "a somewhat distant analogy" to his own use of "irrational time signatures" as a sort of rhythmic dissonance. It is disputed whether the use of these signatures makes metric relationships clearer or more obscure to the musician; it is always possible to write a passage using non-irrational signatures by specifying a relationship between some note length in the previous bar and some other in the succeeding one. Sometimes, successive metric relationships between bars are so convoluted that the pure use of irrational signatures would quickly render the notation extremely hard to penetrate. Good examples, written entirely in conventional signatures with the aid of between-bar specified metric relationships, occur a number of times in John Adams' opera "Nixon in China" (1987), where the sole use of irrational signatures would quickly produce massive numerators and denominators.


title: Time signature

text: Historically, this device has been prefigured wherever composers wrote tuplets. For example, a bar of 3 triplet quarter notes could be written as a bar of . Henry Cowell's piano piece "Fabric" (1920) employs separate divisions of the bar (1 to 9) for the three contrapuntal parts, using a scheme of shaped noteheads to visually clarify the differences, but the pioneering of these signatures is largely due to Brian Ferneyhough, who says that he finds that "such 'irrational' measures serve as a useful buffer between local changes of event density and actual changes of base tempo". Thomas Adès has also used them extensively—for example in "Traced Overhead" (1996), the second movement of which contains, among more conventional meters, bars in such signatures as , and .


title: Time signature

text: A gradual process of diffusion into less rarefied musical circles seems underway. For example, John Pickard's "Eden", commissioned for the 2005 finals of the National Brass Band Championships of Great Britain, contains bars of and .


title: Time signature

text: Notationally, rather than using Cowell's elaborate series of notehead shapes, the same convention has been invoked as when normal tuplets are written; for example, one beat in is written as a normal quarter note, four quarter notes complete the bar, but the whole bar lasts only of a reference whole note, and a beat of one (or of a normal quarter note). This is notated in exactly the same way that one would write if one were writing the first four quarter notes of five quintuplet quarter notes.


title: Time signature

text: These video samples show two time signatures combined to make a polymeter, since , say, in isolation, is identical to .


title: Time signature

text: Some composers have used fractional beats: for example, the time signature appears in Carlos Chávez's Piano Sonata No. 3 (1928) IV, m. 1. Both and appear in the fifth movement of Percy Grainger's Lincolnshire Posy.


title: Time signature

text: Music educator Carl Orff proposed replacing the lower number of the time signature with an actual note image, as shown at right. This system eliminates the need for compound time signatures, which are confusing to beginners. While this notation has not been adopted by music publishers generally (except in Orff's own compositions), it is used extensively in music education textbooks. Similarly, American composers George Crumb and Joseph Schwantner, among others, have used this system in many of their works. Émile Jaques-Dalcroze proposed this in his 1920 collection, "Le Rythme, la musique et l'éducation".


title: Time signature

text: Another possibility is to extend the barline where a time change is to take place above the top instrument's line in a score and to write the time signature there, and there only, saving the ink and effort that would have been spent writing it in each instrument's staff. Henryk Górecki's "Beatus Vir" is an example of this. Alternatively, music in a large score sometimes has time signatures written as very long, thin numbers covering the whole height of the score rather than replicating it on each staff; this is an aid to the conductor, who can see signature changes more easily.


title: Time signature

text: In the mensural notation of the 14th, 15th and 16th centuries there are no bar lines, and the four basic "mensuration signs" indicate the normal ratio of duration between different note values. Unlike modern notation, the subdivisions could be either 2:1 or 3:1. The relation between the "breve" and the "semibreve" was called tempus, and could be perfect (triple 3:1 indicated by circle) or imperfect (duple 2:1, with broken circle), while the relation between the "semibreve" and the "minim" was called prolatio and could be major (3:1 or compound, indicated by dot) or minor (2:1 or simple meter).


title: Time signature

text: N.B.: In mensural notation actual note values depend not only on the prevailing mensuration, but on rules for imperfection and alteration, with ambiguous cases using a dot of separation, similar in appearance but not always in effect to the modern dot of augmentation.


title: Time signature

text: Besides showing the organization of beats with musical meter, the mensuration signs discussed above have a second function, which is showing tempo relationships between one section to another, which modern notation can only specify with tuplets or metric modulations. This is a fraught subject, because the usage has varied with both time and place: Charles Hamm was even able to establish a rough chronology of works based on three distinct usages of mensural signs over the career of Guillaume Dufay (1397(?) – 1474). By the end of the sixteenth century Thomas Morley was able to satirize the confusion in an imagined dialogue:


title: Time signature

text: it was a world to hear them wrangle, every one defending his own for the best. "What? You keep not time in your proportions." "You sing them false. What proportion is this?" "Sesquipaltry." "Nay, you sing you know not what; it would seem you came lately from a barber's shop where you had 'Gregory Walker' or a Curranta played in the new Proportions by them lately found out, called 'Sesquiblinda' and 'Sesquihearkenafter'."


title: Time signature

text: In general though, a slash or the numeral 2 shows a doubling of tempo, and paired numbers (either side by side or one atop another) show ratios instead of beats per measure over note value: in early music contexts for example is unrelated to 'third-notes'.


title: Time signature

text: In particular, when the sign was encountered, the tactus (beat) changed from the usual whole note (semibreve) to the double whole note (breve), a circumstance called "alla breve". This term has been sustained to the present day, and though now it means the beat is a half note (minim), in contradiction to the literal meaning of the phrase, it still indicates that the "tactus" has changed from a short to a doubled value.


title: Time signature

text: Certain composers delighted in creating mensuration canons, "puzzle" compositions that were intentionally difficult to decipher.


title: Time signature

text: Irregular bars are a change in time signature normally for only one bar. Such a bar is most often a bar of 3/4, 5/4 or 2/4 in a 4/4 composition, or a bar of 4/4 in a 3/4 composition, or a bar of 5/8 in a 6/8 composition.


title: Time signature

text: If a song is entirely in 4/4 a change to 3/4 will make the song feel like it has skipped a beat, the opposite is true for 5/4 where it feels like the song adds a beat. If a song changes to 2/4 is will make it feel like that bar is half as long as all the others


title: Time signature

text: Some popular examples include "Golden Brown" by The Stranglers (4/4 in a 3/4 composition), "I Love Rock 'n' Roll" originally by the Arrows (3/4 in a 4/4 composition), "Hey Ya!" by Outkast (2/4 in a 4/4 composition), and "Wuthering Heights" by Kate Bush (different kinds of irregular bars in a 4/4 composition).


title: Emperor of China

text: Huangdi (), translated into English as Emperor, was the superlative title held by monarchs of China who ruled various imperial regimes in Chinese history. In traditional Chinese political theory, the emperor was considered the Son of Heaven and the autocrat of all under Heaven. Under the Han dynasty, Confucianism replaced Legalism as the official political theory and succession in most cases theoretically followed agnatic primogeniture. The lineage of emperors descended from a paternal family line constituted a dynasty.


title: Emperor of China

text: The absolute authority of the emperor came with a variety of governing duties and moral obligations; failure to uphold these was thought to remove the dynasty's Mandate of Heaven and to justify its overthrow. In practice, emperors sometimes avoided the strict rules of succession and dynasties' ostensible "failures" were detailed in official histories written by their successful replacements. The power of the emperor was also limited by the imperial bureaucracy, which was staffed by scholar-officials and in some dynasties eunuchs. An emperor was also constrained by filial obligations to his ancestors' policies and dynastic traditions, such as those first detailed in the Ming dynasty's "Ancestral Instructions".


title: Emperor of China

text: During the Zhou dynasty (c. 1046 BC – 256 BC), Chinese feudal rulers with power over their particular fiefdoms were called "gong" () but, as the power of the Shang and Zhou kings (,  "wang") waned, the dukes began to usurp that title for themselves. In 221 BCE, after the then-king of Qin completed the conquest of the various kingdoms of the Warring States period, he adopted a new title to reflect his prestige as a ruler greater than the rulers before him. He called himself "Shi Huangdi", the First Emperor. Before this, "Huang" () and "Di" () were the nominal "titles" of eight rulers of Chinese mythology or prehistory: The three "Huang" (, "august, sovereign") were godly rulers credited with feats like ordering the sky and forming the first humans out of clay; the five "Di" (, , also often translated as "emperor" but also meaning "the God of Heaven") were cultural heroes credited with the invention of agriculture, clothing, astrology, music, etc. In the 3rd century BCE, the two titles had not previously been used together. Because of the god-like powers of the "Huang", the folk worship of the "Di", and the latter's use in the name of the God of Heaven Shangdi, however, the First Emperor's title would have been understood as implying "The August Ancestor", "The Holy Ruler", or "The Divine Lord". On that account, some modern scholars translate the title as "thearch".


title: Emperor of China

text: On occasion, the father of the ascended emperor was still alive. Such an emperor was titled the "Taishang Huang" (), the "Grand Imperial Sire". The practice was initiated by the First Emperor, who gave the title as a posthumous name to his own father. Liu Bang, who established the Han dynasty, was the first to become emperor while his father yet lived. It was said he granted the title during his father's life because he would not be done obeisance to by his own father, a commoner.


title: Emperor of China

text: Owing to political fragmentation, over the centuries, it has not been uncommon to have numerous claimants to the title of "Emperor of All China". The Chinese political concept of the Mandate of Heaven essentially legitimized those claimants who emerged victorious. The proper list was considered those made by the official dynastic histories; the compilation of a history of the preceding dynasty was considered one of the hallmarks of legitimacy, along with symbols such as the Nine Ding or the Heirloom Seal of the Realm. As with the First Emperor, it was very common also to retroactively grant posthumous titles to the ancestors of the victors; even in Chinese historiography, however, such grants were not considered to elevate emperors prior to the successful declaration of a new dynasty.


title: Emperor of China

text: The Yuan and Qing dynasties were founded by successful invaders; as part of their rule over China, however, they also went through the rituals of formally declaring a new dynasty and taking on the Chinese title of "Huangdi", in addition to the titles of their respective people. Thus, Kublai Khan was simultaneously khagan of the Mongols and emperor of China.


title: Emperor of China

text: In 1911, the title of Prime Minister of the Imperial Cabinet was created to rule alongside the emperor, as part of an attempt to turn China into a constitutional monarchy.


title: Emperor of China

text: The Xuantong Emperor (Puyi) of the Qing dynasty was the de jure last emperor of China, abdicated on 12 February 1912, thus ending the imperial tradition after more than 2,100 years.


title: Emperor of China

text: Yuan Shikai, former President of the Republic of China, attempted to restore a monarchy with himself as the Hongxian Emperor, however his reign as emperor ended on 22 March 1916.


title: Emperor of China

text: Puyi was briefly restored for almost two weeks during a coup in 1917 but was overthrown again shortly after. He later became the emperor of Manchukuo, a Japanese puppet state, and was captured by the Red Army as a prisoner of war after World War II and held in Chita, Soviet Union. He was returned to China and imprisoned in Fushun War Criminals Management Centre, and after he was released lived until 1967. The current head of the family of the ex-Qing dynasty is Jin Yuzhang, who has worked for various local councils on China and makes no claim to the imperial throne.


title: Emperor of China

text: Confucian historiography holds that there can only be one legitimate Son of Heaven at any given time. However, identifying the "legitimate" emperor during times of division is not always uncontroversial, and therefore the exact number of legitimate emperors depends on where one stands on a number of succession disputes. The two most notable such controversies are whether Wei or Shu was the legitimate dynasty during the Three Kingdoms, and at what point the Song dynasty ceased to be the legitimate dynasty in favor of the Yuan dynasty. The Qing view, reported to Europe by the Jesuits, was that there had been 150 emperors from the First Emperor to the Kangxi Emperor. Adding the eight uncontroversial emperors that followed the Kangxi Emperor would give a grand total of 158 emperors from the First Emperor to Puyi.


title: Emperor of China

text: By one count, from the Qin dynasty to the Qing dynasty, there were a total 557 individuals who at one point or another claimed the title "emperor", several of them simultaneously. Some, such as Li Zicheng, Huang Chao, and Yuan Shu, declared themselves the emperors, Son of Heaven and founded their own empires as a rival government to challenge the legitimacy of and overthrow the existing emperor. Among the most famous emperors were Qin Shi Huang of the Qin dynasty, the emperors Gaozu and Wu of the Han dynasty, Emperor Taizong of the Tang dynasty, Kublai Khan of the Yuan dynasty, the Hongwu Emperor of the Ming dynasty, and the Kangxi Emperor of the Qing dynasty.


title: Emperor of China

text: The emperor's words were considered sacred edicts () and his written proclamations "directives from above" (). In theory, the emperor's orders were to be obeyed immediately. He was elevated above all commoners, nobility and members of the Imperial family. Addresses to the emperor were always to be formal and self-deprecatory, even by the closest of family members.


title: Emperor of China

text: In practice, however, the power of the emperor varied between different emperors and different dynasties. Generally, in the Chinese dynastic cycle, emperors founding a dynasty usually consolidated the empire through absolute rule: examples include Qin Shi Huang of the Qin, Emperor Gaozu of Han, Emperor Guangwu of Han, Emperor Taizong of the Tang, Kublai Khan of the Yuan, and the Kangxi Emperor of the Qing. These emperors ruled as absolute monarchs throughout their reign, maintaining a centralized grip on the country. During the Song dynasty, the emperor's power was significantly overshadowed by the power of the chancellor.


title: Emperor of China

text: The emperor's position, unless deposed in a rebellion, was always hereditary, usually by agnatic primogeniture. As a result, many emperors ascended the throne while still children. During minority reigns, the Empress Dowager (i.e., the emperor's mother) would usually possess significant political power. In fact, the vast majority of female rulers throughout Chinese Imperial history came to power by ruling as regents on behalf of their sons; prominent examples include the Empress Lü of the Han dynasty, as well as Empress Dowager Cixi and Empress Dowager Ci'an of the Qing dynasty, who for a time ruled jointly as co-regents. Where Empresses Dowager were too weak to assume power, court officials often seized control. Court eunuchs had a significant role in the power structure, as emperors often relied on a few of them as confidants, which gave them access to many court documents. In a few places, eunuchs wielded vast power; one of the most powerful eunuchs in Chinese history was Wei Zhongxian during the Ming dynasty. Occasionally, other nobles seized power as regents. The actual area ruled by the Emperor of China varied from dynasty to dynasty. In some cases, such as during the Southern Song dynasty, political power in East Asia was effectively split among several governments; nonetheless, the political fiction that there was but one ruler was maintained.


title: Emperor of China

text: The title of "emperor" was hereditary, traditionally passed on from father to son in each dynasty. There are also instances where the throne is assumed by a younger brother, should the deceased emperor have no male offspring. By convention in most dynasties, the eldest son born to the Empress () succeeded to the throne. In some cases when the empress did not bear any children, the emperor would have a child with another of his many wives (all children of the emperor were said also to be the children of the empress, regardless of birth mother). In some dynasties the succession of the empress' eldest son was disputed, and because many emperors had large numbers of progeny, there were wars of succession between rival sons. In an attempt to resolve after-death disputes, the emperor, while still living, often designated a Crown Prince (). Even such a clear designation, however, was often thwarted by jealousy and distrust, whether it was the crown prince plotting against the emperor, or brothers plotting against each other. Some emperors, like the Yongzheng Emperor, after abolishing the position of Crown Prince, placed the succession papers in a sealed box, only to be opened and announced after his death.


title: Emperor of China

text: Unlike, for example, the Japanese monarchy, Chinese political theory allowed for a change in the ruling house. This was based on the concept of the "Mandate of Heaven". The theory behind this was that the Chinese emperor acted as the "Son of Heaven" and held a mandate to rule over everyone else in the world; but only as long as he served the people well. If the quality of rule became questionable because of repeated natural disasters such as flood or famine, or for other reasons, then rebellion was justified. This important concept legitimized the dynastic cycle or the change of dynasties.


title: Emperor of China

text: This principle made it possible even for peasants to found new dynasties, as happened with the Han and Ming dynasties, and for the establishment of conquest dynasties such as the Mongol-led Yuan dynasty and Manchu-led Qing dynasty. It was moral integrity and benevolent leadership that determined the holder of the "Mandate of Heaven".


title: Emperor of China

text: There has been only one lawful female reigning emperor in China, Empress Zetian, who briefly replaced the Tang dynasty with her own Zhou dynasty. Many women, however, did become "de facto" leaders, usually as Empress Dowager. Prominent examples include Empress Dowager Lü of the Han dynasty, Empress Dowager Liu of the Sung dynasty and Empress Dowager Cixi of the Qing dynasty.


title: Emperor of China

text: As the emperor had, by law, an absolute position not to be challenged by anyone else, his or her subjects were to show the utmost respect in his or her presence, whether in direct conversation or otherwise. When approaching the Imperial throne, one was expected to kowtow before the emperor. In a conversation with the emperor, it was considered a crime to compare oneself to the emperor in any way. It was taboo to refer to the emperor by his or her given name, even for the emperor's own mother, who instead was to use "Huángdì" (), or simply "Ér" (, "son", for male emperor). The emperor was never to be addressed as "you". Anyone who spoke to the emperor was to address him or her as "Bìxià" (, the "Bottom of the Steps"), corresponding to "Your Imperial Majesty"; "Huángshàng" (, lit. "Radiant Highness"); "Shèngshàng" (, "Holy Highness"); or "Tiānzǐ" (, "Son of Heaven"). The emperor could also be alluded to indirectly through reference to the imperial dragon symbology. Servants often addressed the emperor as "Wànsuìyé" (, lit. "Lord of Ten Thousand Years"). The emperor referred to himself or herself as "zhèn" (), the original Chinese first-person singular arrogated by the First Emperor, functioning as an equivalent to the "Royal We", or, self-deprecatingly, "Guǎrén" (, the "Morally-Deficient One") in front of his or her subjects.


title: Emperor of China

text: In contrast to the Western convention of referring to a sovereign using a regnal name (e.g. George V) or by a personal name (e.g. Queen Victoria), a governing emperor was to be referred to simply as "Huángdì Bìxià" (, "Majesty|His/Her Majesty the Emperor") or "Dāngjīn Huángshàng" (, "The Present Emperor Above") when spoken about in the third person. Under the Qing, the emperor was usually styled "His Imperial Majesty the Emperor of the Great Qing Dynasty, Son of Heaven, Lord of Ten Thousand Years" although this varied considerably.


title: Emperor of China

text: Generally, emperors also ruled with an era name (). Since the adoption of era name by Emperor Wu of Han and up until the Ming dynasty, the sovereign conventionally changed the era name semi-regularly during his or her reign. During the Ming and Qing dynasties, emperors simply chose one era name for their entire reign, and people often referred to past emperors with that title. In earlier dynasties, the emperors were known with a temple name () given after their death. Most emperors were also given a posthumous name (, Shìhào), which was sometimes combined with the temple name (e.g. Emperor Shèngzǔ Rén for the Kangxi Emperor). The passing of an emperor was referred to as "Jiàbēng" (, lit. "collapse of the [imperial] chariot") and an emperor that had just died was referred to as "Dàxíng Huángdì" (), literally "the Emperor of the Great Journey."


title: Emperor of China

text: The imperial family was made up of the emperor and the empress () as the primary consort and Mother of the Nation (). In addition, the emperor would typically have several other consorts and concubines (), ranked by importance into a harem, in which the Empress was supreme. Every dynasty had its set of rules regarding the numerical composition of the harem. During the Qing dynasty, for example, imperial convention dictated that at any given time there should be one Empress, one "Huang Guifei", two "Guifei", four "fei" and six "pin", plus an unlimited number of other consorts and concubines. Although the emperor had the highest status by law, by tradition and precedent the mother of the emperor, "i.e.", the empress dowager (), usually received the greatest respect in the palace and was the decision maker in most family affairs. At times, especially when a young emperor was on the throne, she was the "de facto" ruler. The emperor's children, the princes () and princesses (), were often referred to by their order of birth, "e.g.", Eldest Prince, Third Princess, etc. The princes were often given titles of peerage once they reached adulthood. The emperor's brothers and uncles served in court by law, and held equal status with other court officials (). The emperor was always elevated above all others despite any chronological or generational superiority.


title: Emperor of China

text: Recent scholarship is wary of applying present-day ethnic categories to historical situations. Most Chinese emperors have been considered members of the Han ethnicity, but there were also many Chinese emperors who were of non-Han ethnic origins. The most successful of these were the Khitans (Liao dynasty), Jurchens (Jin dynasty), Mongols (Yuan dynasty), and Manchus (Qing dynasty). The orthodox historical view sees these as dynasties as sinicized polities as they adopted Han culture, claimed the Mandate of Heaven, and performed the traditional imperial obligations such as annual sacrifices to Heaven (as Tian or Shangdi) for rain and prosperity. The revisionist New Qing History school, however, argues that the interaction between politics and ethnicity was far more complex and that elements of these dynasties differed from and altered "native Chinese" traditions concerning imperial rule.


title: IEEE 754

text: The IEEE Standard for Floating-Point Arithmetic (IEEE 754) is a technical standard for floating-point arithmetic established in 1985 by the Institute of Electrical and Electronics Engineers (IEEE). The standard addressed many problems found in the diverse floating-point implementations that made them difficult to use reliably and portably. Many hardware floating-point units use the IEEE 754 standard.


title: IEEE 754

text: IEEE 754-2008, published in August 2008, includes nearly all of the original IEEE 754-1985 standard, plus the IEEE 854-1987 Standard for Radix-Independent Floating-Point Arithmetic. The current version, IEEE 754-2019, was published in July 2019. It is a minor revision of the previous version, incorporating mainly clarifications, defect fixes and new recommended operations.


title: IEEE 754

text: The first standard for floating-point arithmetic, IEEE 754-1985, was published in 1985. It covered only binary floating-point arithmetic.


title: IEEE 754

text: A new version, IEEE 754-2008, was published in August 2008, following a seven-year revision process, chaired by Dan Zuras and edited by Mike Cowlishaw. It replaced both IEEE 754-1985 (binary floating-point arithmetic) and IEEE 854-1987 Standard for Radix-Independent Floating-Point Arithmetic. The binary formats in the original standard are included in this new standard along with three new basic formats, one binary and two decimal. To conform to the current standard, an implementation must implement at least one of the basic formats as both an arithmetic format and an interchange format.


title: IEEE 754

text: The international standard ISO/IEC/IEEE 60559:2011 (with content identical to IEEE 754-2008) has been approved for adoption through ISO/IEC JTC 1/SC 25 under the ISO/IEEE PSDO Agreement and published.


title: IEEE 754

text: The current version, IEEE 754-2019 published in July 2019, is derived from and replaces IEEE 754-2008, following a revision process started in September 2015, chaired by David G. Hough and edited by Mike Cowlishaw. It incorporates mainly clarifications (e.g. "totalOrder") and defect fixes (e.g. "minNum"), but also includes some new recommended operations (e.g. "augmentedAddition").


title: IEEE 754

text: The international standard ISO/IEC 60559:2020 (with content identical to IEEE 754-2019) has been approved for adoption through ISO/IEC JTC 1/SC 25 and published.


title: IEEE 754

text: An IEEE 754 "format" is a "set of representations of numerical values and symbols". A format may also include how the set is encoded.


title: IEEE 754

text: For example, if "b" = 10, "p" = 7, and "emax" = 96, then "emin" = −95, the significand satisfies 0 ≤ "c" ≤ , and the exponent satisfies −101 ≤ "q" ≤ 90. Consequently, the smallest non-zero positive number that can be represented is 1×10, and the largest is 9999999×10 (9.999999×10), so the full range of numbers is −9.999999×10 through 9.999999×10. The numbers −"b" and "b" (here, −1×10 and 1×10) are the smallest (in magnitude) "normal numbers"; non-zero numbers between these smallest numbers are called subnormal numbers.


title: IEEE 754

text: Some numbers may have several possible exponential format representations. For instance, if "b" = 10, and "p" = 7, then −12.345 can be represented by −12345×10, −123450×10, and −1234500×10. However, for most operations, such as arithmetic operations, the result (value) does not depend on the representation of the inputs.


title: IEEE 754

text: For the decimal formats, any representation is valid, and the set of these representations is called a "cohort". When a result can have several representations, the standard specifies which member of the cohort is chosen.


title: IEEE 754

text: For the binary formats, the representation is made unique by choosing the smallest representable exponent allowing the value to be represented exactly. Further, the exponent is not represented directly, but a bias is added so that the smallest representable exponent is represented as 1, with 0 used for subnormal numbers. For numbers with an exponent in the normal range (the exponent field being neither all ones nor all zeros), the leading bit of the significand will always be 1. Consequently, a leading 1 can be implied rather than explicitly present in the memory encoding, and under the standard the explicitly represented part of the significand will lie between 0 and 1. This rule is called "leading bit convention", "implicit bit convention", or "hidden bit convention". This rule allows the binary format to have an extra bit of precision. The leading bit convention cannot be used for the subnormal numbers as they have an exponent outside the normal exponent range and scale by the smallest represented exponent as used for the smallest normal numbers.


title: IEEE 754

text: Due to the possibility of multiple encodings (at least in formats called "interchange formats"), a NaN may carry other information: a sign bit (which has no meaning, but may be used by some operations) and a "payload", which is intended for diagnostic information indicating the source of the NaN (but the payload may have other uses, such as "NaN-boxing").


title: IEEE 754

text: The standard defines five basic formats that are named for their numeric base and the number of bits used in their interchange encoding. There are three binary floating-point basic formats (encoded with 32, 64 or 128 bits) and two decimal floating-point basic formats (encoded with 64 or 128 bits). The binary32 and binary64 formats are the "single" and "double" formats of IEEE 754-1985 respectively. A conforming implementation must fully implement at least one of the basic formats.


title: IEEE 754

text: The standard also defines "interchange formats", which generalize these basic formats. For the binary formats, the leading bit convention is required. The following table summarizes the smallest interchange formats (including the basic ones).


title: IEEE 754

text: Note that in the table above, the minimum exponents listed are for normal numbers; the special subnormal number representation allows even smaller numbers to be represented (with some loss of precision). For example, the smallest positive number that can be represented in binary64 is 2; contributions to the −1074 figure include the E min value −1022 and all but one of the 53 significand bits (2 = 2).


title: IEEE 754

text: Decimal digits is "digits" × log "base". This gives an approximate precision in number of decimal digits.


title: IEEE 754

text: Decimal E max is "Emax" × log "base". This gives an approximate value of the maximum decimal exponent.


title: IEEE 754

text: The binary32 (single) and binary64 (double) formats are two of the most common formats used today. The figure below shows the absolute precision for both formats over a range of values. This figure can be used to select an appropriate format given the expected value of a number and the required precision.


title: IEEE 754

text: The standard specifies optional extended and extendable precision formats, which provide greater precision than the basic formats. An extended precision format extends a basic format by using more precision and more exponent range. An extendable precision format allows the user to specify the precision and exponent range. An implementation may use whatever internal representation it chooses for such formats; all that needs to be defined are its parameters ("b", "p", and "emax"). These parameters uniquely describe the set of finite numbers (combinations of sign, significand, and exponent for the given radix) that it can represent.


title: IEEE 754

text: The standard recommends that language standards provide a method of specifying "p" and "emax" for each supported base "b". The standard recommends that language standards and implementations support an extended format which has a greater precision than the largest basic format supported for each radix "b". For an extended format with a precision between two basic formats the exponent range must be as great as that of the next wider basic format. So for instance a 64-bit extended precision binary number must have an 'emax' of at least 16383. The x87 80-bit extended format meets this requirement.


title: IEEE 754

text: Interchange formats are intended for the exchange of floating-point data using a bit string of fixed length for a given format.


title: IEEE 754

text: For the exchange of binary floating-point numbers, interchange formats of length 16 bits, 32 bits, 64 bits, and any multiple of 32 bits ≥ 128 are defined. The 16-bit format is intended for the exchange or storage of small numbers (e.g., for graphics).


title: IEEE 754

text: The encoding scheme for these binary interchange formats is the same as that of IEEE 754-1985: a sign bit, followed by "w" exponent bits that describe the exponent offset by a "bias", and "p" − 1 bits that describe the significand. The width of the exponent field for a "k"-bit format is computed as "w" = round(4 log("k")) − 13. The existing 64- and 128-bit formats follow this rule, but the 16- and 32-bit formats have more exponent bits (5 and 8 respectively) than this formula would provide (3 and 7 respectively).


title: IEEE 754

text: As with IEEE 754-1985, the biased-exponent field is filled with all 1 bits to indicate either infinity (trailing significand field = 0) or a NaN (trailing significand field ≠ 0). For NaNs, quiet NaNs and signaling NaNs are distinguished by using the most significant bit of the trailing significand field exclusively, and the payload is carried in the remaining bits.


title: IEEE 754

text: For the exchange of decimal floating-point numbers, interchange formats of any multiple of 32 bits are defined. As with binary interchange, the encoding scheme for the decimal interchange formats encodes the sign, exponent, and significand. Two different bit-level encodings are defined, and interchange is complicated by the fact that some external indicator of the encoding in use may be required.


title: IEEE 754

text: The two options allow the significand to be encoded as a compressed sequence of decimal digits using densely packed decimal or, alternatively, as a binary integer. The former is more convenient for direct hardware implementation of the standard, while the latter is more suited to software emulation on a binary computer. In either case, the set of numbers (combinations of sign, significand, and exponent) that may be encoded is identical, and special values (±zero with the minimum exponent, ±infinity, quiet NaNs, and signaling NaNs) have identical encodings.


title: IEEE 754

text: The standard defines five rounding rules. The first two rules round to a nearest value; the others are called "directed roundings":


title: IEEE 754

text: At the extremes, a value with a magnitude strictly less than formula_1 will be rounded to the minimum or maximum finite number (depending on the value's sign). Any numbers with exactly this magnitude are considered ties; this choice of tie may be conceptualized as the midpoint between formula_2 and formula_3, which, were the exponent not limited, would be the next representable floating-point numbers larger in magnitude. Numbers with a magnitude strictly larger than formula_4 are rounded to the corresponding infinity.


title: IEEE 754

text: Unless specified otherwise, the floating-point result of an operation is determined by applying the rounding function on the infinitely precise (mathematical) result. Such an operation is said to be "correctly rounded". This requirement is called "correct rounding".


title: IEEE 754

text: The standard provides comparison predicates to compare one floating-point datum to another in the supported arithmetic format. Any comparison with a NaN is treated as unordered. −0 and +0 compare as equal.


title: IEEE 754

text: The standard provides a predicate "totalOrder", which defines a total ordering on canonical members of the supported arithmetic format. The predicate agrees with the comparison predicates when one floating-point number is less than the other. The "totalOrder" predicate does not impose a total ordering on all encodings in a format. In particular, it does not distinguish among different encodings of the same floating-point representation, as when one or both encodings are non-canonical. IEEE 754-2019 incorporates clarifications of "totalOrder".


title: IEEE 754

text: For the binary interchange formats whose encoding follows the IEEE 754-2008 recommendation on placement of the NaN signaling bit, the comparison is identical to one that type puns the floating-point numbers to a sign–magnitude integer (assuming a payload ordering consistent with this comparison), an old trick for FP comparison without an FPU.


title: IEEE 754

text: The standard defines five exceptions, each of which returns a default value and has a corresponding status flag that is raised when the exception occurs. No other exception handling is required, but additional non-default alternatives are recommended (see ).


title: IEEE 754

text: These are the same five exceptions as were defined in IEEE 754-1985, but the "division by zero" exception has been extended to operations other than the division.


title: IEEE 754

text: Some decimal floating-point implementations define additional exceptions, which are not part of IEEE 754:


title: IEEE 754

text: Additionally, operations like quantize when either operand is infinite, or when the result does not fit the destination format, will also signal invalid operation exception.


title: IEEE 754

text: The standard recommends optional exception handling in various forms, including presubstitution of user-defined default values, and traps (exceptions that change the flow of control in some way) and other exception handling models that interrupt the flow, such as try/catch. The traps and other exception mechanisms remain optional, as they were in IEEE 754-1985.


title: IEEE 754

text: Clause 9 in the standard recommends additional mathematical operations that language standards should define. None are required in order to conform to the standard.


title: IEEE 754

text: The , and functions were not part of the IEEE 754-2008 standard because they were deemed less necessary. , were mentioned, but this was regarded as an error. All three were added in the 2019 revision.


title: IEEE 754

text: The recommended operations also include setting and accessing dynamic mode rounding direction, and implementation-defined vector reduction operations such as sum, scaled product, and dot product, whose accuracy is unspecified by the standard.


title: IEEE 754

text: , "augmented arithmetic operations" for the binary formats are also recommended. These operations, specified for addition, subtraction and multiplication, produce a pair of values consisting of a result correctly rounded to nearest in the format and the error term, which is representable exactly in the format. At the time of publication of the standard, no hardware implementations are known, but very similar operations were already implemented in software using well-known algorithms. The history and motivation for their standardization are explained in a background document.


title: IEEE 754

text: As of 2019, the formerly required "minNum", "maxNum", "minNumMag", and "maxNumMag" in IEEE 754-2008 are now deprecated due to their non-associativity. Instead, two sets of new minimum and maximum operations are recommended. The first set contains "minimum", "minimumNumber", "maximum" and "maximumNumber". The second set contains "minimumMagnitude", "minimumMagnitudeNumber", "maximumMagnitude" and "maximumMagnitudeNumber". The history and motivation for this change are explained in a background document.


title: IEEE 754

text: The standard recommends how language standards should specify the semantics of sequences of operations, and points out the subtleties of literal meanings and optimizations that change the value of a result. By contrast, the previous 1985 version of the standard left aspects of the language interface unspecified, which led to inconsistent behavior between compilers, or different optimization levels in an optimizing compiler.


title: IEEE 754

text: Programming languages should allow a user to specify a minimum precision for intermediate calculations of expressions for each radix. This is referred to as "preferredWidth" in the standard, and it should be possible to set this on a per-block basis. Intermediate calculations within expressions should be calculated, and any temporaries saved, using the maximum of the width of the operands and the preferred width if set. Thus, for instance, a compiler targeting x87 floating-point hardware should have a means of specifying that intermediate calculations must use the double-extended format. The stored value of a variable must always be used when evaluating subsequent expressions, rather than any precursor from before rounding and assigning to the variable.


title: IEEE 754

text: The IEEE 754-1985 version of the standard allowed many variations in implementations (such as the encoding of some values and the detection of certain exceptions). IEEE 754-2008 has reduced these allowances, but a few variations still remain (especially for binary formats). The reproducibility clause recommends that language standards should provide a means to write reproducible programs (i.e., programs that will produce the same result in all implementations of a language) and describes what needs to be done to achieve reproducible results.


title: IEEE 754

text: The standard requires operations to convert between basic formats and "external character sequence" formats. Conversions to and from a decimal character format are required for all formats. Conversion to an external character sequence must be such that conversion back using round to nearest, ties to even will recover the original number. There is no requirement to preserve the payload of a quiet NaN or signaling NaN, and conversion from the external character sequence may turn a signaling NaN into a quiet NaN.


title: IEEE 754

text: Algorithms, with code, for correctly rounded conversion from binary to decimal and decimal to binary are discussed by Gay, and for testing by Paxson and Kahan.


title: IEEE 754

text: The standard recommends providing conversions to and from "external hexadecimal-significand character sequences", based on C99's hexadecimal floating point literals. Such a literal consists of an optional sign (codice_1 or codice_2), the indicator "0x", a hexadecimal number with or without a period, an exponent indicator "p", and a decimal exponent with optional sign. The syntax is not case-sensitive. The decimal exponent scales by powers of 2, so for example codice_3 is 1/256.


title: Leibniz integral rule

text: In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form


title: Leibniz integral rule

text: where formula_2 and the integral are functions dependent on formula_3 the derivative of this integral is expressible as


title: Leibniz integral rule

text: where the partial derivative formula_5 indicates that inside the integral, only the variation of formula_6 with formula_7 is considered in taking the derivative.


title: Leibniz integral rule

text: In the special case where the functions formula_8 and formula_9 are constants formula_10 and formula_11 with values that do not depend on formula_3 this simplifies to:


title: Leibniz integral rule

text: If formula_10 is constant and formula_15, which is another common situation (for example, in the proof of Cauchy's repeated integration formula), the Leibniz integral rule becomes:


title: Leibniz integral rule

text: This important result may, under certain conditions, be used to interchange the integral and partial differential operators, and is particularly useful in the differentiation of integral transforms. An example of such is the moment generating function in probability theory, a variation of the Laplace transform, which can be differentiated to generate the moments of a random variable. Whether Leibniz's integral rule applies is essentially a question about the interchange of limits.


title: Leibniz integral rule

text: Stronger versions of the theorem only require that the partial derivative exist almost everywhere, and not that it be continuous. This formula is the general form of the Leibniz integral rule and can be derived using the fundamental theorem of calculus. The (first) fundamental theorem of calculus is just the particular case of the above formula where formula_17, formula_18, and formula_19.


title: Leibniz integral rule

text: If both upper and lower limits are taken as constants, then the formula takes the shape of an operator equation:


title: Leibniz integral rule

text: where formula_21 is the partial derivative with respect to formula_7 and formula_23 is the integral operator with respect to formula_24 over a fixed interval. That is, it is related to the symmetry of second derivatives, but involving integrals as well as derivatives. This case is also known as the Leibniz integral rule.


title: Leibniz integral rule

text: The Leibniz integral rule can be extended to multidimensional integrals. In two and three dimensions, this rule is better known from the field of fluid dynamics as the Reynolds transport theorem:


title: Leibniz integral rule

text: where formula_27 is a scalar function, and denote a time-varying connected region of R and its boundary, respectively, formula_28 is the Eulerian velocity of the boundary (see Lagrangian and Eulerian coordinates) and is the unit normal component of the surface element.


title: Leibniz integral rule

text: The general statement of the Leibniz integral rule requires concepts from differential geometry, specifically differential forms, exterior derivatives, wedge products and interior products. With those tools, the Leibniz integral rule in "n" dimensions is


title: Leibniz integral rule

text: where is a time-varying domain of integration, "ω" is a "p"-form, formula_30 is the vector field of the velocity, formula_31 denotes the interior product with formula_32, "d""ω" is the exterior derivative of "ω" with respect to the space variables only and formula_33 is the time derivative of "ω".


title: Leibniz integral rule

text: Here, the ambient manifold on which the differential form formula_35 lives includes both space and time.


title: Leibniz integral rule

text: Something remarkable about this form, is that it can account for the case when formula_36 changes its shape and size over time, since such deformations are fully determined by formula_39.


title: Leibniz integral rule

text: Let formula_49 be an open subset of formula_50, and formula_36 be a measure space. Suppose formula_52 satisfies the following conditions:


title: Leibniz integral rule

text: We use Fubini's theorem to change the order of integration. For every and , such that and both and are within , we have:


title: Leibniz integral rule

text: Note that the integrals at hand are well defined since formula_66 is continuous at the closed rectangle formula_67 and thus also uniformly continuous there; thus its integrals by either "dt" or "dx" are continuous in the other variable and also integrable by it (essentially this is because for uniformly continuous functions, one may pass the limit through the integration sign, as elaborated below).


title: Leibniz integral rule

text: "F" is differentiable with derivative formula_70, so we can take the limit where approaches zero. For the left hand side this limit is:


title: Leibniz integral rule

text: If the integrals at hand are Lebesgue integrals, we may use the bounded convergence theorem (valid for these integrals, but not for Riemann integrals) in order to show that the limit can be passed through the integral sign.


title: Leibniz integral rule

text: Note that this proof is weaker in the sense that it only shows that "f"("x","t") is Lebesgue integrable, but not that it is Riemann integrable. In the former (stronger) proof, if "f"("x","t") is Riemann integrable, then so is "f"("x","t") (and thus is obviously also Lebesgue integrable).


title: Leibniz integral rule

text: Substitute equation () into equation (). The difference of two integrals equals the integral of the difference, and 1/"h" is a constant, so


title: Leibniz integral rule

text: We claim that the passage of the limit under the integral sign is valid by the bounded convergence theorem (a corollary of the dominated convergence theorem). For each "δ" > 0, consider the difference quotient


title: Leibniz integral rule

text: For "t" fixed, the mean value theorem implies there exists "z" in the interval ["x", "x" + "δ"] such that


title: Leibniz integral rule

text: Continuity of "f"("x", "t") and compactness of the domain together imply that "f"("x", "t") is bounded. The above application of the mean value theorem therefore gives a uniform (independent of formula_24) bound on formula_78. The difference quotients converge pointwise to the partial derivative "f" by the assumption that the partial derivative exists.


title: Leibniz integral rule

text: The above argument shows that for every sequence {"δ"} → 0, the sequence formula_79 is uniformly bounded and converges pointwise to "f". The bounded convergence theorem states that if a sequence of functions on a set of finite measure is uniformly bounded and converges pointwise, then passage of the limit under the integral is valid. In particular, the limit and integral may be exchanged for every sequence {"δ"} → 0. Therefore, the limit as "δ" → 0 may be passed through the integral sign.


title: Leibniz integral rule

text: For a continuous real valued function "g" of one real variable, and real valued differentiable functions formula_80 and formula_81 of one real variable,


title: Leibniz integral rule

text: By the First Fundamental Theorem of Calculus, formula_89. Therefore, substituting this result above, we get the desired equation:


title: Leibniz integral rule

text: Because formula_92 does not depend on the limits of integration, it may be moved out from under the integral sign, and the above form may be used with the Product rule, i.e.,


title: Leibniz integral rule

text: where "a" and "b" are functions of "α" that exhibit increments Δ"a" and Δ"b", respectively, when "α" is increased by Δ"α". Then,


title: Leibniz integral rule

text: A form of the mean value theorem, formula_96, where "a" < "ξ" < "b", may be applied to the first and last integrals of the formula for Δ"φ" above, resulting in


title: Leibniz integral rule

text: Divide by Δ"α" and let Δ"α" → 0. Notice "ξ" → "a" and "ξ" → "b". We may pass the limit through the integral sign:


title: Leibniz integral rule

text: The general form of Leibniz's Integral Rule with variable limits can be derived as a consequence of the basic form of Leibniz's Integral Rule, the multivariable chain rule, and the First Fundamental Theorem of Calculus. Suppose formula_100 is defined in a rectangle in the formula_101 plane, for formula_102 and formula_103. Also, assume formula_100 and the partial derivative formula_105 are both continuous functions on this rectangle. Suppose formula_106 are differentiable real valued functions defined on formula_107, with values in formula_108 (i.e. for every formula_109). Now, set


title: Leibniz integral rule

text: Since the functions formula_113 are all differentiable (see the remark at the end of the proof), by the Multivariable Chain Rule, it follows that formula_114 is differentiable, and its derivative is given by the formula:


title: Leibniz integral rule

text: Now, note that for every formula_102, and for every formula_117, we have that formula_118, because when taking the partial derivative with respect to formula_119 of formula_120, we are keeping formula_121 fixed in the expression formula_122; thus the basic form of Leibniz's Integral Rule with constant limits of integration applies. Next, by the First Fundamental Theorem of Calculus, we have that formula_123; because when taking the partial derivative with respect to formula_121 of formula_120, the first variable formula_119 is fixed, so the fundamental theorem can indeed be applied.


title: Leibniz integral rule

text: There is a technical point in the proof above which is worth noting: applying the Chain Rule to formula_114 requires that formula_120 already be differentiable. This is where we use our assumptions about formula_100. As mentioned above, the partial derivatives of formula_120 are given by the formulas formula_118 and formula_123. Since formula_135 is continuous, its integral is also a continuous function, and since formula_100 is also continuous, these two results show that both the partial derivatives of formula_120 are continuous. Since continuity of partial derivatives implies differentiability of the function, formula_120 is indeed differentiable.


title: Leibniz integral rule

text: At time "t" the surface Σ in Figure 1 contains a set of points arranged about a centroid formula_139. The function formula_140 can be written as


title: Leibniz integral rule

text: with formula_142 independent of time. Variables are shifted to a new frame of reference attached to the moving surface, with origin at formula_139. For a rigidly translating surface, the limits of integration are then independent of time, so:


title: Leibniz integral rule

text: where the limits of integration confining the integral to the region Σ no longer are time dependent so differentiation passes through the integration to act on the integrand only:


title: Leibniz integral rule

text: This equation expresses the material derivative of the field, that is, the derivative with respect to a coordinate system attached to the moving surface. Having found the derivative, variables can be switched back to the original frame of reference. We notice that (see article on curl)


title: Leibniz integral rule

text: The sign of the line integral is based on the right-hand rule for the choice of direction of line element "ds. To establish this sign, for example, suppose the field F points in the positive "z"-direction, and the surface Σ is a portion of the "xy"-plane with perimeter ∂Σ. We adopt the normal to Σ to be in the positive "z"-direction. Positive traversal of ∂Σ is then counterclockwise (right-hand rule with thumb along "z"-axis). Then the integral on the left-hand side determines a "positive" flux of F through Σ. Suppose Σ translates in the positive "x"-direction at velocity v. An element of the boundary of Σ parallel to the "y"-axis, say "ds, sweeps out an area vt" × "ds in time "t". If we integrate around the boundary ∂Σ in a counterclockwise sense, vt" × "ds points in the negative "z"-direction on the left side of ∂Σ (where "ds points downward), and in the positive "z"-direction on the right side of ∂Σ (where "ds points upward), which makes sense because Σ is moving to the right, adding area on the right and losing it on the left. On that basis, the flux of F is increasing on the right of ∂Σ and decreasing on the left. However, the dot product . Consequently, the sign of the line integral is taken as negative.


title: Leibniz integral rule

text: which is the quoted result. This proof does not consider the possibility of the surface deforming as it moves.


title: Leibniz integral rule

text: Suppose "a" and "b" are constant, and that "f"("x") involves a parameter "α" which is constant in the integration but may vary to form different integrals. Assume that "f"("x", "α") is a continuous function of "x" and "α" in the compact set {("x", "α") : "α" ≤ "α" ≤ "α" and "a" ≤ "x" ≤ "b"}, and that the partial derivative "f"("x", "α") exists and is continuous. If one defines:


title: Leibniz integral rule

text: then formula_154 may be differentiated with respect to "α" by differentiating under the integral sign, i.e.,


title: Leibniz integral rule

text: By the Heine–Cantor theorem it is uniformly continuous in that set. In other words, for any "ε" > 0 there exists Δ"α" such that for all values of "x" in ["a", "b"],


title: Leibniz integral rule

text: Given formula_173, this integral reduction formula can be used to compute all of the values of formula_174 for formula_175. Integrals like formula_142 and formula_177 may also be handled using the Weierstrass substitution.


title: Leibniz integral rule

text: When "φ" = 1 this equals the original integral. However, this more general integral may be differentiated with respect to formula_154:


title: Leibniz integral rule

text: Now, fix "φ", and consider the vector field on formula_189 defined by formula_190. Further, choose the positive oriented parametrization of the unit circle formula_191 given by formula_192, formula_193, so that formula_194. Then the final integral above is precisely


title: Leibniz integral rule

text: the line integral of formula_196 over formula_197. By Green's Theorem, this equals the double integral


title: Leibniz integral rule

text: where formula_199 is the closed unit disc. Its integrand is identically 0, so formula_200 is likewise identically zero. This implies that "f"("φ") is constant. The constant may be determined by evaluating formula_201 at formula_202:


title: Leibniz integral rule

text: There are innumerable other integrals that can be solved using the technique of differentiation under the integral sign. For example, in each of the following cases, the original integral may be replaced by a similar integral having a new parameter formula_179:


title: Leibniz integral rule

text: The first integral, the Dirichlet integral, is absolutely convergent for positive "α" but only conditionally convergent when formula_207. Therefore, differentiation under the integral sign is easy to justify when formula_208, but proving that the resulting formula remains valid when formula_207 requires some careful work.


title: Leibniz integral rule

text: The measure-theoretic version of differentiation under the integral sign also applies to summation (finite or infinite) by interpreting summation as counting measure. An example of an application is the fact that power series are differentiable in their radius of convergence.


title: Leibniz integral rule

text: Differentiation under the integral sign is mentioned in the late physicist Richard Feynman's best-selling memoir "Surely You're Joking, Mr. Feynman!" in the chapter "A Different Box of Tools". He describes learning it, while in high school, from an old text, "Advanced Calculus" (1926), by Frederick S. Woods (who was a professor of mathematics in the Massachusetts Institute of Technology). The technique was not often taught when Feynman later received his formal education in calculus, but using this technique, Feynman was able to solve otherwise difficult integration problems upon his arrival at graduate school at Princeton University:


title: Leibniz integral rule

text: One thing I never did learn was contour integration. I had learned to do integrals by various methods shown in a book that my high school physics teacher Mr. Bader had given me. One day he told me to stay after class. "Feynman," he said, "you talk too much and you make too much noise. I know why. You're bored. So I'm going to give you a book. You go up there in the back, in the corner, and study this book, and when you know everything that's in this book, you can talk again." So every physics class, I paid no attention to what was going on with Pascal's Law, or whatever they were doing. I was up in the back with this book: "Advanced Calculus", by Woods. Bader knew I had studied "Calculus for the Practical Man" a little bit, so he gave me the real works—it was for a junior or senior course in college. It had Fourier series, Bessel functions, determinants, elliptic functions—all kinds of wonderful stuff that I didn't know anything about. That book also showed how to differentiate parameters under the integral sign—it's a certain operation. It turns out that's not taught very much in the universities; they don't emphasize it. But I caught on how to use that method, and I used that one damn tool again and again. So because I was self-taught using that book, I had peculiar methods of doing integrals. The result was, when guys at MIT or Princeton had trouble doing a certain integral, it was because they couldn't do it with the standard methods they had learned in school. If it was contour integration, they would have found it; if it was a simple series expansion, they would have found it. Then I come along and try differentiating under the integral sign, and often it worked. So I got a great reputation for doing integrals, only because my box of tools was different from everybody else's, and they had tried all their tools on it before giving the problem to me.


title: Nissan 370Z

text: The Nissan 370Z (known as the Fairlady Z Z34 in Japan) is a 2-door, 2-seater sports car (S-segment in Europe) manufactured by Nissan Motor Company. It was announced on October 29, 2006, and was first shown at an event in Los Angeles ahead of the 2008 Greater LA Auto Show, before being officially unveiled at the show itself. The 370Z is the sixth generation of the Nissan Z-car line, succeeding the 350Z.


title: Nissan 370Z

text: The 370Z marks the last production car with a naturally aspirated and high-rev V6 coupled to a manual transmission. The 2020 model year was the final model year for the 370Z. The line was continued by the Nissan Z (RZ34) on a modified version of the same platform.


title: Nissan 370Z

text: Almost every piece and component of the 370Z has been redesigned from the previous 350Z. The wheelbase is shorter at and an overall length shorter at . The overall width has been increased by , the rear track by , and overall height reduced by . The smaller exterior dimensions and use of more lightweight materials helped reduce weight.


title: Nissan 370Z

text: The 370Z features a front aluminum subframe, aluminum-alloy engine cradle, aluminum door panels, an all-aluminum hood, and an aluminum hatch. Front body torsional rigidity is improved by 10 percent with an extensively revised body structure, which includes a new front suspension cradle to reduce front body lateral bending, new rear structural reinforcements, and an underbody "V-bar" to help reduce rear lateral bending. Rear body torsional rigidity is improved by up to 22 percent and rear body vertical bending rigidity is improved by up to 30 percent. Additional enhancements include the use of a carbon fiber composite radiator housing and strengthening of the rear fender and hatch areas. The new structure weighs slightly less than the 350Z.


title: Nissan 370Z

text: The 370Z uses a front double wishbone suspension, with forged aluminum control arms and steering knuckle. The rear multi-link suspension uses a forged aluminum upper control arm, lower arm and radius rod, the toe control rod is steel and wheel carrier assembly is aluminum. The refreshed 2013 model uses new dampers with the Sport package models. The brakes have been changed from the more expensive Brembo racing brakes to Nissan branded brakes which are manufactured by Akebono.


title: Nissan 370Z

text: The vehicle was unveiled in the 2008 Los Angeles Auto Show, with sales of the North American model beginning at Nissan dealers in early 2009.


title: Nissan 370Z

text: Standard and optional equipment includes 19-inch RAYS wheels, Bluetooth, Sirius/XM satellite radio, heated electric seats, viscous limited slip differential, Bose sound system with dual subwoofers and 6-CD changer, and automatic climate control.


title: Nissan 370Z

text: The touring model adds heated and cooling leather-appointed power net seats, a 6-CD Bose audio system with 8 speakers (with dual subwoofer) and MP3/WMA playback, XM Satellite Radio, Bluetooth Hands-free Phone System, HomeLink Universal Transceiver, and aluminium-trimmed pedals.


title: Nissan 370Z

text: 370Z Roadster went on sale in late summer 2009 as 2010 model year vehicle. Early models include the 370Z and 370Z Touring, with Sport Package and Navigation packages for the 370Z Touring.


title: Nissan 370Z

text: The Roadster was discontinued in the US market after the 2019 model year. In Canada, however, it continued to be sold up to its 2020 model year.


title: Nissan 370Z

text: The NISMO version was introduced for the 2009 model year. The engine produces at 7,400 rpm and of torque at 5,200 rpm with revised ECU settings and exhaust setup. Transmission choice is limited to a 6-speed manual with Nissan's new "SynchroRev Match" function. Handling is improved via stiffened springs and stabilizer bars compared to the 370Z Coupe. Other equipment includes 19-inch RAYS forged aluminium-alloy wheels with Yokohama ADVAN Sport Y-rated tires (P245/40ZR19 front, P285/35ZR19 rear), 14.0-inch front and 13.8-inch rear vented rotors, NISMO Sport Brakes with 4-piston front and 2-piston rear aluminium calipers, and a fully integrated chin spoiler.


title: Nissan 370Z

text: For 2015, the Nissan 370Z NISMO has seen various updates: an automatic (seven-speed) transmission, a built-in navigation system, and tweaks to its restyled front and rear fascia and body trim.


title: Nissan 370Z

text: The Fairlady Z 40th Anniversary is a limited version of the Fairlady Z coupe Version ST for the Japanese market. It was built to commemorate the 40th anniversary of the original Fairlady Z. It includes:


title: Nissan 370Z

text: The 370Z Yellow is a limited edition of the 370Z GT Pack coupe for the British market, inspired by the European GT4 race car. It includes an Ultimate Yellow body colour, 19-inch RAYS forged alloy wheels, black leather and suede interior upholstery, integrated satellite navigation system, illuminated entry plates, special mats and a Bose audio with 8 speakers and a 6-CD autochanger.


title: Nissan 370Z

text: A North American version, was also available with a different paint color called Chicane Yellow for 2009 only, is otherwise identical to 370Zs. Chicane Yellow was also available the following model year for an extra $500 in North America as the only extra cost color with other colors available for no additional charge.


title: Nissan 370Z

text: The 370Z Nürburgring Edition is a limited (80 units) version of the 370Z coupe for the German market. It includes a Premium Ultimate Yellow body colour, special decals, a more sonorous Cobra exhaust system, a numbered plaque, 19-inch OZ alloy wheels, Dunlop SP Sport Maxx GT tires (255/40 R19 front, 285/35 R19 back) and a €150 gift card redeemable at the Nürburgring.


title: Nissan 370Z

text: The 370Z 40th Anniversary Edition is a limited (1,000 units) version based on the 370Z Coupe Touring, built to commemorate the 40th anniversary of the Datsun 240Z arriving in North America. It includes a manual transmission, Sport Package (including SynchroRev Match, front chin spoiler, rear spoiler, Nissan Sport Brakes, 19-inch RAYS forged aluminium-alloy wheels and viscous limited-slip differential), special "40th Anniversary Graphite" exterior color, red leather interior upholstery, a high-luster smoke wheel finish, red brake calipers, 40th Anniversary badges on the rear hatch and front strut tower brace, red door panel inserts, 40th Anniversary seatback and floormat embroidery, and red stitching on the center stack, shift boot and kneepads. It also has a leather steering wheel with red stitching, a plaque of authenticity, and a commemorative premium satin car cover.


title: Nissan 370Z

text: The 370Z Black Edition is a limited (370 units) version of the 370Z Coupe commemorating the 40th Anniversary of the Datsun 240Z launch in the US market. It was built for the European market. It includes unique red leather & suede appointed interior (seats & door trim) with embossed 40th Anniversary logo seats, smooth leather wrapped steering wheel with red stitching, red stitching on the centre cluster, centre console and knee pads; Nissan's Connect Premium combined touch-screen satellite navigation and audio system with USB connectivity, dark grey 19" aluminium forged RAYS wheels, red brake calipers, 40th Anniversary emblem on the back door.


title: Nissan 370Z

text: The vehicle went on sale from all Nissan High Performance Centres and throughout the dealer network with deliveries starting in April.


title: Nissan 370Z

text: The BRE 370Z is a version of the 2010 Nissan NISMO 370Z Coupe built to commemorate the 40th Anniversary of the Z's First National Championship. the 370Z commemorative car was built by STILLEN and it includes a red, white and blue BRE paint scheme designed by Pete Brock.


title: Nissan 370Z

text: The vehicle was unveiled at the BRE Reunion dinner at the Classic Motorsports Mitty at Road Atlanta on Thursday, April 29, 2010.


title: Nissan 370Z

text: The 370Z GT Edition is a version of the 370Z Coupé commemorating Nissan's global success in GT racing and 40 years of the Z in the UK. It is built for the UK market and it includes a choice of 3 body colours, (Pearl White, Kuro (metallic) black and Black Rose), grey 'GT' stripes running along the flanks of the car, 19 inch RAYS forged alloy wheels in dark anthracite colour, retuned dampers to provide the ideal balance between ride comfort and high-speed handling, a rear view parking camera as part of the Premium Connect satellite navigation system, a tyre pressure monitoring system, seven-speed automatic option with snow mode and revised underbody insulation to reduce road noise.


title: Nissan 370Z

text: All models (including the 2015 Nismo) include either a 7-speed automatic transmission with paddle shifters or a 6-speed manual transmission. SynchroRev Match system for 6-speed manual transmission is included with the Sport Package.


title: Nissan 370Z

text: The Nissan 370Z is featured in Electronic Arts' "" video game which was released on November 18, 2008. This allowed people to drive the 370Z in-game prior to the vehicle's actual release. Nissan claimed it to be the first time an automaker had a vehicle world premiere by partnering with a video game company. It was featured in the Xbox 360 racing simulator "Forza Motorsport 3", and featured in the PlayStation 3 driving simulator game "Gran Turismo 5", it was also featured in "". The 370Z also made its debut on the arcade racing game "Wangan Midnight Maximum Tune 3DX". The car also appeared in the "Asphalt" franchise, and "Real Racing 3".


title: Nissan 370Z

text: A version of Megatron Fairlady Z was produced by Takara TOMY, as part of the Transformers Alternity toy line.


title: Nissan 370Z

text: As part of the UK market launch, a TV commercial featuring a Nissan 370Z vs racing chickens was produced.