"Midnight"", described by Phillips as ""the most different record I ever heard"", was probably recorded at a different session. Phillips had not yet set up Sun Records and regularly leased his recordings to the Chess label in Chicago. The record was issued as Chess 1479 on 15 September 1951, with ""Moanin' at Midnight"" as the A-side and ""How Many More Years"" as the B-side. ""Moanin' at Midnight"" entered the R&B chart on 10 November 1951, and was followed four weeks later by ""How Many More Years"", which became the more popular side and rose to no.4 on the chart. The"
"a real number and the first equation would imply that . Therefore, and . In other words: a quaternion squares to −1 if and only if it is a vector quaternion with norm 1. By definition, the set of all such vectors forms the unit sphere. Only negative real quaternions have infinitely many square roots. All others have just two (or one in the case of 0). The identification of the square roots of minus one in H was given by Hamilton but was frequently omitted in other texts. By 1971 the sphere was included by Sam Perlis in his"
"because the quaternions are not commutative, the order of the factors in the multiplication formula becomes important—if the last factor in the multiplication formula were formula_25 rather than formula_26, the formula for multiplication of an element by its conjugate would not yield a real number. For exactly the same reasons as before, the conjugation operator yields a norm and a multiplicative inverse of any nonzero element. This algebra was discovered by John T. Graves in 1843, and is called the octonions or the ""Cayley numbers"". As an octonion consists of two independent quaternions, they form an 8-dimensional vector space over"
"Quarter (United States coin) The quarter, short for quarter dollar, is a United States coin worth 25 cents, one-fourth of a dollar. It has a diameter of .955 inches (24.26 mm) and a thickness of .069 inches (1.75 mm). The coin sports the profile of George Washington on its obverse, and its reverse design has changed frequently. It has been produced on and off since 1796 and consistently since 1831. The choice of as a denomination—as opposed to the more common elsewhere—originated with the practice of dividing Spanish milled dollars into eight wedge-shaped segments. ""Two bits"" (that is, two ""pieces"
"a hundred, let alone a thousand million, and still 7 digits short of a googol). Better yet, ""one centyllion"" (formula_3) would have 2+1, or 507,0602;4009,1291:7605,9868;1282,1505, or about a half tryllion digits, whereas a conventional ""centillion"" has only 304 digits. For a more extensive table, see Myriad system. The corresponding Chinese numerals are given, with the traditional form listed before the simplified form. Today these numerals are still in use, but are used for different values. -yllion -yllion is a proposal from Donald Knuth for the terminology and symbols of an alternate decimal superbase system. In it, he adapts the familiar"
"ordered pairs and quaternions as quadruples, then the correspondence is In the complex numbers, C, there are just two numbers, ""i"" and −""i"", whose square is −1 . In H there are infinitely many square roots of minus one: the quaternion solution for the square root of −1 is the unit sphere in R. To see this, let be a quaternion, and assume that its square is −1. In terms of ""a"", ""b"", ""c"", and ""d"", this means To satisfy the last three equations, either or ""b"", ""c"", and ""d"" are all 0. The latter is impossible because ""a"" is"
", , is 0, the corresponding term is omitted; if , , , are all zero, the quaternion is the ""zero quaternion"", denoted 0; if one of , , equals 1, the corresponding term is written simply ', ', or '. Hamilton describes a quaternion formula_5, as consisting of a scalar part and a vector part. The quaternion formula_6 is called the ""vector part"" (sometimes ""imaginary part"") of , and is the ""scalar part"" (sometimes ""real part"") of . A quaternion that equals its real part (that is, its vector part is zero) is called a ""scalar"", and is identified"
"systems, in which units stand for 10, 10, 10, 10, ..., 10, and so on. Today the corresponding characters are used for 10, 10, 10, 10, and so on. In Knuth's ""-yllion"" proposal: Each new number name is the square of the previous one — therefore, each new name covers twice as many digits. Knuth continues borrowing the traditional names changing ""illion"" to ""yllion"" on each one. Abstractly, then, ""one n-yllion"" is formula_1. ""One trigintyllion"" (formula_2) would have 2+1, or 42;9496,7297, or nearly forty-three myllion (4300 million) digits (by contrast, a conventional ""trigintillion"" has merely 94 digits — not even"
"квадриллио́н (10, quadrillion), квинтиллио́н (10, quintillion), etc. (continued as short scale). They decline in the same way as миллио́н. Russian uses words (billion) and numerals with -ard endings only in historical texts or literal translations. Also, (billiard) is a noun meaning a cue sport. Note for superscript case notations: small letters denote singular forms, capitals denote plural. is masculine (important for ""51""); both and are inanimate (important for Accusative). Blue digits are indicatives of case endings, marked by blue letters. Collective numerals () are used in Russian (and many other Slavic languages) instead of usual cardinal ones in specific lexical"
"6, 7, 8, 8, 10, 13, 15, 16, 20} are {7, 9, 15}. If also required, the zeroth quartile is 3 and the fourth quartile is 20. Consider an ordered population of 11 data values {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20}. What are the 4-quantiles (the ""quartiles"") of this dataset? So the first, second and third 4-quantiles (the ""quartiles"") of the dataset {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20} are {7, 9, 15}. If also required, the zeroth quartile is 3 and the fourth quartile is 20. When one has a"
"in at most 6 iterations, except for a small set of initial numbers which converge instead to 0. The set of numbers that converge to zero depends on whether leading zeros are discarded (the usual formulation) or are retained (as in Kaprekar's original formulation). In the usual formulation, there are 60 three-digit numbers that converge to zero, for example 211. However, in Kaprekar's original formulation the leading zeros are retained, and only repdigits such as 111 or 222 map to zero. Below is a flowchart. Leading zeros are retained, however the only difference when leading zeros are discarded is that"
"in the study of the representation of numbers as sums of squares. The group of eight basic unit quaternions, positive and negative, the quaternion group, is also the simplest non-commutative Sylow group. The study of integral quaternions began with Rudolf Lipschitz in 1886, whose system was later simplified by Leonard Eugene Dickson; but the modern system was published by Adolf Hurwitz in 1919. The difference between them consists of which quaternions are accounted integral: Lipschitz included only those quaternions with integral coordinates, but Hurwitz added those quaternions ""all four"" of whose coordinates are half-integers. Both systems are closed under subtraction"
"its norm produces a unit quaternion U""q"" called the versor of ""q"": Every quaternion has a polar decomposition formula_40. Using conjugation and the norm makes it possible to define the reciprocal of a non-zero quaternion. The product of a quaternion with its reciprocal should equal 1, and the considerations above imply that the product of formula_41 and formula_42 (in either order) is 1. So the reciprocal of ""q"" is defined to be This makes it possible to divide two quaternions ""p"" and ""q"" in two different ways (when q is non-zero). That is, their quotient can be either ""p q"""
"Split-quaternion In abstract algebra, the split-quaternions or coquaternions are elements of a 4-dimensional associative algebra introduced by James Cockle in 1849 under the latter name. Like the quaternions introduced by Hamilton in 1843, they form a four dimensional real vector space equipped with a multiplicative operation. Unlike the quaternion algebra, the split-quaternions contain nontrivial zero divisors, nilpotent elements, and idempotents. (For example, is an idempotent zero-divisor, and is nilpotent.) As a mathematical structure, they form an algebra over the real numbers, which is isomorphic to the algebra of 2 × 2 real matrices. For other names for split-quaternions see the"
"1,000,000,000 1,000,000,000 (one billion, short scale; one thousand million or milliard, yard, long scale) is the natural number following 999,999,999 and preceding 1,000,000,001. One billion can also be written as b or bn. In scientific notation, it is written as 1 × 10. The metric prefix giga indicates 1,000,000,000 times the base unit. Its symbol is G. One billion years may be called ""eon""/""aeon"" in astronomy or geology. Previously in British English (but not in American English), the word ""billion"" referred exclusively to a million millions (1,000,000,000,000). However, this is no longer as common as earlier, and the word has"
"six digits, with the comment: Ou qui veult le premier point peult signiffier million Le second point byllion Le tiers point tryllion Le quart quadrillion Le cinq quyllion Le six sixlion Le sept. septyllion Le huyt ottyllion Le neuf nonyllion et ainsi des ault' se plus oultre on vouloit preceder (Or if you prefer the first mark can signify million, the second mark byllion, the third mark tryllion, the fourth quadrillion, the fifth quyillion, the sixth sixlion, the seventh septyllion, the eighth ottyllion, the ninth nonyllion and so on with others as far as you wish to go). Adam and"
"Quadrilateral In Euclidean plane geometry, a quadrilateral is a polygon with four edges (or sides) and four vertices or corners. Sometimes, the term quadrangle is used, by analogy with triangle, and sometimes tetragon for consistency with pentagon (5-sided), hexagon (6-sided) and so on. The origin of the word ""quadrilateral"" is the two Latin words ""quadri"", a variant of four, and ""latus"", meaning ""side"". Quadrilaterals are simple (not self-intersecting) or complex (self-intersecting), also called crossed. Simple quadrilaterals are either convex or concave. The interior angles of a simple (and planar) quadrilateral ""ABCD"" add up to 360 degrees of arc, that is"
"10 = 0 (where the 2 is the tens digit of 24 from the sum for the final digit). This set of numbers ending ...404 may therefore contain one or more missing-digit sums. Similar reasoning can be applied to sums in which two, three and more digits are deleted from the original number. 14358846, 14400000, 15000000, 28758846, 28800000, 29358846, 29400000, 1107488889, 1107489042, 1111088889, 1111089042, 3277800000, 3281400000, 4388888889, 4388889042, 4392488889, 4392489042, 4500000000, 5607488889, 5607489042, 5611088889, 5611089042, 7777800000, 7781400000, 8888888889, 8888889042, 8892488889, 8892489042, 10000000000, 20000000000, 30000000000, 40000000000, 50000000000, 60000000000, 70000000000, 80000000000, 90000000000 Missing-digit sum Missing-digit sums are integer numbers that are equal"
"numbers, zeroes between significant digits, and zeroes indicated to be significant. Leading and trailing zeroes are not significant because they exist only to show the scale of the number. Therefore, 1,230,400 usually has five significant figures: 1, 2, 3, 0, and 4; the final two zeroes serve only as placeholders and add no precision to the original number. When a number is converted into normalized scientific notation, it is scaled down to a number between 1 and 10. All of the significant digits remain, but the place holding zeroes are no longer required. Thus 1,230,400 would become 1.2304 × 10."
"""c"", ""d"") = ""a"" + ""bi + ""cj + ""d""k. Unlike the complex numbers, the quaternions are an example of a non-commutative algebra: for instance, (0,1,0,0) · (0,0,1,0) = (0,0,0,1) but (0,0,1,0) · (0,1,0,0) = (0,0,0,−1). The quaternions were soon followed by several other hypercomplex number systems, which were the early examples of algebras over a field. Previous examples are associative algebras. An example of a non-associative algebra is a three dimensional vector space equipped with the cross product. This is a simple example of a class of nonassociative algebras, which is widely used in mathematics and physics, the Lie"
"also called the right part. Every quaternion is equal to a versor multiplied by the tensor of the quaternion. Denoting the versor of a quaternion by and the tensor of a quaternion by we have A right quaternion is a quaternion whose scalar component is zero, The angle of a right quaternion is 90 degrees. A right quaternion can also be thought of as a vector plus a zero scalar. Right quaternions may be put in what was called the standard trinomial form. For example, if Q is a right quaternion, it may be written as: Four operations are of"
"some threshold formula_1 must be at least proportional to the square root of formula_1, for otherwise there would not be enough of them to provide winning moves from all the hot numbers. Cold numbers tend to end in 0, 2, 4, 5, 7, or 9. Cold values that end with other digits are quite uncommon. This holds in particular for cold numbers ending in 6. Out of all the over 180,000 cold numbers less than 40 million, only one ends in a 6: 11,356. No two cold numbers can differ by a square, because if they did then a move"
"Quaternion In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions are generally represented in the form: where ""a"", ""b"", ""c"", and ""d"" are real numbers, and i, j, and k are the fundamental ""quaternion units"". Quaternions find uses in both pure"
"Zero flag The zero flag is a single bit flag that is a central feature on most conventional CPU architectures (including x86, ARM, PDP-11, 68000, 6502, and numerous others). It is often stored in a dedicated register, typically called status register or flag register, along with other flags. The zero flag is typically abbreviated Z or ZF or similar in most documentation and assembly languages. Along with a carry flag, a sign flag and an overflow flag, the zero flag is used to check the result of an arithmetic operation, including bitwise logical instructions. It is set to 1, or"
"Billionth One billionth is the reciprocal of a billion, which means it has one of two definitions depending on whether the ""long scale"" or ""short scale"" definition is being used. In the short (or American) scale, a billionth is equal to 0.000 000 001, or 1 x 10 in scientific notation or standard form. The prefix for this number is nano, and is abbreviated as ""n"" (for example, in electronics, one nanofarad would be written as 1 nF). In the long (or English) scale, a billionth is equal to 0.000 000 000 001, or 1 x 10 in scientific notation"
"The biggest blocks have just two digits. After 8 digits, a plus sign ""+"" is inserted in the code as a delimeter to aid with visual parsing. After 10 digits at each subdivision, subblocks are coded in a single code digit as follows: Consider for example zooming in on the Merlion in Singapore (N 1.286785, E 103.854503). It lies in the block around the equator bounded by –10° and +10° and between 100° and 120° East. It has offsets 80° from the South Pole and 280° from the anti-meridian, or 4 and 14 as first base-20 digit, coded as 6"
"the generators, though pointed out that the formula has a widely copied misprint in it. A generic quaternary cubic can be written as a sum of 5 cubes of linear forms, unique up to multiplication by cube roots of unity. This was conjectured by Sylvester in 1851, and proven 10 years later by Clebsch. The union of the 5 planes where these 5 linear forms vanish is called the Sylvester pentahedron. Quaternary cubic In mathematics, a quaternary cubic form is a degree 3 homogeneous polynomial in four variables. The zeros form a cubic surface in 3-dimensional projective space. and studied"
"be 0. In September 2010, a Yahoo! employee used the company's Hadoop application on one thousand computers over a 23-day period to compute 256 bits of at the two-quadrillionth (2×10th) bit, which also happens to be zero. Because is closely related to the circle, it is found in many formulae from the fields of geometry and trigonometry, particularly those concerning circles, spheres, or ellipses. Other branches of science, such as statistics, physics, Fourier analysis, and number theory, also include in some of their important formulae. appears in formulae for areas and volumes of geometrical shapes based on circles, such as"
"numbers past one trillion in the short scale system, in ascending powers of 1000, are as follows: quadrillion, quintillion, sextillion, septillion, octillion, nonillion, decillion, undecillion, duodecillion, tredecillion, quattuordecillion, quindecillion, sexdecillion, septendecillion, octodecillion, novemdecillion and vigintillion (which is 10 to the 63rd power, or a one followed by 63 zeros). The highest number in this series listed in modern dictionaries is centillion, which is 10 to the 303rd power. The interim powers of one thousand between vigintillion and centillion do not have standardized names, nor do any higher powers, but there are many ad hoc extensions in use. The highest number"
"constructed. Continuing in this manner, he eventually arrived at the orders of the myriad-myriadth period. The largest number named by Archimedes was the last number in this period, which is Another way of describing this number is a one followed by (short scale) eighty quadrillion (80·10) zeroes. Archimedes' system is reminiscent of a positional numeral system with base 10, which is remarkable because the ancient Greeks used a very simple system for writing numbers, which employs 27 different letters of the alphabet for the units 1 through 9, the tens 10 through 90 and the hundreds 100 through 900. Archimedes"
"zero. The IBM 7070, IBM 7072, and IBM 7074 computers used this code to represent each of the ten decimal digits in a machine word, although they numbered the bit positions 0-1-2-3-4, rather than with weights. Each word also had a sign flag, encoded using a two-out-of-three code, that could be A Alphanumeric, − Minus, or + Plus. When copied to a digit, the three bits were placed in bit positions 0-3-4. (Thus producing the numeric values 3, 6 and 9, respectively.) A variant is the U.S. Post Office POSTNET barcode, used to represent the ZIP+4 code for automated mail"
"Quartan prime In mathematics, a quartan prime is a prime number of the form ""x"" + ""y"", where ""x"" > 0, ""y"" > 0 (and ""x"" and ""y"" are integers). The odd quartan primes are of the form 16""n"" + 1. For example, 17 is the smallest odd quartan prime: 1 + 2 = 1 + 16 = 17. With the exception of 2 (""x"" = ""y"" = 1), one of ""x"" and ""y"" will be odd, and the other will be even. If both are odd or even, the resulting integer will be even, and 2 is the only"
"126 computers from 18 different countries. The highest calculation, the one quadrillionth digit, took 1.2 million CPU hours and 1,734 computers from 56 different countries. Total resources: 1,885 computers donated 1.3 million CPU hours. The average computer that was used to calculate would have taken 148 years to complete the calculations alone. While the PiHex project calculated the least significant digits of pi ever attempted in any base, the second place is held by Peter Trueb who computed some 22+ trillion digits in 2016 and third place by ""houkouonchi"" who derived the 13.3 trillionth digit in base 10. PiHex PiHex"