So we need to end up with whole numbers here.
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The rankings were close, and final results had to be rounded up to whole numbers.
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Today, he still gravitates toward problems that have their roots in basic equations about whole numbers.
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It's called Shor's algorithm, and is a method of factoring very large whole numbers using quantum hardware.
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In other words, xᵏ and j³ are two consecutive whole numbers that both happen to be powers.
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I was thinking of a rule that the numbers must be any three decreasing, positive whole numbers.
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The framers of the Constitution made clear that everyone counted in apportioning Congress (if not in whole numbers).
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Whole numbers were excised, though Prokofiev was able to salvage some of the divertissements elsewhere in the score.
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" "It doesn't particularly mean a whole lot to a professional investor, like crossing the whole numbers on the Dow.
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The math also suggests a reason why electric charge is quantized in discrete units—essentially, because whole numbers are.
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He's happiest, he said, when his abstract constructions lead him back around to small discoveries about ordinary whole numbers.
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Among the whole vast, infinite universe of whole numbers, 8 and 9 are the only two that are consecutive powers.
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Outside of Matthew Dellavedova, the whole numbers-as-letters marketing fetish is the worst thing to ever happen to sports.
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It's not equal to the ratio of any two whole numbers, so an approximation -- 22/7 -- is used in many calculations.
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For example, define a set of numbers such as the integers, all the whole numbers from minus infinity to positive infinity.
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Computers love whole numbers, and so do I. Even the weird NTSC numbers in use due to certain technical constraints divide nicely.
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The overall pattern is that low whole numbers ranked best in the survey and the higher the number, the less well it ranked.
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According to Mr. Guzzetta, INTEGERS are "always" NATURAL or WHOLE numbers, "sometimes" RATIONAL (I know how they feel) numbers and "never" IMAGINARY numbers.
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"The women's memorial is one of whole numbers of memorial projects that were part of the late 20th-century boom," Mr. Savage said.
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Whole numbers are easy because it's easy to represent a whole number in our base-10 counting system as a binary (base-2) number.
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As opposed to integers or whole numbers, floating point numbers—with decimal points—are crucial to the calculations running through the neural networks involved with deep learning.
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But what if you want your denominators to be drawn from some (still infinite) subset of the whole numbers, like all prime numbers, or all perfect squares?
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With 32 bits, or 212 digits, we can represent a huge range of whole numbers (integers or ints, in computer science), all the way up to 21000.
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Prime numbers are integers (whole numbers) that can only be divided by themselves or the number 1, and they appear along the number line in a highly erratic way.
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The way Democrats ensure this is they start with just the whole numbers (avoiding the decimals for now) — so Sanders gets 215, Biden 153, Warren 215, and Buttigieg 215.
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At the Paper Mill Playhouse, where "Bandstand" had its premiere in 53, 25 minutes — including whole numbers and scenes — were cut from the show in one day, he said.
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For example, for Alice to take two large whole numbers and multiply them is relatively easy; for Eve to take the result and recover the original numbers seems much harder.
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Nearly 80% of children in grade four (20163- or 10-year-olds) cannot read and understand sentences in any language; 61% of pupils a year older cannot add or subtract whole numbers.
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Zhou told financial media Caixin in late May that whole numbers "do not really much matter" to the exchange rate, and 7 is not necessarily the threshold for yuan to the dollar.
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Players will get new scores in season 2, on a scale of 1 to 5,000, so no more frustrating fractional changes when you win big; your skill rating will change by whole numbers.
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But during the first decade of the 19th century he took Proust's concept and showed not only that elements reacted in fixed proportions by weight, but also that those proportions were ratios of small whole numbers.
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To determine its list of The 50 Best Places to Interview in 2017, Glassdoor calculated an interview score for each company (where calculations extend beyond the thousandth place but are displayed as whole numbers for simplicity).
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In the 1940s, Otto E. Neugebauer and Abraham J. Sachs, mathematics historians, pointed out that the other three columns were essentially Pythagorean triples — sets of integers, or whole numbers, that satisfy the equation a2 + b0003 = c2.
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For some equations of this type, it is fruitful to study whether they have solutions among alternative number systems called p-adic numbers, which, like the real numbers, are built by filling in the gaps between whole numbers and fractions.
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The planets line up, with the ratio of their orbits falling very close to whole numbers: The two inner planets have a ratio of 3:5, since the super-Earth takes three days to orbit its star and the closer sub-Neptune takes five days.
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The connection between mathematics and art has been present since the ancient Greeks, who believed that aesthetics were guided by whole numbers and used other rational relationships, such as the golden ratio, in their art, said Professor Dan Rockmore, director of the Neukom Institute for Computational Science at Dartmouth College.
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As is the case for the related atomic mass when expressed in daltons, the relative isotopic mass numbers of nuclides other than carbon-12 are not whole numbers, but are always close to whole numbers. This is discussed more fully below.
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Like whole numbers, fractions obey the commutative, associative, and distributive laws, and the rule against division by zero.
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Early Fortran compilers only allowed whole numbers as labels. Beginning with Fortran-90, alphanumeric labels have also been allowed.
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The game is intended for Grades 2-8 and teaches operations involving whole numbers, integers, fractions, decimals, and rational numbers.
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If fractions now are considered there are an infinite number of fractions between any of the two whole numbers, suggesting that the infinity of fractions is bigger than the infinity of whole numbers. Yet Cantor was still able to pair each such fraction to a whole number 1 - 1/1; 2 - 2/1; 3 - 1/2 ... etc.
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In general, the negativity or positivity of a number is referred to as its sign. Every real number other than zero is either positive or negative. The non-negative whole numbers are referred to as natural numbers (i.e., 0, 1, 2, 3...), while the positive and negative whole numbers (together with zero) are referred to as integers.
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Math Mysteries is a collection of five math-related educational video games for the Windows and Macintosh platforms, developed and published by Tom Snyder Productions. The games were designed to fit the NCTM standards at their time of development. The series consists of Math Mysteries: Measurements, Math Mysteries: Whole Numbers, Math Mysteries: Fractions, Math Mysteries: Advanced Whole Numbers and Math Mysteries: Advanced Fractions.
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Containers over 3 litres must be bottled in quantities of whole numbers of litres. No other sizes may be bottled. Spirits must also be sold in metric quantities.
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Simple tiles are generated by Möbius triangles with whole numbers p,q,r, while Schwarz triangles allow rational numbers p,q,r and allow star polygon faces, and have overlapping elements.
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This version again used whole numbers: #1 Sept. 16, 1910 - #143 June 7, 1913. See Walter Goldwater, Radical Periodicals in America, 1890-1950. New Haven: Yale University Library 1964; pg. 7.
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Whole Numbers Play the Basics was released by Carpark Records on September 17, 2002 in the United States. The album was released on compact disc, vinyl record, and as a digital download.
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In this Monro took a broad view of the sphere of poetry, devoting whole numbers to children's rhymes and to songs by Walter de la Mare complete with scores.The Modernist Lab at Yale University Retrieved 17 December 2014.
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There are 3 levels of difficulty. The game teaches skills including: word problems, estimation, geometry, equations, modelling, whole numbers, money, fractions and decimals. These are presented as activities that help solve the game's puzzles rather than tiresome, repetitive exercises.
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The last numbers of the comic were dedicated to themes that did not dealt with the comic story itself, including 8 whole numbers filled with a philosophical and metaphysical indoctrination manual called "Manual del guerrero Kundalini" (Kundalini Warrior Guide).
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Beckmann, S. (2014). The twenty- third ICMI study: primary mathematics study on whole numbers. International Journal of STEM Education, 1(1), 1-8. Chicago However, throughout the world, addition is taught by the end of the first year of elementary school.
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PHP stores whole numbers in a platform-dependent range. This range is typically that of 32-bit signed integers. Integer variables can be assigned using decimal (positive and negative), octal and hexadecimal notations. Real numbers are also stored in a platform-specific range.
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US convention is to use whole numbers when even (e.g. "three in twelve") or the nearest single or two-digit fraction when not (e.g. either "five and a quarter in twelve" or "five point two-five in twelve", each expressed numerically as 5.25:12).
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Whole Numbers Play the Basics is an album by Erik Kowalski under the alias of Casino Versus Japan. It was released in 2002 by Carpark Records. On its release, the album was praised by AllMusic, the BBC and Uncut while being panned by Pitchfork.
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Languages which use these often impose the constraint that the line numbers must increase in value in each following line, but may not require that they be consecutive. For example, in BASIC: 10 LET X = 3 20 PRINT X In other languages such as C and Ada, a label is an identifier, usually appearing at the start of a line and immediately followed by a colon. For example, in C: Success: printf("The operation was successful. "); The language ALGOL 60 allowed both whole numbers and identifiers as labels (both linked by colons to the following statement), but few if any other ALGOL variants allowed whole numbers.
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As for the assignment of 1 or 100 to the index substances, this makes no difference to the rankings themselves, only to whether the values are displayed as whole numbers or decimal points. Glucose remains about three- quarters as sweet as sucrose whether displayed as 75 or 0.75.
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1, 2, 3...). Larger hooks are referenced by increasing whole numbers followed by a slash and a zero (e.g. 1/0 (one aught), 2/0, 3/0...) as their size increases. The numbers represent relative sizes, normally associated with the gap (the distance from the point tip to the shank).
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In 1618, Robert Fludd devised a mundane monochord (also celestial or divine monochord) that linked the Ptolemaic universe to musical intervals. "Was it [Mersenne's discoveries through use of the monochord (1637)] physical intuition or a Pythagorean confidence in the importance of small whole numbers? ... It was the latter."Gozza, Paolo; ed. (2013).
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Furthermore, they are usually read aloud as if they were whole numbers (e.g. 1.000, "a thousand" or 0.500, "five hundred"). In this case, the name "winning percentage" is actually a misnomer, since it is not expressed as a percentage. A winning percentage such as .536 ("five thirty-six") expressed as a percentage would be 53.6%.
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The addition of two whole numbers results in the total amount or sum of those values combined. The example in the adjacent picture shows a combination of three apples and two apples, making a total of five apples. This observation is equivalent to the mathematical expression (i.e., "3 add 2 is equal to 5").
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This means the brightest white (a value of 255) is only 255 levels brighter than the darkest shade above pure black (i.e.: value of 0). #Lighting calculations were integer based, which didn't offer as much accuracy because the real world is not confined to whole numbers. On December 24, 2002, Microsoft released a new version of DirectX.
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List of Maya numerals from 0 to 19 with underneath two vertically oriented examples The Mayas used a positional base-twenty (vigesimal) numerical system which only included whole numbers. For simple counting operations, a bar and dot notation was used. The dot represents 1 and the bar represents 5. A shell was used to represent zero.
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The adversary is presumed to have manufactured a series of tanks marked with consecutive whole numbers, beginning with serial number 1. Additionally, regardless of a tank's date of manufacture, history of service, or the serial number it bears, the distribution over serial numbers becoming revealed to analysis is uniform, up to the point in time when the analysis is conducted.
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It is worth noting that "the number of sales" is fundamentally countable and therefore discrete. A continuous simulation of sales implies the possibility of fractional sales e.g. 1/3 of a sale. For that reason, a continuous simulation of sales does not model reality but nevertheless may make useful predictions that match a discrete simulation's predictions for whole numbers of sales.
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Fudge dice The faces of most dice are labelled using sequences of whole numbers, usually starting at one, expressed with either pips or digits. However, there are some applications that require results other than numbers. Examples include letters for Boggle, directions for Warhammer Fantasy Battle, Fudge dice, playing card symbols for poker dice, and instructions for sexual acts using sex dice.
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Beginning with the fifth edition, it is intended that subsequent revisions will be added more often, to keep up with research in the field. It is notable that DSM-5 uses Arabic rather than Roman numerals. Beginning with DSM-5, the APA will use decimals to identify incremental updates (e.g., DSM-5.1, DSM-5.2) and whole numbers for new editions (e.g.
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The Dewey Decimal Classification organizes library materials by discipline or field of study. Main divisions include philosophy, social sciences, science, technology, and history. The scheme comprises ten classes, each divided into ten divisions, each having ten sections. The system's notation uses Indo- Arabic numbers, with three whole numbers making up the main classes and sub- classes and decimals designating further divisions.
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In chemistry, the law of multiple proportions states that if two elements form more than one compound between them, then the ratios of the masses of the second element which combine with a fixed mass of the first element will always be ratios of small whole numbers. This law is sometimes called Dalton's Law, named after John Dalton, the chemist who first expressed it. For example, Dalton knew that the element carbon forms two oxides by combining with oxygen in different proportions. A fixed mass of carbon, say 100 grams, may react with 133 grams of oxygen to produce one oxide, or with 266 grams of oxygen to produce the other. The ratio of the masses of oxygen that can react with 100 grams of carbon is 266:133 = 2:1, a ratio of small whole numbers.
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The range of a variable is given as the set of possible values that that variable can hold. In the case of an integer, the variable definition is restricted to whole numbers only, and the range will cover every number within its range (including the maximum and minimum). For example, the range of a signed 16-bit integer variable is all the integers from −32,768 to +32,767.
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The division of two whole numbers does not necessarily result in a whole number. For example, 1 divided by 4 equals 1/4, which is neither even nor odd, since the concepts even and odd apply only to integers. But when the quotient is an integer, it will be even if and only if the dividend has more factors of two than the divisor..
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An additional multiplication with the radix value occurs for each additional digit, so the numeral "201" represents a value of two-hundred- and-one (equal to ). The elementary level of study typically includes understanding the value of individual whole numbers using Arabic numerals with a maximum of seven digits, and performing the four basic operations using Arabic numerals with a maximum of four digits each.
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This discovery overturned Lavoisier's definition of acids as compounds of oxygen. Davy was a popular lecturer and able experimenter. Joseph Louis Gay-Lussac, who stated that the ratio between the volumes of the reactant gases and the products can be expressed in simple whole numbers. French chemist Joseph Louis Gay-Lussac shared the interest of Lavoisier and others in the quantitative study of the properties of gases.
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In mathematics, the notion of a divisor originally arose within the context of arithmetic of whole numbers. With the development of abstract rings, of which the integers are the archetype, the original notion of divisor found a natural extension. Divisibility is a useful concept for the analysis of the structure of commutative rings because of its relationship with the ideal structure of such rings.
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Therefore, a debut album would just be known as their first (or 1집), while their fifth would simply be their fifth. The actual title of the album is secondary. Under this system, "real" albums are given whole numbers, while special or concept albums are labeled "half" albums. Because this album was not new material but old material made new, it is not considered "real" material.
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The ALU is capable of performing two classes of operations: arithmetic and logic. The set of arithmetic operations that a particular ALU supports may be limited to addition and subtraction, or might include multiplication, division, trigonometry functions such as sine, cosine, etc., and square roots. Some can only operate on whole numbers (integers) while others use floating point to represent real numbers, albeit with limited precision.
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Road sign The Military Load Classification (MLC) is a system of standards used by NATO to classify the safe amount of load a surface can withstand. Load- carrying capacity is shown in whole numbers for vehicles, bridges, roads, and routes. Vehicles are classified by weight, type, and effect on routes. Bridges, roads, and routes are classified by physical characteristics, type and flow of traffic, effects of weather, and other special conditions.
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Season sixteen of Dancing with the Stars premiered on March 18, 2013. Tom Bergeron and Brooke Burke Charvet returned as hosts, while Carrie Ann Inaba, Len Goodman and Bruno Tonioli returned as judges. The Harold Wheeler orchestra and singers also returned to provide the music throughout the season. Scoring returned to using traditional whole numbers, instead of continuing with the fractional scores introduced for the previous (all-star) season.
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The goal of the puzzle is to place all the towers onto the base so as to form a level cube with each of the six colors appearing once, and only once, in each row and column. The 36 cube was invented by Dr. Derrick Niederman, a PhD. at MIT. He came up with the idea while writing a book on whole numbers, after unearthing an 18th-century mathematical hypothesis.
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Al-Kashi's book, Key to Arithmetic, was written at the beginning of the 15th century and was the stimulus for the systematic application of decimals to whole numbers and fractions thereof. But nobody established their daily use before Stevin. He felt that this innovation was so significant, that he declared the universal introduction of decimal coinage, measures and weights to be merely a question of time. His notation is rather unwieldy.
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In January 2000, Go Hawaii was released on CD by Wobblyhead (later on double- vinyl LP by City Centre Offices, 2001). By the summer of 2002, the track "It's Very Sunny" was used in a Hummer television commercial. His third album Whole Numbers Play the Basics, followed in September, 2002 on Carpark Records. The song "Manic Thru Tone" was used in MTV's "Choose Or Lose" campaign in the fall of 2002.
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In fact, software charges are why the MSU measurement exists at all. IBM publishes MSU ratings for every mainframe server model, including the zSeries and System z9 ranges. For example, a zSeries z890 Model 110 is a 4 MSU system. MSU ratings are always rounded to whole numbers. IBM enforces an MSU rule called the “technology dividend”: each new mainframe model has a 10% lower MSU rating for the same level of system capacity.
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In each right triangle, Pythagoras's theorem establishes the length of the hypotenuse in terms of this unit. If a hypotenuse is related to the unit by the square root of a positive integer that is not a perfect square, it is a realization of a length incommensurable with the unit, such as , , . For more detail, see Quadratic irrational. Incommensurable lengths conflicted with the Pythagorean school's concept of numbers as only whole numbers.
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The charge limits for the Eurotariff and the wholesale average charge should be calculated to the maximum number of decimal places permitted by the official exchange rate. This sets the maximum that can be charged in the national currency. Providers may wish in practice to quote charges in whole numbers of currency units, especially at the retail level, although this in practice is not compulsory. In this case, the numbers should be rounded down.
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An enlarged edition of his Doctrine of Decimal Arithmetick, the preparation of which had engaged his attention during about a year before his death, appeared in 1685. It had originally been printed in 1664 on a quarter of a sheet for portability in a letter-case. His Arithmetic in whole Numbers and Fractions, both Vulgar and Decimal, with Tables for the Forbearance and Rebate of Money, &c.;, was published by Thomas Plant in 1688.
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Multiplication can also be visualized as counting objects arranged in a rectangle (for whole numbers), or as finding the area of a rectangle whose sides have some given lengths. The area of a rectangle does not depend on which side is measured first—a consequence of the commutative property. The product of two measurements is a new type of measurement. For example, multiplying the lengths of the two sides of a rectangle gives its area.
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This is called an "angle-based" right triangle. A "side-based" right triangle is one in which the lengths of the sides form ratios of whole numbers, such as 3 : 4 : 5, or of other special numbers such as the golden ratio. Knowing the relationships of the angles or ratios of sides of these special right triangles allows one to quickly calculate various lengths in geometric problems without resorting to more advanced methods.
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Multiplication is often defined for natural numbers, then extended to whole numbers, fractions, and irrational numbers. However, abstract algebra has a more general definition of multiplication as a binary operation on some objects that may or may not be numbers. Notably, one can multiply complex numbers, vectors, matrices, and quaternions. Some educators believe that seeing multiplication exclusively as repeated addition during elementary education can interfere with later understanding of these aspects of multiplication.
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In number theory, the numbers of the form x2 \+ xy + y2 for integer x, y are called the Loeschian numbers. These numbers are named after August Lösch. They are the norms of the Eisenstein integers. They are a set of whole numbers, including zero, and having prime factorization in which all primes congruent to 2 mod 3 have even powers (there is no restriction of primes congruent to 0 or 1 mod 3).
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The long and short scales are two of several naming systems for integer powers of ten which use some of the same terms for different magnitudes. For whole numbers smaller than 1,000,000,000 (109), such as one thousand or one million, the two scales are identical. For larger numbers, starting with 109, the two systems differ. For identical names, the long scale proceeds by powers of one million, whereas the short scale proceeds by powers of one thousand.
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The pop duo was a regular participant of the 'Little Blue Light' and humorous programs of those years, where they played short reprises and whole numbers. Galina Brezhneva personally invited Vladimirov and Tonkov to address the wives of government members at an informal concert. Duet Veronika Mavrikievna and Avdotya Nikitichna traveled a lot with tours around the USSR, there were also trips to Afghanistan during the war of 1979/1989. The duo broke up in 1982.
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A label is an explicit name or number assigned to a fixed position within the source code, and which may be referenced by control flow statements appearing elsewhere in the source code. A label marks a position within source code, and has no other effect. Line numbers are an alternative to a named label used in some languages (such as BASIC). They are whole numbers placed at the start of each line of text in the source code.
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Richardson believed that the time saved in looking through catalogs at a single line and cataloging new texts with a single line increased productivity and allowed resources to be entered into the library system and accessed more easily. Unfortunately, not everyone at Princeton agreed with Richardson’s method of cataloging. His system, often referred to as the “Princeton System” relied on whole numbers and minute specifications for classification, resulting in a lengthy cataloging number.Richardson, Ernest C. “Classification: Theoretical and Practical”.
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It was added to the NRHP on July 24, 1974 as NRIS number 74001086. Located on the southern corner of the junction of Old and New Franklin Ditches, just east of the city of Cooter,Chapman, Carl H., and Anderson, Leo O. "Campbell Site: A Late Mississippi Town Site and Cemetery in Southeast Missouri". Missouri Archaeologist 17.2-3 Whole Numbers (1955). the site has yielded prolific numbers of quartz pebbles and stone tools made of flint.
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For the tested students from the high school graduating class of 2019, 37 percent met ACT's College Readiness Benchmarks in at least three of the four subject areas the ACT tests—English, math, reading, and science. Scores are reported on a 1–36 scale, with a composite score that represents the average scores from each of the four subject area tests. All ACT scores are reported as whole numbers (e.g., a score of 23.5 rounds up to 24).
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Methods used to teach subtraction to elementary school vary from country to country, and within a country, different methods are adopted at different times. In what is known in the United States as traditional mathematics, a specific process is taught to students at the end of the 1st year (or during the 2nd year) for use with multi-digit whole numbers, and is extended in either the fourth or fifth grade to include decimal representations of fractional numbers.
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Eligible candidates receive points based on a number of criteria, including awards and decorations, Enlisted Performance Report (EPR) points, Promotion Fitness Examination (PFE) points, and Specialty Knowledge Test (SKT) points. Fractions of points are awarded for certain categories, resulting in scores that are not whole numbers. Candidates with the highest numbers of points, up to the promotion allowance in each career field, are promoted. The score of the last person promoted is known as the cutoff.
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Almost half could not add 218 and 191 compared to 73% internationally. Ministry of Education figures show the number of 12-year-olds who were able to answer simple multiplication questions correctly dropped from "47% in 2001 — the year new maths teaching methods were introduced — to 37% in 2009". The problem flows on to high schools, where "there are still students who have difficulty with the very basics such as knowledge about whole numbers and decimals".
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Both methods take account of the positions of the bishops first, and ignore the distinction between the king and rooks. Once the positions of the bishops, knights and queen are known, there is only one possibility for the remaining three squares. In the places where division of whole numbers is done, it is always done giving a quotient (designated q1,q2,..) and a remainder (designated r1,r2 ..). There are 16 ways to put two bishops on opposite colored squares.
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In mathematics, and especially general topology, the prime integer topology and the relatively prime integer topology are examples of topologies on the set of positive whole numbers, i.e. the set }. To give the set Z+ a topology means to say which subsets of Z+ are "open", and to do so in a way that the following axioms are met: # The union of open sets is an open set. # The finite intersection of open sets is an open set.
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Once some facts are committed to memory, children begin to derive unknown facts from known ones. For example, a child asked to add six and seven may know that and then reason that is one more, or 13. Such derived facts can be found very quickly and most elementary school students eventually rely on a mixture of memorized and derived facts to add fluently. Different nations introduce whole numbers and arithmetic at different ages, with many countries teaching addition in pre-school.
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Thus, repeated addition extends to the whole numbers (0, 1, 2, 3, 4, ...). The first challenge to the belief that multiplication is repeated addition appears when students start working with fractions. From the mathematical point of view, multiplication as repeated addition can be extended into fractions. For example, : 7/4 \times 5/6 literally calls for “one and three-fourths of the five-sixths.” This is later significant because students are taught that, in word problems, the word “of” usually indicates a multiplication.
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Diagram of a hockey field Most hockey field dimensions were originally fixed using whole numbers of imperial measures. Nevertheless, metric measurements are now the official dimensions as laid down by the International Hockey Federation (FIH) in the "Rules of Hockey". The pitch is a rectangular field. At each end is a goal high and wide, as well as lines across the field from each end-line (generally referred to as the 23-metre lines or the 25-yard lines) and in the center of the field.
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For example, if a party wins one-third of the votes then it should gain about one-third of the seats. In general, exact proportionality is not possible because these divisions produce fractional numbers of seats. As a result, several methods, of which the D'Hondt method is one, have been devised which ensure that the parties' seat allocations, which are of whole numbers, are as proportional as possible. Although all of these methods approximate proportionality, they do so by minimizing different kinds of disproportionality.
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This is used in five countries as part of mixed systems. The Dowdall system, a multi-member constituency variation on the Borda count, is used in Nauru for parliamentary elections and sees voters rank the candidates depending on how many seats there are in their constituency. First preference votes are counted as whole numbers; the second preference votes divided by two, third preferences by three; this continues to the lowest possible ranking.Nauru Parliament: Electoral system IPU The totals achieved by each candidate determine the winners.
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In 1998 The Methodist Church in Ireland embarked on a period of reflection on its position within Irish Society which it called 'Dreaming Dreams'. Although in many areas of the country the Church is increasing in numbers it is aware that as a whole numbers are decreasing in church membership across the country in every denomination. The church has since published its 'ConneXions' plan. The core vision of ConneXions is that each local Church will reflect the life of Christ in its own area.
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"The frequency ratio of every Pythagorean interval is a ratio between a power of two and a power of three...confirming the Pythagorean requirements that all intervals be associated with ratios of whole numbers." For instance, the perfect fifth with ratio 3/2 (equivalent to 31/21) and the perfect fourth with ratio 4/3 (equivalent to 22/31) are Pythagorean intervals. All the intervals between the notes of a scale are Pythagorean if they are tuned using the Pythagorean tuning system. However, some Pythagorean intervals are also used in other tuning systems.
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The law of multiple proportions is best demonstrated using simple compounds. For example, if one tried to demonstrate it using the hydrocarbons decane (chemical formula C10H22) and undecane (C11H24), one would find that 100 grams of carbon could react with 18.46 grams of hydrogen to produce decane or with 18.31 grams of hydrogen to produce undecane, for a ratio of hydrogen masses of 121:120, which is hardly a ratio of "small" whole numbers. The law fails with non- stoichiometric compounds and also doesn't work well with polymers and oligomers.
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A chemical formula identifies each constituent element by its chemical symbol and indicates the proportionate number of atoms of each element. In empirical formulae, these proportions begin with a key element and then assign numbers of atoms of the other elements in the compound, by ratios to the key element. For molecular compounds, these ratio numbers can all be expressed as whole numbers. For example, the empirical formula of ethanol may be written C2H6O because the molecules of ethanol all contain two carbon atoms, six hydrogen atoms, and one oxygen atom.
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Off-key is musical content that is not at the expected frequency or pitch period, either with respect to some absolute reference frequency, or in a ratiometric sense (i.e. through removal of exactly one degree of freedom, such as the frequency of a keynote), or pitch intervals not well-defined in the ratio of small whole numbers. The term may also refer to a person or situation being out of step with what is considered normal or appropriate. A single note deliberately played or sung off-key can be called an "off-note".
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Structuralism is a position holding that mathematical theories describe structures, and that mathematical objects are exhaustively defined by their places in such structures, consequently having no intrinsic properties. For instance, it would maintain that all that needs to be known about the number 1 is that it is the first whole number after 0. Likewise all the other whole numbers are defined by their places in a structure, the number line. Other examples of mathematical objects might include lines and planes in geometry, or elements and operations in abstract algebra.
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In German, whole numbers (smaller than 1 million) can be expressed as single words, which makes (777,777) a 65 letter word. In combination with or, as an inflected noun, , all numbers can be written as one word. A 79 letter word, , was named the longest published word in the German language by the 1972 Guinness Book of World Records, but longer words are possible. The word was the name of a prewar Viennese club for subordinate officials of the headquarters of the electrical division of the company named the , "Danube steam boat operation company".
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Stevin wrote a 35-page booklet called De Thiende ("the art of tenths"), first published in Dutch in 1585 and translated into French as La Disme. The full title of the English translation was Decimal arithmetic: Teaching how to perform all computations whatsoever by whole numbers without fractions, by the four principles of common arithmetic: namely, addition, subtraction, multiplication, and division. The concepts referred to in the booklet included unit fractions and Egyptian fractions. Muslim mathematicians were the first to utilize decimals instead of fractions on a large scale.
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A quadratic form (not quadratic equation) is any polynomial in which each term has variables appearing exactly twice. The general form of such an equation is ax2 \+ bxy + cy2. (All coefficients must be whole numbers.) A given quadratic form is said to represent a natural number if substituting specific numbers for the variables gives the number. Gauss and those who followed found that if we change variables in certain ways, the new quadratic form represented the same natural numbers as the old, but in a different, more easily interpreted form.
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In mathematical astronomy, his fame is due to the introduction of the concentric spheres, and his early contributions to understanding the movement of the planets. His work on proportions shows insight into real numbers; it allows rigorous treatment of continuous quantities and not just whole numbers or even rational numbers. When it was revived by Tartaglia and others in the 16th century, it became the basis for quantitative work in science for a century, until it was replaced by Richard Dedekind. Craters on Mars and the Moon are named in his honor.
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A contrast is defined as the sum of each group mean multiplied by a coefficient for each group (i.e., a signed number, cj). In equation form, L = c_1 \bar X_1 + c_2 \bar X_2 + \cdots + c_k \bar X_k \equiv \sum_j c_j \bar X_j, where L is the weighted sum of group means, the cj coefficients represent the assigned weights of the means (these must sum to 0 for orthogonal contrasts), and \bar Xj represents the group means. Coefficients can be positive or negative, and fractions or whole numbers, depending on the comparison of interest.
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In 1950s American mathematician Paul Cohen took up the challenge of Cantor's Continuum Hypothesis which asks "is there is or isn't there an infinite set of number bigger than the set of whole numbers but smaller than the set of all decimals". Cohen found that there existed two equally consistent mathematical worlds. In one world the Hypothesis was true and there did not exist such a set. Yet there existed a mutually exclusive but equally consistent mathematical proof that Hypothesis was false and there was such a set.
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First, a point is assigned with a mass (often a whole number, but it depends on the problem) in the way that other masses are also whole numbers. The principle of calculation is that the foot of a cevian is the addition (defined above) of the two vertices (they are the endpoints of the side where the foot lie). For each cevian, the point of concurrency is the sum of the vertex and the foot. Each length ratio may then be calculated from the masses at the points.
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The proof that the square root of 2 () is irrational (i.e. cannot be expressed as a fraction of two whole numbers) was discovered by the ancient Greeks, and is perhaps the earliest known example of a proof by infinite descent. Pythagoreans discovered that the diagonal of a square is incommensurable with its side, or in modern language, that the square root of two is irrational. Little is known with certainty about the time or circumstances of this discovery, but the name of Hippasus of Metapontum is often mentioned.
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The passage in The Whetstone of Witte introducing the equals sign. Page 238 in the pdf file. The Whetstone of Witte is the shortened title of Robert Recorde's mathematics book published in 1557, the full title being The whetstone of witte, whiche is the seconde parte of Arithmetike: containyng thextraction of Rootes: The Coßike practise, with the rule of Equation: and the woorkes of Surde Nombers. The book covers topics including whole numbers, the extraction of roots and irrational numbers.. The work is notable for containing the first recorded use of the equals sign.
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Atoms and molecules as depicted in John Dalton's A New System of Chemical Philosophy vol. 1 (1808) In the early 1800s, John Dalton compiled experimental data gathered by himself and other scientists and discovered a pattern now known as the "law of multiple proportions". He noticed that in chemical compounds which contain a particular chemical element, the content of that element in these compounds will differ by ratios of small whole numbers. This pattern suggested to Dalton that each chemical element combines with others by some basic and consistent unit of mass.
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The trio also had a great-grandfather who was Chinese. "By the time I was eight, I was already working up whole numbers for our family's little weekend shows," Ronnie Spector later recalled. "Then Estelle would get up onstage and do a song, or she'd join Nedra or my cousin Elaine and me in a number we'd worked out in three-part harmony." Furthering their interest in show business, Estelle was enrolled at Startime, a popular dancing school in the 1950s, while Ronnie became fascinated with Frankie Lymon and the Teenagers.
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In 1919–1920, Rutherford found that nitrogen and other light elements ejected a proton, which he called a "hydrogen atom", when hit with α (alpha) particles. This result showed Rutherford that hydrogen nuclei were a part of nitrogen nuclei (and by inference, probably other nuclei as well). Such a construction had been suspected for many years on the basis of atomic weights which were whole numbers of that of hydrogen; see Prout's hypothesis. Hydrogen was known to be the lightest element, and its nuclei presumably the lightest nuclei.
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Calculating 9 × 8, and 7 × 6 Multiplying two whole numbers, each from 6 to 10 can be achieved using fingers and thumbs as follows: # Number the fingers and thumbs from 10 to 6, then 6 to 10 from left to right, as in the figure. # Bend the finger or thumb on each hand corresponding to each number, and all the fingers between them. # The number of bent fingers or thumbs gives the tens digit. # To the above is added the product of the unbent fingers or thumbs on the left and right sides.
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An apportionment paradox exists when the rules for apportionment in a political system produce results which are unexpected or seem to violate common sense. To apportion is to divide into parts according to some rule, the rule typically being one of proportion. Certain quantities, like milk, can be divided in any proportion whatsoever; others, such as horses, cannot—only whole numbers will do. In the latter case, there is an inherent tension between the desire to obey the rule of proportion as closely as possible and the constraint restricting the size of each portion to discrete values.
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If this is the case, the phase is defined by the angle, φ(t), described by the segment joining the center of rotation and the projection of the trajectory point onto the plane. In other cases it is still possible to define a phase by means of techniques provided by the theory of signal processing, such as the Hilbert transform. In any case, if φ1(t) and φ2(t) denote the phases of the two coupled oscillators, synchronization of the phase is given by the relation nφ1(t)=mφ2(t) with m and n whole numbers.
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These engaging and interactive games have the ability to teach kids about the some physiological functions of the body. One example is that these games can help show kids how their heart reacts to different activities by using the heart rate monitor within the game. One study took the game Semideus to see if it could help to improve performance on rational number tasks, the understanding of whole numbers and mathematical thinking in general. The study concluded if kids were introduced to games that have math well integrated into the gameplay then it kids then it will help them with their skills.
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The Algebra is perhaps Abu Kamil's most influential work, which he intended to supersede and expand upon that of Al-Khwarizmi. Whereas the Algebra of al- Khwarizmi was geared towards the general public, Abu Kamil was addressing other mathematicians, or readers familiar with Euclid's Elements. In this book Abu Kamil solves systems of equations whose solutions are whole numbers and fractions, and accepted irrational numbers (in the form of a square root or fourth root) as solutions and coefficients to quadratic equations. The first chapter teaches algebra by solving problems of application to geometry, often involving an unknown variable and square roots.
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His work on isotopes also led to his formulation of the whole number rule which states that "the mass of the oxygen isotope being defined [as 16], all the other isotopes have masses that are very nearly whole numbers," a rule that was used extensively in the development of nuclear energy. The exact mass of many isotopes was measured leading to the result that hydrogen has a 1% higher mass than expected by the average mass of the other elements. Aston speculated about the subatomic energy and the use of it in 1936. Isotopes and Mass-spectra and Isotopes are his most well-known books.
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Not only did he gather magnetic measurements at various altitudes, but he also took pressure, temperature, and humidity measurements and samples of air, which he later analyzed chemically. In 1808 Gay-Lussac announced what was probably his single greatest achievement: from his own and others' experiments he deduced that gases at constant temperature and pressure combine in simple numerical proportions by volume, and the resulting product or products—if gases—also bear a simple proportion by volume to the volumes of the reactants. In other words, gases under equal conditions of temperature and pressure react with one another in volume ratios of small whole numbers.
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This neuromuscular depression is due to less neurotransmitter release during stimulation. In order for depletion not to occur, there must be a balance between repletion and depletion which can happen at low stimulation frequencies of less than 30 Hz. When a vesicle releases its neurotransmitters via exocytosis, it empties its entire contents into the synaptic cleft. Neurotransmitter release from vesicles is therefore stated to be quantal because only whole numbers of vesicles can be released. In 1970, Bernard Katz from the University of London won the Nobel Prize for Physiology or Medicine for statistically determining the quantal size of acetylcholine vesicles based on noise analysis in the neuromuscular junction.
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For example, there are two types of tin oxide: one is a black powder that is 88.1% tin and 11.9% oxygen, and the other is a white powder that is 78.7% tin and 21.3% oxygen. Adjusting these figures, in the black oxide there is about 13.5 g of oxygen for every 100 g of tin, and in the white oxide there is about 27 g of oxygen for every 100 g of tin. 13.5 and 27 form a ratio of 1:2, a ratio of small whole numbers. In these oxides, for every tin atom there are one or two oxygen atoms respectively (SnO and SnO2).
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The large rectangle is composed of 20 squares, each having dimensions of 1 by 1. Area of a cloth ; Multiplication (often denoted by the cross symbol , by the mid-line dot operator , by juxtaposition, or, on computers, by an asterisk ) is one of the four elementary mathematical operations of arithmetic, with the other ones being addition, subtraction and division. The result of a multiplication operation is called a product. The multiplication of whole numbers may be thought of as a repeated addition; that is, the multiplication of two numbers is equivalent to adding as many copies of one of them, the multiplicand, as the quantity of the other one, the multiplier.
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In sports statistics, certain percentages such as those for winning or win-loss records and saves in field or ice hockey and association football are almost always expressed as a decimal proportion to three places in AmE and are usually read aloud as if they are whole numbers, e.g. (0).500 or "five hundred", giving rise to the phrase "games/matches over five hundred", whereas in BrE they are also expressed but as true percentages instead, after multiplying the decimal by 100%, that is, 50% or "fifty per cent" and "games/matches over 50% or 50 per cent". However, "games/matches over 50% or 50 percent" is also found in AmE.
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John Dalton studied and expanded upon this previous work and defended a new idea, later known as the law of multiple proportions: if the same two elements can be combined to form a number of different compounds, then the ratios of the masses of the two elements in their various compounds will be represented by small whole numbers. This is a common pattern in chemical reactions that was observed by Dalton and other chemists at the time. Example 1 — tin oxides: Dalton identified two oxides of tin. One is a grey powder in which for every 100 parts of tin there is 13.5 parts of oxygen.
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In this context commensurability means that the ratio of the two planets' mean motions is very nearly equal to a ratio between a pair of small whole numbers. Two periods of Saturn's orbit around the Sun almost equal five of Jupiter's. The corresponding difference between multiples of the mean motions, , corresponds to a period of nearly 900 years, and it occurs as a small divisor in the integration of a very small perturbing force with this same period. As a result, the integrated perturbations with this period are disproportionately large, about 0.8° degrees of arc in orbital longitude for Saturn and about 0.3° for Jupiter.
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If a Condorcet winner exists, they will be elected. If not, (there is a Condorcet cycle) then the preference with the smallest majority will be eliminated. Nanson's method can be adapted to handle incomplete ballots (including "plumping") and equal rankings ("bracketing"), though he describes two different methods to handle these cases: a theoretically correct method involving fractions of a vote, and a practical method involving whole numbers (which has the side effect of diminishing the voting power of voters who plump or bracket). This then allows the use of Approval-style voting for uninformed voters who merely wish to approve of some candidates and disapprove of others.
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Among the earliest musical traditions, musical consonance was thought to arise in a quasi- mystical manner from ratios of small whole numbers. (For instance, Pythagoras made observations relating to this, and the ancient Chinese Guqin contains a dotted scale representing the harmonic series.) The source of these ratios, in the pattern of vibrations known as the harmonic series, was exposed by Joseph Sauveur the early 18th century and even more clearly by Helmholtz in the 1860s. In 1965, Plomp and Levelt showed that this relationship could be generalized beyond the harmonic series, although they did not elaborate in detail. In the 1990s, Sethares began exploring Plomp and Levelt's generalization, both mathematically and musically.
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The equation of time must be taken into account to ensure that the positions of the hour-lines are independent of the time of year when they are marked. An easy way to do this is to set a clock or watch so it shows "sundial time" which is standard time, plus the equation of time on the day in question. The hour-lines on the sundial are marked to show the positions of the shadow of the style when this clock shows whole numbers of hours, and are labelled with these numbers of hours. For example, when the clock reads 5:00, the shadow of the style is marked, and labelled "5" (or "V" in Roman numerals).
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By the time of his analysis, it was known from the Pythagorean diatonic scale that whole numbers alone accounted for musical intervals on a scale. Archytas's work on musical scales included a thorough proof that no mean proportional numbers, like the ones used in his solving of the double cube problem, exist between basic music intervals (the difference in pitch between two sounds). This is to say that the basic interval, does not include any mean proportional number, and cannot then be divided in half. The octave can be doubled without violating this rule, as multiplying a whole number by 2 will always result in a whole number, and can therefore be equated by two mean proportional ratios.
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When the percentage of points to discard does not yield a whole number, the trimmed mean may be defined by interpolation, generally linear interpolation, between the nearest whole numbers. For example, if you need to calculate the 15% trimmed mean of a sample containing 10 entries, strictly this would mean discarding 1 point from each end (equivalent to the 10% trimmed mean). If interpolating, one would instead compute the 10% trimmed mean (discarding 1 point from each end) and the 20% trimmed mean (discarding 2 points from each end), and then interpolating, in this case averaging these two values. Similarly, if interpolating the 12% trimmed mean, one would take the weighted average: weight the 10% trimmed mean by 0.8 and the 20% trimmed mean by 0.2.
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After posting a 5.04 goals against average (GAA) with no wins and two losses during his fill-in stint, he was sent back to the QMJHL to develop further. After the 2000–01 season, he was a consistent goaltender for the Sabres as his play in the crease improved drastically. Biron, along with Rob Ray and Dominik Hašek, was one of the three Sabres against whom, in three consecutive years, the NHL made a specific rule. After NHL statisticians discovered a bug in their new stat-tracking software, the "Biron rule" restricted jersey numbers to whole numbers between 1 and 99 (later limited to numbers between 1 and 98 after the league-wide retirement of number 99 for Wayne Gretzky).
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The r-process is responsible for our natural cohort of radioactive elements, such as uranium and thorium, as well as the most neutron-rich isotopes of each heavy element. The rp-process (rapid proton) involves the rapid absorption of free protons as well as neutrons, but its role and its existence are less certain. Explosive nucleosynthesis occurs too rapidly for radioactive decay to decrease the number of neutrons, so that many abundant isotopes with equal and even numbers of protons and neutrons are synthesized by the silicon quasi- equilibrium process. During this process, the burning of oxygen and silicon fuses nuclei that themselves have equal numbers of protons and neutrons to produce nuclides which consist of whole numbers of helium nuclei, up to 15 (representing 60Ni).
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The final episode considers the great unsolved problems that confronted mathematicians in the 20th century. On 8 August 1900 David Hilbert gave a historic talk at the International Congress of Mathematicians in Paris. Hilbert posed twenty-three then unsolved problems in mathematics which he believed were of the most immediate importance. Hilbert succeeded in setting the agenda for 20thC mathematics and the programme commenced with Hilbert's first problem. Georg Cantor considered the infinite set of whole numbers 1, 2, 3 ... ∞ which he compared with the smaller set of numbers 10, 20, 30 ... ∞. Cantor showed that these two infinite sets of numbers actually had the same size as it was possible to pair each number up; 1 - 10, 2 - 20, 3 - 30 ... etc.
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On this scale, for sand the value of Φ varies from −1 to +4, with the divisions between sub- categories at whole numbers. Close up of black volcanic sand from Perissa, Santorini, Greece The most common constituent of sand, in inland continental settings and non-tropical coastal settings, is silica (silicon dioxide, or SiO2), usually in the form of quartz, which, because of its chemical inertness and considerable hardness, is the most common mineral resistant to weathering. The composition of mineral sand is highly variable, depending on the local rock sources and conditions. The bright white sands found in tropical and subtropical coastal settings are eroded limestone and may contain coral and shell fragments in addition to other organic or organically derived fragmental material, suggesting that sand formation depends on living organisms, too.
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The International Bureau of Weights and Measures states that "when there are only four digits before or after the decimal marker, it is customary not to use a space to isolate a single digit". Likewise, some manuals of style state that thousands separators should not be used in normal text for numbers from 1,000 to 9,999 inclusive where no decimal fractional part is shown (in other words, for four-digit whole numbers), whereas others use thousands separators, and others use both. For example, APA style stipulates a thousands separator for "most figures of 1,000 or more" except for page numbers, binary digits, temperatures, etc. There are always "common-sense" country-specific exceptions to digit grouping, such as year numbers, postal codes and ID numbers of predefined nongrouped format, which style guides usually point out.
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The degree to which four suits in one hand, one suit in four hands, or all of the hands and suits are dealt in long and short holdings. Long and short holdings constitute "lots of distribution" and three-card holdings in particular constitute "no distribution". :: Specific. Either way, four whole numbers that sum to 13 are commonly used to denote a distribution briefly, such as 4333 or 4-3-3-3 for a hand comprising one four-card suit and three three-card suits; or for a suit with one four-card holding and three three-card holdings in the four hands. Also 22 or 2-2 for the opposing distribution of spades when one pair holds nine of them; or for one hand's distribution in the minors when it holds nine in the Majors.
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Daniel Thomas Gillespie (15 August 1938 – 19 April 2017) was a physicist who is best known for his derivation in 1976 of the stochastic simulation algorithm (SSA), also called the Gillespie algorithm. The SSA is a procedure for numerically simulating the time evolution of the molecular populations in a chemically reacting system in a way that takes account of the fact that molecules react in whole numbers and in a largely random way. Since the late 1990s, the SSA has been widely used to simulate chemical reactions inside living cells, where the small molecular populations of some reactant species often invalidate the differential equations of traditional deterministic chemical kinetics. Gillespie's original derivation of the SSA began by considering how chemical reactions actually occur in a well-stirred dilute gas.
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Numbers can count (3 apples), order (the 3rd apple), or measure (3.5 feet high); as the history of mathematics has progressed from counting on our fingers to modelling quantum mechanics, multiplication has been generalized to more complicated and abstract types of numbers, and to things that are not numbers (such as matrices) or do not look much like numbers (such as quaternions). ;Integers :N\times M is the sum of N copies of M when N and M are positive whole numbers. This gives the number of things in an array N wide and M high. Generalization to negative numbers can be done by :N\times (-M) = (-N)\times M = - (N\times M) and :(-N)\times (-M) = N\times M :The same sign rules apply to rational and real numbers.
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There exist multiple correlations, none of which are of very high quality. Use of SPT data for direct prediction of liquefaction potential suffers from roughness of correlations and from the need to "normalize" SPT data to account for overburden pressure, sampling technique, and other factors. Additionally, the method cannot collect accurate data for weak soil layers for several reasons: # The results are limited to whole numbers for a specific driving interval, but with very low blow counts, the granularity of the results, and the possibility of a zero result, makes handling the data cumbersome. # In loose sands and very soft clays, the act of driving the sampler will significantly disturb the soil, including by soil liquefaction of loose sands, giving results based on the disturbed soil properties rather than the intact soil properties.
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For over 2,000 years, Euclid's Elements stood as a perfectly solid foundation for mathematics, as its methodology of rational exploration guided mathematicians, philosophers, and scientists well into the 19th century. The Middle Ages saw a dispute over the ontological status of the universals (platonic Ideas): Realism asserted their existence independently of perception; conceptualism asserted their existence within the mind only; nominalism denied either, only seeing universals as names of collections of individual objects (following older speculations that they are words, "logoi"). René Descartes published La Géométrie (1637), aimed at reducing geometry to algebra by means of coordinate systems, giving algebra a more foundational role (while the Greeks embedded arithmetic into geometry by identifying whole numbers with evenly spaced points on a line). Descartes' book became famous after 1649 and paved the way to infinitesimal calculus.
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The origins of the Daily Worker begin with the weekly Ohio Socialist published by the Socialist Party of Ohio in Cleveland from 1917 to November 1919. The Ohio party joined the nascent Communist Labor Party of America at the 1919 Emergency National Convention. The Ohio Socialist only used whole numbers. Its final issue was #94 November 19, 1919. The Toiler continued this numbering, even though a typographical error made its debut issue #85 November 26, 1919. Beginning sometime in 1921 the volume number IV was added, perhaps reflecting the publications fourth year in print, though its issue numbers continued the whole number scheme. The final edition of the Toiler was Vol IV #207 January 28, 1922. The Worker continued the Toilers numbering during its run Vol. IV #208 February 2, 1922 to Vol.
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In music theory, this theorem implies that if a tuning system is generated by some number of consecutive multiples of a given interval, reduced to a cyclic sequence by considering two tones to be equivalent when they differ by whole numbers of octaves, then there are at most three different intervals between consecutive tones of the scale. For instance, the Pythagorean tuning is constructed in this way from multiples of a perfect fifth. It has only two distinct intervals representing its semitones, but if it were extended by one more step then the sequence of intervals between its tones would include a third shorter interval, the Pythagorean comma. In the theory of Sturmian words, the theorem implies that the words of a given length n that appear within a given Sturmian word have at most three distinct frequencies.
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For Aristotle and Euclid, relations were conceived as whole numbers (Michell, 1993). John Wallis later conceived of ratios of magnitudes as real numbers as reflected in the following: :When a comparison in terms of ratio is made, the resultant ratio often [namely with the exception of the 'numerical genus' itself] leaves the genus of quantities compared, and passes into the numerical genus, whatever the genus of quantities compared may have been. (John Wallis, Mathesis Universalis) That is, the ratio of magnitudes of any quantity, whether volume, mass, heat and so on, is a number. Following this, Newton then defined number, and the relationship between quantity and number, in the following terms: "By number we understand not so much a multitude of unities, as the abstracted ratio of any quantity to another quantity of the same kind, which we take for unity" (Newton, 1728).
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Compounds that have a monocyclic, planar conjugated system containing (4n + 2) π-electrons for whole numbers n are aromatic and exhibit an unusual stability. The classic example benzene has a system of six π electrons, which, together with the planar ring of C–C σ bonds containing 12 electrons and radial C–H σ bonds containing six electrons, forms the thermodynamically and kinetically stable benzene ring, the common core of the benzenoid aromatic compounds. For benzene itself, there are two equivalent conjugated contributing Lewis structures (the so-called Kekulé structures) that predominate.While the two Kekulé resonance forms contribute to most (>90%) of the π bond energy, there are also a number of other minor contributors to the wavefunction in the valence bond treatment, including the three Dewar resonance forms, and even smaller contributions from various ionic and singlet diradical forms.
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Sinclair's brief to the pair was fairly non-specific but primarily concerned remedying a key defect of the ZX80 so that the new machine could be used for practical programming and calculations. Vickers later recalled: The new ROM incorporated trigonometric and floating point functions, which its predecessor had lacked – the ZX80 could only deal with whole numbers. Grant came up with one of the ZX81's more novel features, a syntax checker that indicated errors in BASIC code as soon as it was entered (rather than, as was standard at the time, only disclosing coding errors when a program was run). Unfortunately for Vickers, he introduced a briefly notorious error – the so-called "square-root bug" that caused the square root of 0.25 to be returned erroneously as 1.3591409 – as a result of problems with integrating the ZX Printer code into the ROM.
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Also, unlike the Egyptians, Greeks, and Romans, the Babylonians had a true place-value system, where digits written in the left column represented larger values, much as in the decimal system. The power of the Babylonian notational system lay in that it could be used to represent fractions as easily as whole numbers; thus multiplying two numbers that contained fractions was no different than multiplying integers, similar to our modern notation. The notational system of the Babylonians was the best of any civilization until the Renaissance, and its power allowed it to achieve remarkable computational accuracy; for example, the Babylonian tablet YBC 7289 gives an approximation of accurate to five decimal places. The Babylonians lacked, however, an equivalent of the decimal point, and so the place value of a symbol often had to be inferred from the context.
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The first ideas on nucleosynthesis were simply that the chemical elements were created at the beginning of the universe, but no rational physical scenario for this could be identified. Gradually it became clear that hydrogen and helium are much more abundant than any of the other elements. All the rest constitute less than 2% of the mass of the Solar System, and of other star systems as well. At the same time it was clear that oxygen and carbon were the next two most common elements, and also that there was a general trend toward high abundance of the light elements, especially those with isotopes composed of whole numbers of helium-4 nuclei (alpha nuclides). Arthur Stanley Eddington first suggested in 1920, that stars obtain their energy by fusing hydrogen into helium and raised the possibility that the heavier elements may also form in stars.
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349-350 Pierre Ageron, studying superficially8 a copy of Ibn Hamza's manuscript in Ottoman Turkish, kept at the Süleymaniye Kütüphanesi library, and dated to the Hegirian year 1013, highlights an example linking geometric progression and arithmetic progression: the first written in oriental Arabic numerals (۱ ۲ ٤ ۸ ۱٦ ۳۲ ٦٤ ۱۲۸), and the second in alphabetical numbers (ا ب ج د ه و ز ح). In the margin is a figure which gives two graduations of the same segment: a regular one above, and a "logarithmic" graduation below. But, for the latter, the use of alphabetic and therefore whole numbers suggests that Ibn Hamza did not think of inserting non-integers and no approximate logarithm calculation is recorded in the manuscript9. Nevertheless we can note that, in the text in Ottoman Turkish, where Pierre Agero identifies the Arabic words us (exponent), dil'ayn (two sides) and a series of powers of 2 in oriental Arabic numerals and that of the corresponding exponents in numerals alphabetic, he was unable to read the actual text of the book because he did not master Ottoman Turkish.
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STP, a reaction between three cubic meters of hydrogen gas and one cubic meter of nitrogen gas will produce about two cubic meters of ammonia. The law of combining volumes states that, when gases react together they do so in volume which bears simple whole number ratio provided that the temperature and pressure of the reacting gases and their products remain constant The ratio between the volumes of the reactant gases and the gaseous products can be expressed in simple whole numbers. For example, Gay-Lussac found that two volumes of hydrogen and one volume of oxygen would react to form two volumes of gaseous water. Based on Gay-Lussac's results, Amedeo Avogadro hypothesized that, at the same temperature and pressure, equal volumes of gas contain equal numbers of molecules (Avogadro's law). This hypothesis meant that the previously stated result :2 volumes of hydrogen + 1 volume of oxygen = 2 volume of gaseous water could also be expressed as :2 molecules of hydrogen + 1 molecule of oxygen = 2 molecule of water. It can also be expressed in another way of example, 100 mL of hydrogen combine with 50 mL of oxygen to give 100 mL of water vapour.
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