1.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b

2.
Decimal
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This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0

3.
Division by zero
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In mathematics, division by zero is division where the divisor is zero. Such a division can be expressed as a/0 where a is the dividend. In ordinary arithmetic, the expression has no meaning, as there is no number which, multiplied by 0, gives a, and so division by zero is undefined. Since any number multiplied by zero is zero, the expression 0/0 also has no defined value, in computing, a program error may result from an attempt to divide by zero. When division is explained at the elementary level, it is often considered as splitting a set of objects into equal parts. As an example, consider having ten cookies, and these cookies are to be distributed equally to five people at a table, each person would receive 105 =2 cookies. Similarly, if there are ten cookies, and only one person at the table, so, for dividing by zero, what is the number of cookies that each person receives when 10 cookies are evenly distributed amongst 0 people at a table. Certain words can be pinpointed in the question to highlight the problem, the problem with this question is the when. There is no way to evenly distribute 10 cookies to nobody, in mathematical jargon, a set of 10 items cannot be partitioned into 0 subsets. So 100, at least in elementary arithmetic, is said to be either meaningless, similar problems occur if one has 0 cookies and 0 people, but this time the problem is in the phrase the number. A partition is possible, but since the partition has 0 parts, vacuously every set in our partition has a number of elements, be it 0,2,5. If there are, say,5 cookies and 2 people, in any integer partition of a 5-set into 2 parts, one of the parts of the partition will have more elements than the other. But the problem with 5 cookies and 2 people can be solved by cutting one cookie in half, the problem with 5 cookies and 0 people cannot be solved in any way that preserves the meaning of divides. Another way of looking at division by zero is that division can always be checked using multiplication. Considering the 10/0 example above, setting x = 10/0, if x equals ten divided by zero, then x times zero equals ten, but there is no x that, when multiplied by zero, gives ten. If instead of x=10/0 we have x=0/0, then every x satisfies the question what number x, multiplied by zero, the Brahmasphutasiddhanta of Brahmagupta is the earliest known text to treat zero as a number in its own right and to define operations involving zero. The author could not explain division by zero in his texts, according to Brahmagupta, A positive or negative number when divided by zero is a fraction with the zero as denominator. Zero divided by a negative or positive number is zero or is expressed as a fraction with zero as numerator

4.
Dyadic rational
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These are precisely the numbers whose binary expansion is finite. The inch is customarily subdivided in dyadic rather than decimal fractions, similarly, the divisions of the gallon into half-gallons, quarts. The ancient Egyptians also used dyadic fractions in measurement, with denominators up to 64, however, the result of dividing one dyadic fraction by another is not necessarily a dyadic fraction. Addition modulo 1 forms a group, this is the Prüfer 2-group, the set of all dyadic fractions is dense in the real line, any real number x can be arbitrarily closely approximated by dyadic rationals of the form ⌊2 i x ⌋ /2 i. Compared to other dense subsets of the line, such as the rational numbers, the dyadic rationals are in some sense a relatively small dense set. Considering only the addition and subtraction operations of the dyadic rationals gives them the structure of an abelian group. It is called the dyadic solenoid and is an example of a solenoid group, the group operation on these elements multiplies any two sequences componentwise. Each element of the dyadic solenoid corresponds to a character of the dyadic rationals that maps a/2b to the complex number qba, conversely, every character χ of the dyadic rationals corresponds to the element of the dyadic solenoid given by qi = χ. As a topological space the dyadic solenoid is a solenoid, the binary van der Corput sequence is an equidistributed permutation of the positive dyadic rational numbers. Time signatures in Western musical notation traditionally consist of dyadic fractions, non-dyadic time signatures are called irrational in musical terminology, but this usage does not correspond to the irrational numbers of mathematics, because they still consist of ratios of integers. Irrational time signatures in the mathematical sense are very rare, the same is true for the majority of fixed-point datatypes, which also uses powers of two implicitly in the majority of cases. Half-integer, a dyadic rational formed by dividing an odd number by two 2-adic number, a system that extends the dyadic rationals

5.
Farey sequence
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Each Farey sequence starts with the value 0, denoted by the fraction 0⁄1, and ends with the value 1, denoted by the fraction 1⁄1. A Farey sequence is called a Farey series, which is not strictly correct. — Beiler Chapter XVI Farey sequences are named after the British geologist John Farey, Farey conjectured, without offering proof, that each new term in a Farey sequence expansion is the mediant of its neighbours. Fareys letter was read by Cauchy, who provided a proof in his Exercices de mathématique, in fact, another mathematician, Charles Haros, had published similar results in 1802 which were not known either to Farey or to Cauchy. Thus it was an accident that linked Fareys name with these sequences. This is an example of Stiglers law of eponymy, the Farey sequence of order n contains all of the members of the Farey sequences of lower orders. In particular Fn contains all of the members of Fn−1 and also contains an additional fraction for each number that is less than n, thus F6 consists of F5 together with the fractions 1/6 and 5/6. The middle term of a Farey sequence Fn is always 1/2, from this, we can relate the lengths of Fn and Fn−1 using Eulers totient function φ, | F n | = | F n −1 | + φ. Using the fact that |F1| =2, we can derive an expression for the length of Fn, the asymptotic behaviour of |Fn| is, | F n | ∼3 n 2 π2. The index I n = k of a fraction a k, n in the Farey sequence F n = is simply the position that a k, n occupies in the sequence. This is of relevance as it is used in an alternative formulation of the Riemann hypothesis. Various useful properties follow, I n =0, I n =1, I n = /2, I n = | F n | −1, I n = | F n | −1 − I n. Fractions which are neighbouring terms in any Farey sequence are known as a Farey pair and have the following properties, if a/b and c/d are neighbours in a Farey sequence, with a/b < c/d, then their difference c/d − a/b is equal to 1/bd. Since c d − a b = b c − a d b d, thus 1/3 and 2/5 are neighbours in F5, and their difference is 1/15. If b c − a d =1 for positive integers a, b, c and d with a < b and c < d then a/b, thus the first term to appear between 1/3 and 2/5 is 3/8, which appears in F8. The Stern-Brocot tree is a data structure showing how the sequence is built up from 0 and 1, Fractions that appear as neighbours in a Farey sequence have closely related continued fraction expansions. Every fraction has two continued fraction expansions — in one the final term is 1, in the other the term is greater than 1. Farey sequences are useful to find rational approximations of irrational numbers

6.
Ford circle
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In mathematics, a Ford circle is a circle with center at and radius 1 /, where p / q is an irreducible fraction, i. e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y =0, Ford circles are a special case of mutually tangent circles, the base line can be thought of as a circle with infinite radius. Systems of mutually tangent circles were studied by Apollonius of Perga, after whom the problem of Apollonius, in the 17th century René Descartes discovered Descartes theorem, a relationship between the reciprocals of the radii of mutually tangent circles. Ford circles also appear in the Sangaku of Japanese mathematics, a typical problem, which is presented on an 1824 tablet in the Gunma Prefecture, covers the relationship of three touching circles with a common tangent. Given the size of the two outer circles, what is the size of the small circle between them. The answer is equivalent to a Ford circle,1 r middle =1 r left +1 r right, Ford circles are named after the American mathematician Lester R. Ford, Sr. who wrote about them in 1938. The Ford circle associated with the fraction p / q is denoted by C or C, there is a Ford circle associated with every rational number. In addition, the line y =1 is counted as a Ford circle – it can be thought of as the Ford circle associated with infinity, two different Ford circles are either disjoint or tangent to one another. No two interiors of Ford circles intersect, even there is a Ford circle tangent to the x-axis at each point on it with rational coordinates. If C and C are two tangent Ford circles, then the circle through and and that is perpendicular to the x -axis also passes through the point where the two circles are tangent to one another, Ford circles can also be thought of as curves in the complex plane. The modular group of transformations of the plane maps Ford circles to other Ford circles. Ford circles are a sub-set of the circles in the Apollonian gasket generated by the lines y =0 and y =1, by interpreting the upper half of the complex plane as a model of the hyperbolic plane Ford circles can be interpreted as horocycles. In hyperbolic geometry any two horocycles are congruent, when these horocycles are circumscribed by apeirogons they tile the hyperbolic plane with an order-3 apeirogonal tiling. There is a link between the area of Ford circles, Eulers totient function φ, the Riemann zeta function ζ, as no two Ford circles intersect, it follows immediately that the total area of the Ford circles is less than 1. In fact the area of these Ford circles is given by a convergent sum. From the definition, the area is A = ∑ q ≥1 ∑ =11 ≤ p < q π2, since ζ = π4 /90, this finally becomes A =452 ζ π3 ≈0.872284041. Fords Touching Circles at cut-the-knot Weisstein, Eric W. Ford Circle

7.
One half
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One half is the irreducible fraction resulting from dividing one by two, or the fraction resulting from dividing any number by its double. Multiplication by one half is equivalent to division by two, or halving, conversely, division by one half is equivalent to multiplication by two, or doubling, one half appears often in mathematical equations, recipes, measurements, etc. Half can also be said to be one part of something divided into two equal parts, for instance, the area S of a triangle is computed S = 1⁄2 × base × perpendicular height. The Riemann hypothesis states that every nontrivial complex root of the Riemann zeta function has a part equal to 1⁄2. One half has two different decimal expansions, the familiar 0.5 and the recurring 0.49999999 and it has a similar pair of expansions in any even base. It is a trap to believe these expressions represent distinct numbers. Equals 1 for detailed discussion of a related case, in odd bases, one half has no terminating representation, only a single representation with a repeating fractional component, such as 0.11111111. in ternary. 1⁄2 is also one of the few fractions to usually have a key of its own on typewriters and it also has its own code point in some early extensions of ASCII at 171. In Unicode, it has its own unit at U+00BD in the C1 Controls and Latin-1 Supplement block. List of numbers Division by two