1.
Rational number
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In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q. Since q may be equal to 1, every integer is a rational number, the decimal expansion of a rational number always either terminates after a finite number of digits or begins to repeat the same finite sequence of digits over and over. Moreover, any repeating or terminating decimal represents a rational number and these statements hold true not just for base 10, but also for any other integer base. A real number that is not rational is called irrational, irrational numbers include √2, π, e, and φ. The decimal expansion of an irrational number continues without repeating, since the set of rational numbers is countable, and the set of real numbers is uncountable, almost all real numbers are irrational. Rational numbers can be defined as equivalence classes of pairs of integers such that q ≠0, for the equivalence relation defined by ~ if. The rational numbers together with addition and multiplication form field which contains the integers and is contained in any field containing the integers, finite extensions of Q are called algebraic number fields, and the algebraic closure of Q is the field of algebraic numbers. In mathematical analysis, the numbers form a dense subset of the real numbers. The real numbers can be constructed from the numbers by completion, using Cauchy sequences, Dedekind cuts. The term rational in reference to the set Q refers to the fact that a number represents a ratio of two integers. In mathematics, rational is often used as a noun abbreviating rational number, the adjective rational sometimes means that the coefficients are rational numbers. However, a curve is not a curve defined over the rationals. Any integer n can be expressed as the rational number n/1, a b = c d if and only if a d = b c. Where both denominators are positive, a b < c d if and only if a d < b c. If either denominator is negative, the fractions must first be converted into equivalent forms with positive denominators, through the equations, − a − b = a b, two fractions are added as follows, a b + c d = a d + b c b d. A b − c d = a d − b c b d, the rule for multiplication is, a b ⋅ c d = a c b d. Where c ≠0, a b ÷ c d = a d b c, note that division is equivalent to multiplying by the reciprocal of the divisor fraction, a d b c = a b × d c. Additive and multiplicative inverses exist in the numbers, − = − a b = a − b and −1 = b a if a ≠0

2.
Real number
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In mathematics, a real number is a value that represents a quantity along a line. The adjective real in this context was introduced in the 17th century by René Descartes, the real numbers include all the rational numbers, such as the integer −5 and the fraction 4/3, and all the irrational numbers, such as √2. Included within the irrationals are the numbers, such as π. Real numbers can be thought of as points on a long line called the number line or real line. Any real number can be determined by a possibly infinite decimal representation, such as that of 8.632, the real line can be thought of as a part of the complex plane, and complex numbers include real numbers. These descriptions of the numbers are not sufficiently rigorous by the modern standards of pure mathematics. All these definitions satisfy the definition and are thus equivalent. The statement that there is no subset of the reals with cardinality greater than ℵ0. Simple fractions were used by the Egyptians around 1000 BC, the Vedic Sulba Sutras in, c.600 BC, around 500 BC, the Greek mathematicians led by Pythagoras realized the need for irrational numbers, in particular the irrationality of the square root of 2. Arabic mathematicians merged the concepts of number and magnitude into a general idea of real numbers. In the 16th century, Simon Stevin created the basis for modern decimal notation, in the 17th century, Descartes introduced the term real to describe roots of a polynomial, distinguishing them from imaginary ones. In the 18th and 19th centuries, there was work on irrational and transcendental numbers. Johann Heinrich Lambert gave the first flawed proof that π cannot be rational, Adrien-Marie Legendre completed the proof, Évariste Galois developed techniques for determining whether a given equation could be solved by radicals, which gave rise to the field of Galois theory. Charles Hermite first proved that e is transcendental, and Ferdinand von Lindemann, lindemanns proof was much simplified by Weierstrass, still further by David Hilbert, and has finally been made elementary by Adolf Hurwitz and Paul Gordan. The development of calculus in the 18th century used the set of real numbers without having defined them cleanly. The first rigorous definition was given by Georg Cantor in 1871, in 1874, he showed that the set of all real numbers is uncountably infinite but the set of all algebraic numbers is countably infinite. Contrary to widely held beliefs, his first method was not his famous diagonal argument, the real number system can be defined axiomatically up to an isomorphism, which is described hereafter. Another possibility is to start from some rigorous axiomatization of Euclidean geometry, from the structuralist point of view all these constructions are on equal footing

3.
Integer
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An integer is a number that can be written without a fractional component. For example,21,4,0, and −2048 are integers, while 9.75, 5 1⁄2, the set of integers consists of zero, the positive natural numbers, also called whole numbers or counting numbers, and their additive inverses. This is often denoted by a boldface Z or blackboard bold Z standing for the German word Zahlen, ℤ is a subset of the sets of rational and real numbers and, like the natural numbers, is countably infinite. The integers form the smallest group and the smallest ring containing the natural numbers, in algebraic number theory, the integers are sometimes called rational integers to distinguish them from the more general algebraic integers. In fact, the integers are the integers that are also rational numbers. Like the natural numbers, Z is closed under the operations of addition and multiplication, that is, however, with the inclusion of the negative natural numbers, and, importantly,0, Z is also closed under subtraction. The integers form a ring which is the most basic one, in the following sense, for any unital ring. This universal property, namely to be an object in the category of rings. Z is not closed under division, since the quotient of two integers, need not be an integer, although the natural numbers are closed under exponentiation, the integers are not. The following lists some of the properties of addition and multiplication for any integers a, b and c. In the language of algebra, the first five properties listed above for addition say that Z under addition is an abelian group. As a group under addition, Z is a cyclic group, in fact, Z under addition is the only infinite cyclic group, in the sense that any infinite cyclic group is isomorphic to Z. The first four properties listed above for multiplication say that Z under multiplication is a commutative monoid. However, not every integer has an inverse, e. g. there is no integer x such that 2x =1, because the left hand side is even. This means that Z under multiplication is not a group, all the rules from the above property table, except for the last, taken together say that Z together with addition and multiplication is a commutative ring with unity. It is the prototype of all objects of algebraic structure. Only those equalities of expressions are true in Z for all values of variables, note that certain non-zero integers map to zero in certain rings. The lack of zero-divisors in the means that the commutative ring Z is an integral domain

4.
0 (number)
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0 is both a number and the numerical digit used to represent that number in numerals. The number 0 fulfills a role in mathematics as the additive identity of the integers, real numbers. As a digit,0 is used as a placeholder in place value systems, names for the number 0 in English include zero, nought or naught, nil, or—in contexts where at least one adjacent digit distinguishes it from the letter O—oh or o. Informal or slang terms for zero include zilch and zip, ought and aught, as well as cipher, have also been used historically. The word zero came into the English language via French zéro from Italian zero, in pre-Islamic time the word ṣifr had the meaning empty. Sifr evolved to mean zero when it was used to translate śūnya from India, the first known English use of zero was in 1598. The Italian mathematician Fibonacci, who grew up in North Africa and is credited with introducing the system to Europe. This became zefiro in Italian, and was contracted to zero in Venetian. The Italian word zefiro was already in existence and may have influenced the spelling when transcribing Arabic ṣifr, modern usage There are different words used for the number or concept of zero depending on the context. For the simple notion of lacking, the words nothing and none are often used, sometimes the words nought, naught and aught are used. Several sports have specific words for zero, such as nil in football, love in tennis and it is often called oh in the context of telephone numbers. Slang words for zero include zip, zilch, nada, duck egg and goose egg are also slang for zero. Ancient Egyptian numerals were base 10 and they used hieroglyphs for the digits and were not positional. By 1740 BC, the Egyptians had a symbol for zero in accounting texts. The symbol nfr, meaning beautiful, was used to indicate the base level in drawings of tombs and pyramids. By the middle of the 2nd millennium BC, the Babylonian mathematics had a sophisticated sexagesimal positional numeral system, the lack of a positional value was indicated by a space between sexagesimal numerals. By 300 BC, a symbol was co-opted as a placeholder in the same Babylonian system. In a tablet unearthed at Kish, the scribe Bêl-bân-aplu wrote his zeros with three hooks, rather than two slanted wedges, the Babylonian placeholder was not a true zero because it was not used alone

5.
Dedekind cut
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Dedekind cuts are one method of construction of the real numbers. The set B may or may not have a smallest element among the rationals, if B has a smallest element among the rationals, the cut corresponds to that rational. Otherwise, that cut defines a unique number which, loosely speaking. In other words, A contains every rational number less than the cut, an irrational cut is equated to an irrational number which is in neither set. Every real number, rational or not, is equated to one, whenever, then, we have to do with a cut produced by no rational number, we create a new irrational number, which we regard as completely defined by this cut. From now on, therefore, to every definite cut there corresponds a definite rational or irrational number. More generally, a Dedekind cut is a partition of an ordered set into two non-empty parts A and B, such that A is closed downwards and B is closed upwards. It is straightforward to show that a Dedekind cut among the numbers is uniquely defined by the corresponding cut among the rational numbers. Similarly, every cut of reals is identical to the cut produced by a real number. In other words, the line where every real number is defined as a Dedekind cut of rationals is a complete continuum without any further gaps. Dedekind used the German word Schnitt in a visual sense rooted in Euclidean geometry and his theorem asserting the completeness of the real number system is nevertheless a theorem about numbers and not geometry. In David Hilberts axiom system, continuity is provided by the Axiom of Archimedes, in mathematical logic, the identification of the real numbers with the real number line is provided by the Cantor–Dedekind axiom. It is more symmetrical to use the notation for Dedekind cuts and it can be a simplification, in terms of notation if nothing more, to concentrate on one half — say, the lower one — and call any downward closed set A without greatest element a Dedekind cut. If the ordered set S is complete, then, for every Dedekind cut of S, the set B must have an element b, hence we must have that A is the interval. In this case, we say that b is represented by the cut, the important purpose of the Dedekind cut is to work with number sets that are not complete. The cut itself can represent a number not in the collection of numbers. The cut can represent a number b, even though the numbers contained in the two sets A and B do not actually include the number b that their cut represents. Even though there is no value for √2, if the rational numbers are partitioned into A and B this way

6.
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

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