1.
Cavalieri's principle
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If every line parallel to these two lines intersects both regions in line segments of equal length, then the two regions have equal areas. 3-dimensional case, Suppose two regions in three-space are included two parallel planes. If every plane parallel to two planes intersects both regions in cross-sections of equal area, then the two regions have equal volumes. In the other direction, Cavalieris principle grew out of the ancient Greek method of exhaustion, Cavalieris principle was originally called the method of indivisibles, the name it was known by in Renaissance Europe. Archimedes was able to find the volume of a sphere given the volumes of a cone, in the 5th century AD, Zu Chongzhi and his son Zu Gengzhi established a similar method to find a spheres volume. The transition from Cavalieris indivisibles to Evangelista Torricellis and John Walliss infinitesimals was an advance in the history of the calculus. The indivisibles were entities of codimension 1, so that a figure was thought as made out of an infinity of 1-dimensional lines. Meanwhile, infinitesimals were entities of the dimension as the figure they make up, thus. Applying the formula for the sum of a progression, Wallis computed the area of a triangle by partitioning it into infinitesimal parallelograms of width 1/∞. If one knows that the volume of a cone is 13, then one can use Cavalieris principle to derive the fact that the volume of a sphere is 43 π r 3, where r is the radius. That is done as follows, Consider a sphere of radius r, within the cylinder is the cone whose apex is at the center of the sphere and whose base is the base of the cylinder. By the Pythagorean theorem, the plane located y units above the equator intersects the sphere in a circle of area π, the area of the planes intersection with the part of the cylinder that is outside of the cone is also π. The aforementioned volume of the cone is 13 of the volume of the cylinder, Therefore the volume of the upper half of the sphere is 23 of the volume of the cylinder. The volume of the cylinder is base × height = π r 2 ⋅ r = π r 3 Therefore the volume of the upper half-sphere is π r 3 and that of the whole sphere is π r 3. One may initially establish it in a case by partitioning the interior of a triangular prism into three pyramidal components of equal volumes. One may show the equality of three volumes by means of Cavalieris principle. The ancient Greeks used various techniques such as Archimedess mechanical arguments or method of exhaustion to compute these volumes. The cross-section of the ring is a plane annulus, whose area is the difference between the areas of two circles

2.
Complex number
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A complex number is a number that can be expressed in the form a + bi, where a and b are real numbers and i is the imaginary unit, satisfying the equation i2 = −1. In this expression, a is the part and b is the imaginary part of the complex number. If z = a + b i, then ℜ z = a, ℑ z = b, Complex numbers extend the concept of the one-dimensional number line to the two-dimensional complex plane by using the horizontal axis for the real part and the vertical axis for the imaginary part. The complex number a + bi can be identified with the point in the complex plane, a complex number whose real part is zero is said to be purely imaginary, whereas a complex number whose imaginary part is zero is a real number. In this way, the numbers are a field extension of the ordinary real numbers. As well as their use within mathematics, complex numbers have applications in many fields, including physics, chemistry, biology, economics, electrical engineering. The Italian mathematician Gerolamo Cardano is the first known to have introduced complex numbers and he called them fictitious during his attempts to find solutions to cubic equations in the 16th century. Complex numbers allow solutions to equations that have no solutions in real numbers. For example, the equation 2 = −9 has no real solution, Complex numbers provide a solution to this problem. The idea is to extend the real numbers with the unit i where i2 = −1. According to the theorem of algebra, all polynomial equations with real or complex coefficients in a single variable have a solution in complex numbers. A complex number is a number of the form a + bi, for example, −3.5 + 2i is a complex number. The real number a is called the part of the complex number a + bi. By this convention the imaginary part does not include the unit, hence b. The real part of a number z is denoted by Re or ℜ. For example, Re = −3.5 Im =2, hence, in terms of its real and imaginary parts, a complex number z is equal to Re + Im ⋅ i. This expression is known as the Cartesian form of z. A real number a can be regarded as a number a + 0i whose imaginary part is 0

3.
Number
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Numbers that answer the question How many. Are 0,1,2,3 and so on, when used to indicate position in a sequence they are ordinal numbers. To the Pythagoreans and Greek mathematician Euclid, the numbers were 2,3,4,5, Euclid did not consider 1 to be a number. Numbers like 3 +17 =227, expressible as fractions in which the numerator and denominator are whole numbers, are rational numbers and these make it possible to measure such quantities as two and a quarter gallons and six and a half miles. What we today would consider a proof that a number is irrational Euclid called a proof that two lengths arising in geometry have no common measure, or are incommensurable, Euclid included proofs of incommensurability of lengths arising in geometry in his Elements. In the Rhind Mathematical Papyrus, a pair of walking forward marked addition. They were the first known civilization to use negative numbers, negative numbers came into widespread use as a result of their utility in accounting. They were used by late medieval Italian bankers, by 1740 BC, the Egyptians had a symbol for zero in accounting texts. In Maya civilization zero was a numeral with a shape as a symbol. The ancient Egyptians represented all fractions in terms of sums of fractions with numerator 1, for example, 2/5 = 1/3 + 1/15. Such representations are known as Egyptian Fractions or Unit Fractions. The earliest written approximations of π are found in Egypt and Babylon, in Babylon, a clay tablet dated 1900–1600 BC has a geometrical statement that, by implication, treats π as 25/8 =3.1250. In Egypt, the Rhind Papyrus, dated around 1650 BC, astronomical calculations in the Shatapatha Brahmana use a fractional approximation of 339/108 ≈3.139. Other Indian sources by about 150 BC treat π as √10 ≈3.1622 The first references to the constant e were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms calculated from the constant and it is assumed that the table was written by William Oughtred. The discovery of the constant itself is credited to Jacob Bernoulli, the first known use of the constant, represented by the letter b, was in correspondence from Gottfried Leibniz to Christiaan Huygens in 1690 and 1691. Leonhard Euler introduced the letter e as the base for natural logarithms, Euler started to use the letter e for the constant in 1727 or 1728, in an unpublished paper on explosive forces in cannons, and the first appearance of e in a publication was Eulers Mechanica. While in the subsequent years some researchers used the letter c, e was more common, the first numeral system known is Babylonian numeric system, that has a 60 base, it was introduced in 3100 B. C. and is the first Positional numeral system known

4.
Natural number
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In mathematics, the natural numbers are those used for counting and ordering. In common language, words used for counting are cardinal numbers, texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, but in other writings, that term is used instead for the integers. These chains of extensions make the natural numbers canonically embedded in the number systems. Properties of the numbers, such as divisibility and the distribution of prime numbers, are studied in number theory. Problems concerning counting and ordering, such as partitioning and enumerations, are studied in combinatorics, the most primitive method of representing a natural number is to put down a mark for each object. Later, a set of objects could be tested for equality, excess or shortage, by striking out a mark, the first major advance in abstraction was the use of numerals to represent numbers. This allowed systems to be developed for recording large numbers, the ancient Egyptians developed a powerful system of numerals with distinct hieroglyphs for 1,10, and all the powers of 10 up to over 1 million. A stone carving from Karnak, dating from around 1500 BC and now at the Louvre in Paris, depicts 276 as 2 hundreds,7 tens, and 6 ones, and similarly for the number 4,622. A much later advance was the development of the idea that 0 can be considered as a number, with its own numeral. The use of a 0 digit in place-value notation dates back as early as 700 BC by the Babylonians, the Olmec and Maya civilizations used 0 as a separate number as early as the 1st century BC, but this usage did not spread beyond Mesoamerica. The use of a numeral 0 in modern times originated with the Indian mathematician Brahmagupta in 628, the first systematic study of numbers as abstractions is usually credited to the Greek philosophers Pythagoras and Archimedes. Some Greek mathematicians treated the number 1 differently than larger numbers, independent studies also occurred at around the same time in India, China, and Mesoamerica. In 19th century Europe, there was mathematical and philosophical discussion about the nature of the natural numbers. A school of Naturalism stated that the numbers were a direct consequence of the human psyche. Henri Poincaré was one of its advocates, as was Leopold Kronecker who summarized God made the integers, in opposition to the Naturalists, the constructivists saw a need to improve the logical rigor in the foundations of mathematics. In the 1860s, Hermann Grassmann suggested a recursive definition for natural numbers thus stating they were not really natural, later, two classes of such formal definitions were constructed, later, they were shown to be equivalent in most practical applications. The second class of definitions was introduced by Giuseppe Peano and is now called Peano arithmetic and it is based on an axiomatization of the properties of ordinal numbers, each natural number has a successor and every non-zero natural number has a unique predecessor. Peano arithmetic is equiconsistent with several systems of set theory

5.
Quaternion
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In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843, a feature of quaternions is that multiplication of two quaternions is noncommutative. Hamilton defined a quaternion as the quotient of two directed lines in a space or equivalently as the quotient of two vectors. Quaternions are generally represented in the form, a + bi + cj + dk where a, b, c, and d are real numbers, and i, j, and k are the fundamental quaternion units. In practical applications, they can be used other methods, such as Euler angles and rotation matrices, or as an alternative to them. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, in fact, the quaternions were the first noncommutative division algebra to be discovered. The algebra of quaternions is often denoted by H, or in blackboard bold by H and it can also be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. These rings are also Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. The unit quaternions can be thought of as a choice of a structure on the 3-sphere S3 that gives the group Spin. Quaternion algebra was introduced by Hamilton in 1843, carl Friedrich Gauss had also discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the numbers could be interpreted as points in a plane. Points in space can be represented by their coordinates, which are triples of numbers, however, Hamilton had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space. The great breakthrough in quaternions finally came on Monday 16 October 1843 in Dublin, as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i2 = j2 = k2 = ijk = −1, into the stone of Brougham Bridge as he paused on it. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves and this letter was later published in the London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, vol. xxv, pp 489–95. In the letter, Hamilton states, And here there dawned on me the notion that we must admit, in some sense, an electric circuit seemed to close, and a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, Hamiltons treatment is more geometric than the modern approach, which emphasizes quaternions algebraic properties

6.
Non-standard analysis
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The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using epsilon–delta procedures rather than infinitesimals, Non-standard analysis instead reformulates the calculus using a logically rigorous notion of infinitesimal numbers. Non-standard analysis was originated in the early 1960s by the mathematician Abraham Robinson and he wrote, the idea of infinitely small or infinitesimal quantities seems to appeal naturally to our intuition. At any rate, the use of infinitesimals was widespread during the stages of the Differential and Integral Calculus. Robinson continued, However, neither he nor his disciples and successors were able to give a rational development leading up to a system of this sort, as a result, the theory of infinitesimals gradually fell into disrepute and was replaced eventually by the classical theory of limits. The key to our method is provided by the analysis of the relation between mathematical languages and mathematical structures which lies at the bottom of contemporary model theory. In 1973, intuitionist Arend Heyting praised non-standard analysis as a model of important mathematical research. A non-zero element of an ordered field F is infinitesimal if and only if its value is smaller than any element of F of the form 1 n, for n. Ordered fields that have infinitesimal elements are also called non-Archimedean, more generally, non-standard analysis is any form of mathematics that relies on non-standard models and the transfer principle. A field which satisfies the principle for real numbers is a hyperreal field. Robinsons original approach was based on these models of the field of real numbers. His classic foundational book on the subject Non-standard Analysis was published in 1966 and is still in print, on page 88, Robinson writes, The existence of non-standard models of arithmetic was discovered by Thoralf Skolem. Skolems method foreshadows the ultrapower construction Several technical issues must be addressed to develop a calculus of infinitesimals, for example, it is not enough to construct an ordered field with infinitesimals. See the article on numbers for a discussion of some of the relevant ideas. In this section we outline one of the simplest approaches to defining a hyperreal field ∗ R, let R be the field of real numbers, and let N be the semiring of natural numbers. Denote by R N the set of sequences of real numbers, a field ∗ R is defined as a suitable quotient of R N, as follows. Take a nonprincipal ultrafilter F ⊂ P, in particular, F contains the Fréchet filter. There are at least three reasons to consider non-standard analysis, historical, pedagogical, and technical, much of the earliest development of the infinitesimal calculus by Newton and Leibniz was formulated using expressions such as infinitesimal number and vanishing quantity

7.
Thoralf Skolem
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Thoralf Albert Skolem was a Norwegian mathematician who worked in mathematical logic and set theory. Although Skolems father was a school teacher, most of his extended family were farmers. Skolem attended secondary school in Kristiania, passing the university examinations in 1905. He then entered Det Kongelige Frederiks Universitet to study mathematics, also taking courses in physics, chemistry, zoology, in 1913, Skolem passed the state examinations with distinction, and completed a dissertation titled Investigations on the Algebra of Logic. He also traveled with Birkeland to the Sudan to observe the zodiacal light, in 1916 he was appointed a research fellow at Det Kongelige Frederiks Universitet. In 1918, he became a Docent in Mathematics and was elected to the Norwegian Academy of Science, Skolem did not at first formally enroll as a Ph. D. candidate, believing that the Ph. D. was unnecessary in Norway. He later changed his mind and submitted a thesis in 1926, titled Some theorems about integral solutions to certain algebraic equations and his notional thesis advisor was Axel Thue, even though Thue had died in 1922. In 1927, he married Edith Wilhelmine Hasvold, Skolem continued to teach at Det kongelige Frederiks Universitet until 1930 when he became a Research Associate in Chr. This senior post allowed Skolem to conduct research free of administrative, in 1938, he returned to Oslo to assume the Professorship of Mathematics at the university. There he taught the courses in algebra and number theory. Skolems Ph. D. student Øystein Ore went on to a career in the USA, Skolem served as president of the Norwegian Mathematical Society, and edited the Norsk Matematisk Tidsskrift for many years. He was also the editor of Mathematica Scandinavica. After his 1957 retirement, he made trips to the United States. He remained intellectually active until his sudden and unexpected death, for more on Skolems academic life, see Fenstad. Skolem published around 180 papers on Diophantine equations, group theory, lattice theory and he mostly published in Norwegian journals with limited international circulation, so that his results were occasionally rediscovered by others. An example is the Skolem–Noether theorem, characterizing the automorphisms of simple algebras, Skolem published a proof in 1927, but Emmy Noether independently rediscovered it a few years later. Skolem was among the first to write on lattices, in 1912, he was the first to describe a free distributive lattice generated by n elements. In 1919, he showed that every lattice is distributive and, as a partial converse

8.
Dual quaternion
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In mathematics and mechanics, the set of dual quaternions is a Clifford algebra that can be used to represent spatial rigid body displacements. A dual quaternion is a pair of quaternions Â =. Because rigid body displacements are defined by six parameters, dual quaternion parameters include two algebraic constraints, in ring theory, dual quaternions are a ring constructed in the same way as the quaternions, except using dual numbers instead of real numbers as coefficients. A dual quaternion can be represented in the form p + ε q where p and q are ordinary quaternions and ε is the dual unit, unlike quaternions they do not form a division ring. Similar to the way that rotations in 3D space can be represented by quaternions of unit length and this fact is used in theoretical kinematics, and in applications to 3D computer graphics, robotics and computer vision. In 1898 Alexander McAulay used Ω with Ω2 =0 to generate the dual quaternion algebra, however, his terminology of octonions did not stick as todays octonions are another algebra. In Russia, Aleksandr Kotelnikov developed dual vectors and dual quaternions for use in the study of mechanics, in 1891 Eduard Study realized that this associative algebra was ideal for describing the group of motions of three-dimensional space. He further developed the idea in Geometrie der Dynamen in 1901, B. L. van der Waerden called the structure Study biquaternions, one of three eight-dimensional algebras referred to as biquaternions. In order to describe operations with dual quaternions, it is helpful to first consider quaternions, a quaternion is a linear combinations of the basis elements 1, i, j, and k. Hamiltons product rule for i, j, and k is often written as i 2 = j 2 = k 2 = i j k = −1. Compute i = −j k = −i, to obtain j k = i, now because j = j i = −k, we see that this product yields i j = −j i, which links quaternions to the properties of determinants. The vector dot and cross operations can now be used to define the product of A = a0 + A and C = c0 + C as G = A C = = +. A dual quaternion is usually described as a quaternion with dual numbers as coefficients, a dual number is an ordered pair â =. Two dual numbers add componentwise and multiply by the rule â ĉ = =, Dual numbers are often written in the form â = a + εb, where ε is the dual unit that commutes with i, j, k and has the property ε2 =0. The result is that a dual quaternion is the pair of quaternions Â =. Two dual quaternions add componentwise and multiply by the rule, A ^ C ^ = =. It is convenient to write a dual quaternion as the sum of a scalar and a dual vector, Â = â0 + A. This notation allows us to write the product of two dual quaternions as G ^ = A ^ C ^ = = +

9.
Leibniz's notation
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The derivative of a function of a real variable measures the sensitivity to change of the function value with respect to a change in its argument. Derivatives are a tool of calculus. For example, the derivative of the position of an object with respect to time is the objects velocity. The derivative of a function of a variable at a chosen input value. The tangent line is the best linear approximation of the function near that input value, for this reason, the derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. Derivatives may be generalized to functions of real variables. In this generalization, the derivative is reinterpreted as a transformation whose graph is the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables and it can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of variables, the Jacobian matrix reduces to the gradient vector. The process of finding a derivative is called differentiation, the reverse process is called antidifferentiation. The fundamental theorem of calculus states that antidifferentiation is the same as integration, differentiation and integration constitute the two fundamental operations in single-variable calculus. Differentiation is the action of computing a derivative, the derivative of a function y = f of a variable x is a measure of the rate at which the value y of the function changes with respect to the change of the variable x. It is called the derivative of f with respect to x, If x and y are real numbers, and if the graph of f is plotted against x, the derivative is the slope of this graph at each point. The simplest case, apart from the case of a constant function, is when y is a linear function of x. This formula is true because y + Δ y = f = m + b = m x + m Δ x + b = y + m Δ x. Thus, since y + Δ y = y + m Δ x and this gives an exact value for the slope of a line. If the function f is not linear, however, then the change in y divided by the change in x varies, differentiation is a method to find an exact value for this rate of change at any given value of x. The idea, illustrated by Figures 1 to 3, is to compute the rate of change as the value of the ratio of the differences Δy / Δx as Δx becomes infinitely small

10.
Algebraic number
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An algebraic number is any complex number that is a root of a non-zero polynomial in one variable with rational coefficients. All integers and rational numbers are algebraic, as are all roots of integers, the same is not true for all real and complex numbers because they also include transcendental numbers such as π and e. Almost all real and complex numbers are transcendental, the rational numbers, expressed as the quotient of two integers a and b, b not equal to zero, satisfy the above definition because x = a/b is the root of bx − a. The quadratic surds are algebraic numbers, if the quadratic polynomial is monic then the roots are quadratic integers. The constructible numbers are numbers that can be constructed from a given unit length using straightedge. These include all quadratic surds, all numbers, and all numbers that can be formed from these using the basic arithmetic operations. Any expression formed from algebraic numbers using any combination of the arithmetic operations. Polynomial roots that cannot be expressed in terms of the arithmetic operations. This happens with many, but not all, polynomials of degree 5 or higher, gaussian integers, those complex numbers a + bi where both a and b are integers are also quadratic integers. Trigonometric functions of rational multiples of π, that is, the trigonometric numbers, for example, each of cos π/7, cos 3π/7, cos 5π/7 satisfies 8x3 − 4x2 − 4x +1 =0. This polynomial is irreducible over the rationals, and so these three cosines are conjugate algebraic numbers. Likewise, tan 3π/16, tan 7π/16, tan 11π/16, tan 15π/16 all satisfy the irreducible polynomial x4 − 4x3 − 6x2 + 4x +1, and so are conjugate algebraic integers. Some irrational numbers are algebraic and some are not, The numbers √2 and 3√3/2 are algebraic since they are roots of polynomials x2 −2 and 8x3 −3, the golden ratio φ is algebraic since it is a root of the polynomial x2 − x −1. The numbers π and e are not algebraic numbers, hence they are transcendental, the set of algebraic numbers is countable. Hence, the set of numbers has Lebesgue measure zero. Given an algebraic number, there is a monic polynomial of least degree that has the number as a root. This polynomial is called its minimal polynomial, if its minimal polynomial has degree n, then the algebraic number is said to be of degree n. An algebraic number of degree 1 is a rational number, a real algebraic number of degree 2 is a quadratic irrational

11.
Gaussian integer
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In number theory, a Gaussian integer is a complex number whose real and imaginary parts are both integers. The Gaussian integers, with addition and multiplication of complex numbers, form an integral domain. This integral domain is a case of a commutative ring of quadratic integers. It does not have an ordering that respects arithmetic. Formally, the Gaussian integers are the set Z =, where i 2 = −1, note that when they are considered within the complex plane the Gaussian integers may be seen to constitute the 2-dimensional integer lattice. The norm of a Gaussian integer is the square of its value as a complex number. It is the natural number defined as N = a 2 + b 2 = ¯ =, the norm is multiplicative, since the absolute value of complex numbers is multiplicative, i. e. one has N = N N. The latter can also be verified by a straightforward check, the units of Z are precisely those elements with norm 1, i. e. the set. The Gaussian integers form a principal ideal domain with units, for x ∈ Z, the four numbers ±x, ±ix are called the associates of x. As for every principal ideal domain, Z is also a unique factorization domain and it follows that a Gaussian integer is prime if and only if it is irreducible. The prime elements of Z are also known as Gaussian primes, an associate of a Gaussian prime is also a Gaussian prime. The Gaussian primes are symmetric about the real and imaginary axes, the positive integer Gaussian primes are the prime numbers that are congruent to 3 modulo 4. One should not refer to only these numbers as the Gaussian primes, which refers to all the Gaussian primes, many of which do not lie in Z. In other words, a Gaussian integer is a Gaussian prime if and only if either its norm is a prime number, for example,5 = · and 13 = ·. If p =2, we have 2 = = i2, the ring of Gaussian integers is the integral closure of Z in the field of Gaussian rationals Q consisting of the complex numbers whose real and imaginary part are both rational. It is easy to see graphically that every number is no farther than a distance of 22 from some Gaussian integer. Put another way, every number has a maximal distance of 22 N units to some multiple of z, where z is any Gaussian integer, this turns Z into a Euclidean domain. The ring of Gaussian integers was introduced by Carl Friedrich Gauss in his monograph on quartic reciprocity

12.
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