1.
Getal (wiskunde)
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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
2.
Geheel getal
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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
3.
Rationaal getal
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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
4.
Reëel getal
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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
5.
Complex getal
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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
6.
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
7.
P-adisch getal
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The extension is achieved by an alternative interpretation of the concept of closeness or absolute value. In particular, p-adic numbers have the property that they are said to be close when their difference is divisible by a high power of p, the higher the power. P-adic numbers were first described by Kurt Hensel in 1897, though, with hindsight, the p-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of power series methods into number theory. Their influence now extends far beyond this, for example, the field of p-adic analysis essentially provides an alternative form of calculus. More formally, for a prime p, the field Qp of p-adic numbers is a completion of the rational numbers. The field Qp is also given a topology derived from a metric, which is derived from the p-adic order. This metric space is complete in the sense that every Cauchy sequence converges to a point in Qp and this is what allows the development of calculus on Qp, and it is the interaction of this analytic and algebraic structure that gives the p-adic number systems their power and utility. The p in p-adic is a variable and may be replaced with a prime or another placeholder variable, the adic of p-adic comes from the ending found in words such as dyadic or triadic. This section is an introduction to p-adic numbers, using examples from the ring of 10-adic numbers. Although for p-adic numbers p should be a prime, base 10 was chosen to highlight the analogy with decimals, the decadic numbers are generally not used in mathematics, since 10 is not prime, the decadics are not a field. More formal constructions and properties are given below, in the standard decimal representation, almost all real numbers do not have a terminating decimal representation. For example, 1/3 is represented as a non-terminating decimal as follows 13 =0.333333 …, informally, non-terminating decimals are easily understood, because it is clear that a real number can be approximated to any required degree of precision by a terminating decimal. If two decimal expansions differ only after the 10th decimal place, they are close to one another. 10-adic numbers use a similar non-terminating expansion, but with a different concept of closeness, whereas two decimal expansions are close to one another if their difference is a large negative power of 10, two 10-adic expansions are close if their difference is a large positive power of 10. Thus 4739 and 5739, which differ by 103, are close in the 10-adic world, more precisely, a positive rational number r can be expressed as r =, p/q·10e, where p and q are positive integers and q is relatively prime to p and to 10. For each r ≠0 there exists the maximal e such that this representation is possible, let the 10-adic «absolute value» of r be | r |10, =110 e. Certainly, we have to define |0|10, =0, now, taking p/q =1 and e =0,1,2. We have |100|10 =100, |101|10 = 10−1, |102|10 = 10−2, with the consequence that we have lim + ∞ ← e 10 e =0
8.
Surreëel getal
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The surreals share many properties with the reals, including the usual arithmetic operations, as such, they form an ordered field. The surreals also contain all transfinite ordinal numbers, the arithmetic on them is given by the natural operations, research on the go endgame by John Horton Conway led to another definition and construction of the surreal numbers. Conways construction was introduced in Donald Knuths 1974 book Surreal Numbers, How Two Ex-Students Turned on to Pure Mathematics, in his book, which takes the form of a dialogue, Knuth coined the term surreal numbers for what Conway had called simply numbers. Conway later adopted Knuths term, and used surreals for analyzing games in his 1976 book On Numbers and Games. In the Conway construction, the numbers are constructed in stages. Different subsets may end up defining the same number, and may define the number even if L ≠ L′. So strictly speaking, the numbers are equivalence classes of representations of form that designate the same number. In the first stage of construction, there are no previously existing numbers so the representation must use the empty set. This representation, where L and R are both empty, is called 0, subsequent stages yield forms like, =1 =2 =3 and = −1 = −2 = −3 The integers are thus contained within the surreal numbers. Similarly, representations arise like, = 1/2 = 1/4 = 3/4 so that the rationals are contained within the surreal numbers. Thus the real numbers are also embedded within the surreals, but there are also representations like = ω = ε where ω is a transfinite number greater than all integers and ε is an infinitesimal greater than 0 but less than any positive real number. The construction consists of three interdependent parts, the rule, the comparison rule and the equivalence rule. A form is a pair of sets of numbers, called its left set. A form with left set L and right set R is written, when L and R are given as lists of elements, the braces around them are omitted. Either or both of the left and right set of a form may be the empty set, the form with both left and right set empty is also written. The numeric forms are placed in classes, each such equivalence class is a surreal number. The elements of the left and right set of a form are drawn from the universe of the surreal numbers, equivalence Rule Two numeric forms x and y are forms of the same number if and only if both x ≤ y and y ≤ x. An ordering relationship must be antisymmetric, i. e. it must have the property that x = y only when x and y are the same object and this is not the case for surreal number forms, but is true by construction for surreal numbers
9.
Irrationaal getal
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In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
10.
Algebraïsch getal
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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