1.
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
2.
Apple
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The apple tree is a deciduous tree in the rose family best known for its sweet, pomaceous fruit, the apple. It is cultivated worldwide as a tree, and is the most widely grown species in the genus Malus. The tree originated in Central Asia, where its wild ancestor, Apples have been grown for thousands of years in Asia and Europe, and were brought to North America by European colonists. Apples have religious and mythological significance in many cultures, including Norse, Greek, Apple trees are large if grown from seed. Generally apple cultivars are propagated by grafting onto rootstocks, which control the size of the resulting tree, There are more than 7,500 known cultivars of apples, resulting in a range of desired characteristics. Different cultivars are bred for various tastes and uses, including cooking, eating raw, trees and fruit are prone to a number of fungal, bacterial and pest problems, which can be controlled by a number of organic and non-organic means. In 2010, the genome was sequenced as part of research on disease control. Worldwide production of apples in 2014 was 84.6 million tonnes, the apple is a deciduous tree, generally standing 1.8 to 4.6 m tall in cultivation and up to 12 m in the wild. When cultivated, the size, shape and branch density are determined by rootstock selection, the leaves are alternately arranged dark green-colored simple ovals with serrated margins and slightly downy undersides. Blossoms are produced in spring simultaneously with the budding of the leaves, the 3 to 4 cm flowers are white with a pink tinge that gradually fades, five petaled, with an inflorescence consisting of a cyme with 4–6 flowers. The central flower of the inflorescence is called the king bloom, it opens first, the fruit matures in late summer or autumn, and cultivars exist with a wide range of sizes. Commercial growers aim to produce an apple that is 7.0 to 8.3 cm in diameter, due to market preference. Some consumers, especially those in Japan, prefer a larger apple, while apples below 5.7 cm are generally used for making juice and have little market value. The skin of ripe apples is generally red, yellow, green, pink, the skin may also be wholly or partly russeted i. e. rough and brown. The skin is covered in a layer of epicuticular wax. The exocarp is generally pale yellowish-white, though pink or yellow also occur. The original wild ancestor of Malus pumila was Malus sieversii, found growing wild in the mountains of Central Asia in southern Kazakhstan, Kyrgyzstan, Tajikistan, in strains without recent admixture the contribution of the latter predominates. It had about 57,000 genes, the highest number of any plant genome studied to date and this new understanding of the apple genome will help scientists in identifying genes and gene variants that contribute to resistance to disease and drought, and other desirable characteristics
3.
Mathematics
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Mathematics is the study of topics such as quantity, structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope, Mathematicians seek out patterns and use them to formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proof, when mathematical structures are good models of real phenomena, then mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation, measurement, practical mathematics has been a human activity from as far back as written records exist. The research required to solve mathematical problems can take years or even centuries of sustained inquiry, rigorous arguments first appeared in Greek mathematics, most notably in Euclids Elements. Galileo Galilei said, The universe cannot be read until we have learned the language and it is written in mathematical language, and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word. Without these, one is wandering about in a dark labyrinth, carl Friedrich Gauss referred to mathematics as the Queen of the Sciences. Benjamin Peirce called mathematics the science that draws necessary conclusions, David Hilbert said of mathematics, We are not speaking here of arbitrariness in any sense. Mathematics is not like a game whose tasks are determined by arbitrarily stipulated rules, rather, it is a conceptual system possessing internal necessity that can only be so and by no means otherwise. Albert Einstein stated that as far as the laws of mathematics refer to reality, they are not certain, Mathematics is essential in many fields, including natural science, engineering, medicine, finance and the social sciences. Applied mathematics has led to entirely new mathematical disciplines, such as statistics, Mathematicians also engage in pure mathematics, or mathematics for its own sake, without having any application in mind. There is no clear line separating pure and applied mathematics, the history of mathematics can be seen as an ever-increasing series of abstractions. The earliest uses of mathematics were in trading, land measurement, painting and weaving patterns, in Babylonian mathematics elementary arithmetic first appears in the archaeological record. Numeracy pre-dated writing and numeral systems have many and diverse. Between 600 and 300 BC the Ancient Greeks began a study of mathematics in its own right with Greek mathematics. Mathematics has since been extended, and there has been a fruitful interaction between mathematics and science, to the benefit of both. Mathematical discoveries continue to be made today, the overwhelming majority of works in this ocean contain new mathematical theorems and their proofs. The word máthēma is derived from μανθάνω, while the modern Greek equivalent is μαθαίνω, in Greece, the word for mathematics came to have the narrower and more technical meaning mathematical study even in Classical times
4.
Counting
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Counting is the action of finding the number of elements of a finite set of objects. The related term refers to uniquely identifying the elements of a finite set or infinite set by assigning a number to each element. Counting sometimes involves numbers other than one, for example, when counting money, counting out change, counting by twos, there is archeological evidence suggesting that humans have been counting for at least 50,000 years. Counting was primarily used by ancient cultures to keep track of social and economic data such as number of members, prey animals, property. The development of counting led to the development of mathematical notation, numeral systems, counting can occur in a variety of forms. Counting can be verbal, that is, speaking every number out loud to keep track of progress and this is often used to count objects that are present already, instead of counting a variety of things over time. Counting can also be in the form of tally marks, making a mark for each number and this is useful when counting objects over time, such as the number of times something occurs during the course of a day. Tallying is base 1 counting, normal counting is done in base 10, counting can also be in the form of finger counting, especially when counting small numbers. This is often used by children to facilitate counting and simple mathematical operations, finger-counting uses unary notation, and is thus limited to counting 10. Older finger counting used the four fingers and the three bones in each finger to count to the number twelve, other hand-gesture systems are also in use, for example the Chinese system by which one can count 10 using only gestures of one hand. By using finger binary, it is possible to keep a count up to 1023 =210 −1. Various devices can also be used to facilitate counting, such as tally counters. Inclusive counting is usually encountered when dealing with time in the Romance languages, in exclusive counting languages such as English, when counting 8 days from Sunday, Monday will be day 1, Tuesday day 2, and the following Monday will be the eighth day. When counting inclusively, the Sunday will be day 1 and therefore the following Sunday will be the eighth day, for example, the French phrase for fortnight is quinzaine, and similar words are present in Greek, Spanish and Portuguese. In contrast, the English word fortnight itself derives from a fourteen-night, as the archaic sennight does from a seven-night, learning to count is an important educational/developmental milestone in most cultures of the world. Learning to count is a very first step into mathematics. However, some cultures in Amazonia and the Australian Outback do not count, many children at just 2 years of age have some skill in reciting the count list. They can also answer questions of ordinality for small numbers, e. g and they can even be skilled at pointing to each object in a set and reciting the words one after another
5.
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
6.
1 (number)
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1, is a number, a numeral, and the name of the glyph representing that number. It represents a single entity, the unit of counting or measurement, for example, a line segment of unit length is a line segment of length 1. It is also the first of the series of natural numbers. The word one can be used as a noun, an adjective and it comes from the English word an, which comes from the Proto-Germanic root *ainaz. The Proto-Germanic root *ainaz comes from the Proto-Indo-European root *oi-no-, compare the Proto-Germanic root *ainaz to Old Frisian an, Gothic ains, Danish een, Dutch een, German eins and Old Norse einn. Compare the Proto-Indo-European root *oi-no- to Greek oinos, Latin unus, Old Persian aivam, Old Church Slavonic -inu and ino-, Lithuanian vienas, Old Irish oin, One, sometimes referred to as unity, is the first non-zero natural number. It is thus the integer before two and after zero, and the first positive odd number, any number multiplied by one is that number, as one is the identity for multiplication. As a result,1 is its own factorial, its own square, its own cube, One is also the result of the empty product, as any number multiplied by one is itself. It is also the natural number that is neither composite nor prime with respect to division. The Gupta wrote it as a line, and the Nagari sometimes added a small circle on the left. The Nepali also rotated it to the right but kept the circle small and this eventually became the top serif in the modern numeral, but the occasional short horizontal line at the bottom probably originates from similarity with the Roman numeral I. Where the 1 is written with an upstroke, the number 7 has a horizontal stroke through the vertical line. While the shape of the 1 character has an ascender in most modern typefaces, in typefaces with text figures, many older typewriters do not have a separate symbol for 1 and use the lowercase letter l instead. It is possible to find cases when the uppercase J is used,1 cannot be used as the base of a positional numeral system, as the only digit that would be permitted in such a system would be 0. Since the base 1 exponential function always equals 1, its inverse does not exist, there are two ways to write the real number 1 as a recurring decimal, as 1.000. and as 0.999. There is only one way to represent the real number 1 as a Dedekind cut, in a multiplicative group or monoid, the identity element is sometimes denoted 1, but e is also traditional. However,1 is especially common for the identity of a ring. When such a ring has characteristic n not equal to 0,1 is the first figurate number of every kind, such as triangular number, pentagonal number and centered hexagonal number, to name just a few
7.
2 (number)
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2 is a number, numeral, and glyph. It is the number following 1 and preceding 3. The number two has many properties in mathematics, an integer is called even if it is divisible by 2. For integers written in a system based on an even number, such as decimal and hexadecimal. If it is even, then the number is even. In particular, when written in the system, all multiples of 2 will end in 0,2,4,6. In numeral systems based on an odd number, divisibility by 2 can be tested by having a root that is even. Two is the smallest and first prime number, and the only prime number. Two and three are the two consecutive prime numbers. 2 is the first Sophie Germain prime, the first factorial prime, the first Lucas prime, the first Ramanujan prime, and it is an Eisenstein prime with no imaginary part and real part of the form 3n −1. It is also a Stern prime, a Pell number, the first Fibonacci prime, and it is the third Fibonacci number, and the second and fourth Perrin numbers. Despite being prime, two is also a highly composite number, because it is a natural number which has more divisors than any other number scaled relative to the number itself. The next superior highly composite number is six, vulgar fractions with only 2 or 5 in the denominator do not yield infinite decimal expansions, as is the case with all other primes, because 2 and 5 are factors of ten, the decimal base. Two is the number x such that the sum of the reciprocals of the powers of x equals itself. In symbols ∑ k =0 ∞12 k =1 +12 +14 +18 +116 + ⋯ =2. This comes from the fact that, ∑ k =0 ∞1 n k =1 +1 n −1 for all n ∈ R >1, powers of two are central to the concept of Mersenne primes, and important to computer science. Two is the first Mersenne prime exponent, the square root of 2 was the first known irrational number. The smallest field has two elements, in the set-theoretical construction of the natural numbers,2 is identified with the set
8.
3 (number)
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3 is a number, numeral, and glyph. It is the number following 2 and preceding 4. Three is the largest number still written with as many lines as the number represents, to this day 3 is written as three lines in Roman and Chinese numerals. This was the way the Brahmin Indians wrote it, and the Gupta made the three lines more curved, the Nagari started rotating the lines clockwise and ending each line with a slight downward stroke on the right. Eventually they made these strokes connect with the lines below, and it was the Western Ghubar Arabs who finally eliminated the extra stroke and created our modern 3. ٣ While the shape of the 3 character has an ascender in most modern typefaces, in typefaces with text figures the character usually has a descender, as, for example, in some French text-figure typefaces, though, it has an ascender instead of a descender. A common variant of the digit 3 has a flat top and this form is sometimes used to prevent people from fraudulently changing a 3 into an 8. It is usually found on UPC-A barcodes and standard 52-card decks,3 is, a rough approximation of π and a very rough approximation of e when doing quick estimates. The first odd prime number, and the second smallest prime, the only number that is both a Fermat prime and a Mersenne prime. The first unique prime due to the properties of its reciprocal, the second triangular number and it is the only prime triangular number. Both the zeroth and third Perrin numbers in the Perrin sequence, the smallest number of sides that a simple polygon can have. The only prime which is one less than a perfect square, any other number which is n2 −1 for some integer n is not prime, since it is. This is true for 3 as well, but in case the smaller factor is 1. If n is greater than 2, both n −1 and n +1 are greater than 1 so their product is not prime, the number of non-collinear points needed to determine a plane and a circle. Also, Vulgar fractions with 3 in the denominator have a single digit repeating sequences in their decimal expansions,0.000, a natural number is divisible by three if the sum of its digits in base 10 is divisible by 3. For example, the number 21 is divisible by three and the sum of its digits is 2 +1 =3, because of this, the reverse of any number that is divisible by three is also divisible by three. For instance,1368 and its reverse 8631 are both divisible by three and this works in base 10 and in any positional numeral system whose base divided by three leaves a remainder of one. Three of the five regular polyhedra have triangular faces – the tetrahedron, the octahedron, also, three of the five regular polyhedra have vertices where three faces meet – the tetrahedron, the hexahedron, and the dodecahedron
9.
Additive inverse
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In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is known as the opposite, sign change. For a real number, it reverses its sign, the opposite to a number is negative. Zero is the inverse of itself. The additive inverse of a is denoted by unary minus, −a. For example, the inverse of 7 is −7, because 7 + =0. The additive inverse is defined as its inverse element under the operation of addition. As for any operation, double additive inverse has no net effect. For a number and, generally, in any ring, the inverse can be calculated using multiplication by −1. Examples of rings of numbers are integers, rational numbers, real numbers, Additive inverse is closely related to subtraction, which can be viewed as an addition of the opposite, a − b = a +. Conversely, additive inverse can be thought of as subtraction from zero, if such an operation admits an identity element o, then this element is unique. For a given x , if there exists x′ such that x + x′ = o , if + is associative, then an additive inverse is unique. To see this, let x′ and x″ each be additive inverses of x, for example, since addition of real numbers is associative, each real number has a unique additive inverse. All the following examples are in fact abelian groups, complex numbers, on the complex plane, this operation rotates a complex number 180 degrees around the origin. Addition of real- and complex-valued functions, here, the inverse of a function f is the function −f defined by = − f , for all x, such that f + = o . More generally, what precedes applies to all functions with values in a group, sequences, matrices. In a vector space the additive inverse −v is often called the vector of v, it has the same magnitude as the original. Additive inversion corresponds to multiplication by −1
10.
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
11.
Multiplicative inverse
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In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number which when multiplied by x yields the multiplicative identity,1. The multiplicative inverse of a fraction a/b is b/a, for the multiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth, the reciprocal function, the function f that maps x to 1/x, is one of the simplest examples of a function which is its own inverse. In the phrase multiplicative inverse, the qualifier multiplicative is often omitted, multiplicative inverses can be defined over many mathematical domains as well as numbers. In these cases it can happen that ab ≠ ba, then inverse typically implies that an element is both a left and right inverse. The notation f −1 is sometimes used for the inverse function of the function f. For example, the multiplicative inverse 1/ = −1 is the cosecant of x, only for linear maps are they strongly related. The terminology difference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the opposite naming convention, in the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1. With the exception of zero, reciprocals of every real number are real, reciprocals of every rational number are rational, the property that every element other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples. On the other hand, no other than 1 and −1 has an integer reciprocal. In modular arithmetic, the multiplicative inverse of a is also defined. This multiplicative inverse exists if and only if a and n are coprime, for example, the inverse of 3 modulo 11 is 4 because 4 ·3 ≡1. The extended Euclidean algorithm may be used to compute it, the sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless has divisors of zero, i. e. nonzero elements x, y such that xy =0. A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring, the linear map that has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix in the same base. Thus, the two notions of the inverse of a function are strongly related in this case, while they must be carefully distinguished in the general case. A ring in which every element has a multiplicative inverse is a division ring. As mentioned above, the reciprocal of every complex number z = a + bi is complex. In particular, if ||z||=1, then 1 / z = z ¯, consequently, the imaginary units, ±i, have additive inverse equal to multiplicative inverse, and are the only complex numbers with this property
12.
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
13.
Limit (mathematics)
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In mathematics, a limit is the value that a function or sequence approaches as the input or index approaches some value. Limits are essential to calculus and are used to define continuity, derivatives, the concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory. In formulas, a limit is usually written as lim n → c f = L and is read as the limit of f of n as n approaches c equals L. Here lim indicates limit, and the fact that function f approaches the limit L as n approaches c is represented by the right arrow, suppose f is a real-valued function and c is a real number. Intuitively speaking, the lim x → c f = L means that f can be made to be as close to L as desired by making x sufficiently close to c. The first inequality means that the distance x and c is greater than 0 and that x ≠ c, while the second indicates that x is within distance δ of c. Note that the definition of a limit is true even if f ≠ L. Indeed. Now since x +1 is continuous in x at 1, we can now plug in 1 for x, in addition to limits at finite values, functions can also have limits at infinity. In this case, the limit of f as x approaches infinity is 2, in mathematical notation, lim x → ∞2 x −1 x =2. Consider the following sequence,1.79,1.799,1.7999 and it can be observed that the numbers are approaching 1.8, the limit of the sequence. Formally, suppose a1, a2. is a sequence of real numbers, intuitively, this means that eventually all elements of the sequence get arbitrarily close to the limit, since the absolute value | an − L | is the distance between an and L. Not every sequence has a limit, if it does, it is called convergent, one can show that a convergent sequence has only one limit. The limit of a sequence and the limit of a function are closely related, on one hand, the limit as n goes to infinity of a sequence a is simply the limit at infinity of a function defined on the natural numbers n. On the other hand, a limit L of a function f as x goes to infinity, if it exists, is the same as the limit of any sequence a that approaches L. Note that one such sequence would be L + 1/n, in non-standard analysis, the limit of a sequence can be expressed as the standard part of the value a H of the natural extension of the sequence at an infinite hypernatural index n=H. Thus, lim n → ∞ a n = st , here the standard part function st rounds off each finite hyperreal number to the nearest real number. This formalizes the intuition that for very large values of the index. Conversely, the part of a hyperreal a = represented in the ultrapower construction by a Cauchy sequence, is simply the limit of that sequence
14.
Construction of the real numbers
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In mathematics, there are several ways of defining the real number system as an ordered field. The synthetic approach gives a list of axioms for the numbers as a complete ordered field. Under the usual axioms of set theory, one can show that these axioms are categorical, in the sense there is a model for the axioms. Any one of these models must be constructed, and most of these models are built using the basic properties of the rational number system as an ordered field. The synthetic approach axiomatically defines the number system as a complete ordered field. A model for the number system consists of a set R, two distinct elements 0 and 1 of R, two binary operations + and × on R, and a binary relation ≤ on R. In other words, For all x, y, and z in R, x + = + z and x × = × z, for all x and y in R, x + y = y + x and x × y = y × x. For all x, y, and z in R, x × = +, for all x in R, x +0 = x.0 is not equal to 1, and for all x in R, x ×1 = x. For every x in R, there exists an element −x in R, for every x ≠0 in R, there exists an element x−1 in R, such that x × x−1 =1. In other words, For all x in R, x ≤ x, for all x and y in R, if x ≤ y and y ≤ x, then x = y. For all x, y, and z in R, if x ≤ y and y ≤ z, for all x and y in R, x ≤ y or y ≤ x. The field operations + and × on R are compatible with the order ≤, in other words, For all x, y and z in R, if x ≤ y, then x + z ≤ y + z. In other words, If A is a non-empty subset of R, the rational numbers Q satisfy the first three axioms but Q does not satisfy axiom 4. So axiom 4, which requires the order to be Dedekind-complete, is crucial, axiom 4 implies the Archimedean property. Several models for axioms 1-4 are given below, any two models for axioms 1-4 are isomorphic, and so up to isomorphism, there is only one complete ordered Archimedean field. When we say that any two models of the axioms are isomorphic, we mean that for any two models and, there is a bijection f, R → S preserving both the field operations and the order. Explicitly, f is injective and surjective. For all x and y in R, f = f +S f and f = f ×S f, for all x and y in R, x ≤R y if and only if f ≤S f
15.
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
16.
Imaginary unit
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The imaginary unit or unit imaginary number is a solution to the quadratic equation x2 +1 =0. The term imaginary is used there is no real number having a negative square. There are two square roots of −1, namely i and −i, just as there are two complex square roots of every real number other than zero, which has one double square root. In contexts where i is ambiguous or problematic, j or the Greek ι is sometimes used, in the disciplines of electrical engineering and control systems engineering, the imaginary unit is normally denoted by j instead of i, because i is commonly used to denote electric current. For the history of the unit, see Complex number § History. The imaginary number i is defined solely by the property that its square is −1, with i defined this way, it follows directly from algebra that i and −i are both square roots of −1. In polar form, i is represented as 1eiπ/2, having a value of 1. In the complex plane, i is the point located one unit from the origin along the imaginary axis, more precisely, once a solution i of the equation has been fixed, the value −i, which is distinct from i, is also a solution. Since the equation is the definition of i, it appears that the definition is ambiguous. However, no ambiguity results as long as one or other of the solutions is chosen and labelled as i and this is because, although −i and i are not quantitatively equivalent, there is no algebraic difference between i and −i. Both imaginary numbers have equal claim to being the number whose square is −1, the issue can be a subtle one. See also Complex conjugate and Galois group, a more precise explanation is to say that the automorphism group of the special orthogonal group SO has exactly two elements — the identity and the automorphism which exchanges CW and CCW rotations. All these ambiguities can be solved by adopting a rigorous definition of complex number. For example, the pair, in the usual construction of the complex numbers with two-dimensional vectors. The imaginary unit is sometimes written √−1 in advanced mathematics contexts, however, great care needs to be taken when manipulating formulas involving radicals. The radical sign notation is reserved either for the square root function. Similarly,1 i =1 −1 =1 −1 = −11 = −1 = i, the calculation rules a ⋅ b = a ⋅ b and a b = a b are only valid for real, non-negative values of a and b. These problems are avoided by writing and manipulating expressions like i√7, for a more thorough discussion, see Square root and Branch point
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Prime number
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A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a number is called a composite number. For example,5 is prime because 1 and 5 are its only positive integer factors, the property of being prime is called primality. A simple but slow method of verifying the primality of a number n is known as trial division. It consists of testing whether n is a multiple of any integer between 2 and n, algorithms much more efficient than trial division have been devised to test the primality of large numbers. Particularly fast methods are available for numbers of forms, such as Mersenne numbers. As of January 2016, the largest known prime number has 22,338,618 decimal digits, there are infinitely many primes, as demonstrated by Euclid around 300 BC. There is no simple formula that separates prime numbers from composite numbers. However, the distribution of primes, that is to say, many questions regarding prime numbers remain open, such as Goldbachs conjecture, and the twin prime conjecture. Such questions spurred the development of branches of number theory. Prime numbers give rise to various generalizations in other domains, mainly algebra, such as prime elements. A natural number is called a number if it has exactly two positive divisors,1 and the number itself. Natural numbers greater than 1 that are not prime are called composite, among the numbers 1 to 6, the numbers 2,3, and 5 are the prime numbers, while 1,4, and 6 are not prime. 1 is excluded as a number, for reasons explained below. 2 is a number, since the only natural numbers dividing it are 1 and 2. Next,3 is prime, too,1 and 3 do divide 3 without remainder, however,4 is composite, since 2 is another number dividing 4 without remainder,4 =2 ·2. 5 is again prime, none of the numbers 2,3, next,6 is divisible by 2 or 3, since 6 =2 ·3. The image at the right illustrates that 12 is not prime,12 =3 ·4, no even number greater than 2 is prime because by definition, any such number n has at least three distinct divisors, namely 1,2, and n
18.
Number theory
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Number theory or, in older usage, arithmetic is a branch of pure mathematics devoted primarily to the study of the integers. It is sometimes called The Queen of Mathematics because of its place in the discipline. Number theorists study prime numbers as well as the properties of objects out of integers or defined as generalizations of the integers. Integers can be considered either in themselves or as solutions to equations, questions in number theory are often best understood through the study of analytical objects that encode properties of the integers, primes or other number-theoretic objects in some fashion. One may also study real numbers in relation to rational numbers, the older term for number theory is arithmetic. By the early century, it had been superseded by number theory. The use of the arithmetic for number theory regained some ground in the second half of the 20th century. In particular, arithmetical is preferred as an adjective to number-theoretic. The first historical find of a nature is a fragment of a table. The triples are too many and too large to have been obtained by brute force, the heading over the first column reads, The takiltum of the diagonal which has been subtracted such that the width. The tables layout suggests that it was constructed by means of what amounts, in language, to the identity 2 +1 =2. If some other method was used, the triples were first constructed and then reordered by c / a, presumably for use as a table. It is not known what these applications may have been, or whether there could have any, Babylonian astronomy, for example. It has been suggested instead that the table was a source of examples for school problems. While Babylonian number theory—or what survives of Babylonian mathematics that can be called thus—consists of this single, striking fragment, late Neoplatonic sources state that Pythagoras learned mathematics from the Babylonians. Much earlier sources state that Thales and Pythagoras traveled and studied in Egypt, Euclid IX 21—34 is very probably Pythagorean, it is very simple material, but it is all that is needed to prove that 2 is irrational. Pythagorean mystics gave great importance to the odd and the even, the discovery that 2 is irrational is credited to the early Pythagoreans. This forced a distinction between numbers, on the one hand, and lengths and proportions, on the other hand, the Pythagorean tradition spoke also of so-called polygonal or figurate numbers
19.
Partition (number theory)
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In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition, a summand in a partition is also called a part. The number of partitions of n is given by the function p. The notation λ ⊢ n means that λ is a partition of n, Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symmetric polynomials, the symmetric group and in group representation theory in general. For example, the partition 2 +2 +1 might instead be written as the tuple or in the more compact form where the superscript indicates the number of repetitions of a term. There are two common methods to represent partitions, as Ferrers diagrams, named after Norman Macleod Ferrers. Both have several possible conventions, here, we use English notation, with diagrams aligned in the upper-left corner. The partition 6 +4 +3 +1 of the positive number 14 can be represented by the diagram, The 14 circles are lined up in 4 rows. The diagrams for the 5 partitions of the number 4 are listed below, rather than representing a partition with dots, as in the Ferrers diagram, the Young diagram uses boxes or squares. As a type of shape made by adjacent squares joined together, by convention p =1, p =0 for n negative. The first few values of the function are,1,1,2,3,5,7,11,15,22,30,42,56,77,101,135,176,231,297,385,490,627,792,1002,1255,1575,1958,2436,3010,3718,4565,5604. As of June 2013, the largest known prime number that counts a number of partitions is p, the generating function for p is given by, ∑ n =0 ∞ p x n = ∏ k =1 ∞. Expanding each factor on the side as a geometric series. The xn term in this product counts the number of ways to write n = a1 + 2a2 + 3a3 +, where each number i appears ai times. This is precisely the definition of a partition of n, so our product is the generating function. More generally, the function for the partitions of n into numbers from a set A can be found by taking only those terms in the product where k is an element of A. This result is due to Euler, the formulation of Eulers generating function is a special case of a q-Pochhammer symbol and is similar to the product formulation of many modular forms, and specifically the Dedekind eta function
20.
Combinatorics
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Combinatorics is a branch of mathematics concerning the study of finite or countable discrete structures. Many combinatorial questions have historically been considered in isolation, giving an ad hoc solution to a problem arising in some mathematical context. In the later twentieth century, however, powerful and general methods were developed. One of the oldest and most accessible parts of combinatorics is graph theory, Combinatorics is used frequently in computer science to obtain formulas and estimates in the analysis of algorithms. A mathematician who studies combinatorics is called a combinatorialist or a combinatorist, basic combinatorial concepts and enumerative results appeared throughout the ancient world. Greek historian Plutarch discusses an argument between Chrysippus and Hipparchus of a rather delicate enumerative problem, which was shown to be related to Schröder–Hipparchus numbers. In the Ostomachion, Archimedes considers a tiling puzzle, in the Middle Ages, combinatorics continued to be studied, largely outside of the European civilization. The Indian mathematician Mahāvīra provided formulae for the number of permutations and combinations, later, in Medieval England, campanology provided examples of what is now known as Hamiltonian cycles in certain Cayley graphs on permutations. During the Renaissance, together with the rest of mathematics and the sciences, works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field. In modern times, the works of J. J. Sylvester and Percy MacMahon helped lay the foundation for enumerative, graph theory also enjoyed an explosion of interest at the same time, especially in connection with the four color problem. In the second half of the 20th century, combinatorics enjoyed a rapid growth, in part, the growth was spurred by new connections and applications to other fields, ranging from algebra to probability, from functional analysis to number theory, etc. These connections shed the boundaries between combinatorics and parts of mathematics and theoretical science, but at the same time led to a partial fragmentation of the field. Enumerative combinatorics is the most classical area of combinatorics and concentrates on counting the number of combinatorial objects. Although counting the number of elements in a set is a rather broad mathematical problem, fibonacci numbers is the basic example of a problem in enumerative combinatorics. The twelvefold way provides a framework for counting permutations, combinations and partitions. Analytic combinatorics concerns the enumeration of combinatorial structures using tools from complex analysis, in contrast with enumerative combinatorics, which uses explicit combinatorial formulae and generating functions to describe the results, analytic combinatorics aims at obtaining asymptotic formulae. Partition theory studies various enumeration and asymptotic problems related to integer partitions, originally a part of number theory and analysis, it is now considered a part of combinatorics or an independent field. It incorporates the bijective approach and various tools in analysis and analytic number theory, graphs are basic objects in combinatorics
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Primary school
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A primary school or elementary school is a school in which children receive primary or elementary education from the age of about five to twelve, coming after preschool and before secondary school. In most parts of the world, primary education is the first stage of education, and is normally available without charge. The term grade school is used in the US though this term may refer to both primary education and secondary education. The term primary school is derived from the French école primaire, primary school is the preferred term in the United Kingdom and many Commonwealth Nations, and in most publications of the United Nations Educational, Scientific, and Cultural Organization. Elementary school is preferred in some countries, especially in the United States, in the United States, primary school may refer to a school with grades Kindergarten through second grade or third grade. In these municipalities, the school includes grade three through five or grades four to six
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Discrete mathematics
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Discrete mathematics is the study of mathematical structures that are fundamentally discrete rather than continuous. Discrete mathematics therefore excludes topics in mathematics such as calculus. Discrete objects can often be enumerated by integers, more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no definition of the term discrete mathematics. Indeed, discrete mathematics is described less by what is included than by what is excluded, continuously varying quantities, the set of objects studied in discrete mathematics can be finite or infinite. The term finite mathematics is sometimes applied to parts of the field of mathematics that deals with finite sets. Conversely, computer implementations are significant in applying ideas from mathematics to real-world problems. Although the main objects of study in mathematics are discrete objects. In university curricula, Discrete Mathematics appeared in the 1980s, initially as a computer science support course, some high-school-level discrete mathematics textbooks have appeared as well. At this level, discrete mathematics is seen as a preparatory course. The Fulkerson Prize is awarded for outstanding papers in discrete mathematics, the history of discrete mathematics has involved a number of challenging problems which have focused attention within areas of the field. In graph theory, much research was motivated by attempts to prove the four color theorem, first stated in 1852, in logic, the second problem on David Hilberts list of open problems presented in 1900 was to prove that the axioms of arithmetic are consistent. Gödels second incompleteness theorem, proved in 1931, showed that this was not possible – at least not within arithmetic itself, Hilberts tenth problem was to determine whether a given polynomial Diophantine equation with integer coefficients has an integer solution. In 1970, Yuri Matiyasevich proved that this could not be done, at the same time, military requirements motivated advances in operations research. The Cold War meant that cryptography remained important, with fundamental advances such as public-key cryptography being developed in the following decades, operations research remained important as a tool in business and project management, with the critical path method being developed in the 1950s. The telecommunication industry has also motivated advances in mathematics, particularly in graph theory. Formal verification of statements in logic has been necessary for development of safety-critical systems. Computational geometry has been an important part of the computer graphics incorporated into modern video games, currently, one of the most famous open problems in theoretical computer science is the P = NP problem, which involves the relationship between the complexity classes P and NP
23.
Measurement
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Measurement is the assignment of a number to a characteristic of an object or event, which can be compared with other objects or events. The scope and application of a measurement is dependent on the context, however, in other fields such as statistics as well as the social and behavioral sciences, measurements can have multiple levels, which would include nominal, ordinal, interval, and ratio scales. Measurement is a cornerstone of trade, science, technology, historically, many measurement systems existed for the varied fields of human existence to facilitate comparisons in these fields. Often these were achieved by local agreements between trading partners or collaborators, since the 18th century, developments progressed towards unifying, widely accepted standards that resulted in the modern International System of Units. This system reduces all physical measurements to a combination of seven base units. The science of measurement is pursued in the field of metrology, the measurement of a property may be categorized by the following criteria, type, magnitude, unit, and uncertainty. They enable unambiguous comparisons between measurements, the type or level of measurement is a taxonomy for the methodological character of a comparison. For example, two states of a property may be compared by ratio, difference, or ordinal preference, the type is commonly not explicitly expressed, but implicit in the definition of a measurement procedure. The magnitude is the value of the characterization, usually obtained with a suitably chosen measuring instrument. A unit assigns a mathematical weighting factor to the magnitude that is derived as a ratio to the property of a used as standard or a natural physical quantity. An uncertainty represents the random and systemic errors of the measurement procedure, errors are evaluated by methodically repeating measurements and considering the accuracy and precision of the measuring instrument. Measurements most commonly use the International System of Units as a comparison framework, the system defines seven fundamental units, kilogram, metre, candela, second, ampere, kelvin, and mole. Instead, the measurement unit can only ever change through increased accuracy in determining the value of the constant it is tied to and this directly influenced the Michelson–Morley experiment, Michelson and Morley cite Peirce, and improve on his method. With the exception of a few fundamental quantum constants, units of measurement are derived from historical agreements, nothing inherent in nature dictates that an inch has to be a certain length, nor that a mile is a better measure of distance than a kilometre. Over the course of history, however, first for convenience and then for necessity. Laws regulating measurement were originally developed to prevent fraud in commerce.9144 metres, in the United States, the National Institute of Standards and Technology, a division of the United States Department of Commerce, regulates commercial measurements. Before SI units were adopted around the world, the British systems of English units and later imperial units were used in Britain, the Commonwealth. The system came to be known as U. S. customary units in the United States and is still in use there and in a few Caribbean countries. S
24.
Linguistics
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Linguistics is the scientific study of language, and involves an analysis of language form, language meaning, and language in context. Linguists traditionally analyse human language by observing an interplay between sound and meaning, phonetics is the study of speech and non-speech sounds, and delves into their acoustic and articulatory properties. While the study of semantics typically concerns itself with truth conditions, Grammar is a system of rules which governs the production and use of utterances in a given language. These rules apply to sound as well as meaning, and include componential sub-sets of rules, such as those pertaining to phonology, morphology, modern theories that deal with the principles of grammar are largely based within Noam Chomskys ideological school of generative grammar. In the early 20th century, Ferdinand de Saussure distinguished between the notions of langue and parole in his formulation of structural linguistics. According to him, parole is the utterance of speech, whereas langue refers to an abstract phenomenon that theoretically defines the principles. This distinction resembles the one made by Noam Chomsky between competence and performance in his theory of transformative or generative grammar. According to Chomsky, competence is an innate capacity and potential for language, while performance is the specific way in which it is used by individuals, groups. The study of parole is the domain of sociolinguistics, the sub-discipline that comprises the study of a system of linguistic facets within a certain speech community. Discourse analysis further examines the structure of texts and conversations emerging out of a speech communitys usage of language, Stylistics also involves the study of written, signed, or spoken discourse through varying speech communities, genres, and editorial or narrative formats in the mass media. In the 1960s, Jacques Derrida, for instance, further distinguished between speech and writing, by proposing that language be studied as a linguistic medium of communication in itself. Palaeography is therefore the discipline that studies the evolution of scripts in language. Linguistics also deals with the social, cultural, historical and political factors that influence language, through which linguistic, research on language through the sub-branches of historical and evolutionary linguistics also focus on how languages change and grow, particularly over an extended period of time. Language documentation combines anthropological inquiry with linguistic inquiry, in order to describe languages, lexicography involves the documentation of words that form a vocabulary. Such a documentation of a vocabulary from a particular language is usually compiled in a dictionary. Computational linguistics is concerned with the statistical or rule-based modeling of natural language from a computational perspective, specific knowledge of language is applied by speakers during the act of translation and interpretation, as well as in language education – the teaching of a second or foreign language. Policy makers work with governments to implement new plans in education, related areas of study also includes the disciplines of semiotics, literary criticism, translation, and speech-language pathology. Before the 20th century, the philology, first attested in 1716, was commonly used to refer to the science of language
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Nominal number
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Nominal numbers or categorical numbers are numeric codes, meaning numerals used for labelling or identification only. The values of the numerals are irrelevant, and they do not indicate quantity, rank, the term nominal number is quite recent and of limited use. It appears to have originated as a usage in school textbooks derived from the term nominal data. This usage comes from the sense of nominal as name, nominal number can be broadly defined as any numeral used for identification, however it was assigned, or narrowly as a numeral with no information other than identification. For the purposes of naming, the number is often used loosely to refer to any string. For instance, National Insurance numbers, some drivers license numbers, similarly, one can add or subtract ZIP codes, but this is meaningless,12345 −11111 does not have any meaning as a ZIP code. In general, the only operation with nominal numbers is to compare two nominal numbers to see whether they are identical or not. Postal codes, such as ZIP codes Telephone numbers, assigned by various telephone numbering plans, such as the ITU-T E.164, numbers of train or bus routes in public transport Car model names from some car manufacturers, such as BMW or Peugeot, are plain numbers. Numerical identifiers that are nominal numbers narrowly defined, viz, convey no information other than identity, are quite rare and these must be defined either arbitrarily or randomly, and most commonly arise in computer applications, such as dynamic IP addresses assigned by Dynamic Host Configuration Protocol. A more everyday example are sports squad numbers, which do not in general have any meaning beyond identity. In some settings, these are based on position, but in others they are associated with an individual, farber, William, Germain-Williams, Terri L. Paris, Elaine, Thaller, Bernd, Lehmann, Ingmar. 100 Commonly Asked Questions in Math Class
26.
Ishango bone
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The Ishango bone is a bone tool, dated to the Upper Paleolithic era. It is a dark brown length of bone, the fibula of a baboon, with a piece of quartz affixed to one end. It was first thought to be a stick, as it has a series of what has been interpreted as tally marks carved in three columns running the length of the tool. It has also suggested that the scratches might have been to create a better grip on the handle or for some other non-mathematical reason. The Ishango bone was found in 1960 by Belgian Jean de Heinzelin de Braucourt while exploring what was then the Belgian Congo and it was discovered in the area of Ishango near the Semliki River. Lake Edward empties into the Semliki which forms part of the headwaters of the Nile River, the bone was found among the remains of a small community that fished and gathered in this area of Africa. The settlement had been buried in a volcanic eruption, the artifact was first estimated to have originated between 9,000 BC and 6,500 BC. However, the dating of the site where it was discovered was re-evaluated, the Ishango bone is on permanent exhibition at the Royal Belgian Institute of Natural Sciences, Brussels, Belgium. Some mathematicians, scientists and archaeolgists believe the three columns of asymmetrically grouped notches imply that the implement was used to construct a numeral system, the central column begins with three notches and then doubles to 6 notches. The process is repeated for the number 4, which doubles to 8 notches, and then reversed for the number 10 and these numbers may not be purely random and instead suggest some understanding of the principle of multiplication and division by two. The bone may therefore have been used as a tool for simple mathematical procedures. He also writes that no attempt has been made to explain why a tally of something should exhibit multiples of two, prime numbers between 10 and 20, and some numbers that are almost multiples of 10. Claudia Zaslavsky has suggested that this may indicate that the creator of the tool was a woman, during earlier excavations at the Ishango site in 1959, another bone was also found. It is lighter in color and was scraped, thinned, polished, the artifact possibly held a piece of quartz like the more well-known bone or it could have been a tool handle. The 14-cm long bone has 90 notches on six sides, which are categorized as major or minor according to their length, Jean de Heinzelin interpreted the major notches as being units or multiples and the minor notches as fractions or subsidiary. He believed the bone to be an interchange rule between bases 10 and 12, lebombo bone History of mathematics Paleolithic tally sticks Shurkin, J. Engines of the mind, a history of the computer, W. W. Norton & Co.1984. P21 Bogoshi, J. Naidoo, K. and Webb, Africa, The true cradle of mathematical sciences Ishango,22000 and 50 years later, the cradle of mathematics. The On-Line Encyclopedia of Integer Sequences
27.
Museum of Natural Sciences
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The Museum of Natural Sciences is a museum in the Belgian capital of Brussels dedicated to natural history. The museum is a part of the Royal Belgian Institute of Natural Sciences and its most important pieces are 30 fossilized Iguanodon skeletons, which were discovered in 1878 in Bernissart. The dinosaur hall of the museum is the worlds largest museum hall completely dedicated to dinosaurs, another famous piece is the Ishango bone, which was discovered in 1960 by Jean de Heinzelin de Braucourt. Like in most museums, there is a department and a public exhibit department. The museum was founded on 31 March 1846 as a descendant of the Musée de Bruxelles of 1802 and it was based on the collection established by Prince Charles Alexander of Lorraine, which dated from the 18th century. Bernard du Bus de Gisignies, became the first director of the museum in 1846, on this occasion he donated 2474 birds from his own collection to the museum. In 1860, during the construction of new fortifications around Antwerp, the museum also obtained the skeletons from a bowhead whale and a young blue whale, which are still on display in the museum. In 1860 the skeleton of a mammoth was found near Lier and due to the action of François-Joseph Scohy it was preserved and brought to the museum. At that time the only other skeleton of a mammoth on display was in the museum of Saint Petersburg, in 1878, the largest find of Iguanodon fossils to date occurred in a coal mine at Bernissart in Belgium. At least 38 Iguanodon individuals were uncovered, of which 30 have been on display since they were moved here from their home at the Palace of Charles of Lorraine in 1891. Since 2007, a renovated and enlarged dinosaur hall of 4580 m2 is the largest dinosaur hall in the world. The Dinosaur hall, with the world famous Iguanodon skeletons Of Men and Mammoths, about the evolution of man, the Mammal Gallery, recent and extinct mammals, including a thylacine. The North and South Pole, a view of two different worlds in a glass tunnel, the Whale Hall, skeletons of whales, dolphins, walruses, sirenians, seals. A small part of the hall is devoted to the role in the economy. The Shell Gallery houses an aquarium and a complete survey of the lower classes of invertebrates. The nearby North Sea Discovery Room takes you on a tour around beach. The entire collection consists of 9000000 specimens and is one of the three biggest shell collections in the world, a great deal was collected by Philippe Dautzenberg. The Insect Gallery, insects, spiders, crustaceans and other arthropods, the collection consists of 15000000 specimens of which ten thousands are holotypes of great scientific value
28.
Numeral system
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A numeral system is a writing system for expressing numbers, that is, a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. It can be seen as the context that allows the symbols 11 to be interpreted as the symbol for three, the decimal symbol for eleven, or a symbol for other numbers in different bases. The number the numeral represents is called its value, ideally, a numeral system will, Represent a useful set of numbers Give every number represented a unique representation Reflect the algebraic and arithmetic structure of the numbers. For example, the decimal representation of whole numbers gives every nonzero whole number a unique representation as a finite sequence of digits. Etc. all of which have the same meaning except for some scientific, such systems are, however, not the topic of this article. The most commonly used system of numerals is the Hindu–Arabic numeral system, two Indian mathematicians are credited with developing it. Aryabhata of Kusumapura developed the notation in the 5th century. The numeral system and the concept, developed by the Hindus in India, slowly spread to other surrounding countries due to their commercial. The Arabs adopted and modified it, even today, the Arabs call the numerals which they use Rakam Al-Hind or the Hindu numeral system. The Arabs translated Hindu texts on numerology and spread them to the world due to their trade links with them. The Western world modified them and called them the Arabic numerals, hence the current western numeral system is the modified version of the Hindu numeral system developed in India. It also exhibits a great similarity to the Sanskrit–Devanagari notation, which is used in India. The simplest numeral system is the numeral system, in which every natural number is represented by a corresponding number of symbols. If the symbol / is chosen, for example, then the seven would be represented by ///////. Tally marks represent one such system still in common use, the unary system is only useful for small numbers, although it plays an important role in theoretical computer science. Elias gamma coding, which is used in data compression. The unary notation can be abbreviated by introducing different symbols for new values. The ancient Egyptian numeral system was of type, and the Roman numeral system was a modification of this idea
29.
History of ancient Egypt
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The history of ancient Egypt spans the period from the early prehistoric settlements of the northern Nile valley to the Roman conquest, in 30 BC. The Pharaonic Period is dated from the 32nd century BC, when Upper and Lower Egypt were unified, until the country fell under Macedonian rule, note For alternative revisions to the chronology of Egypt, see Egyptian chronology. Egypts history is split into different periods according to the ruling dynasty of each pharaoh. The dating of events is still a subject of research, the conservative dates are not supported by any reliable absolute date for a span of about three millennia. The following is the list according to conventional Egyptian chronology, traces of these early people appear in the form of artifacts and rock carvings along the terraces of the Nile and in the oases. To the Egyptians the Nile meant life and the desert meant death, evidence also indicates human habitation and cattle herding in the southwestern corner of Egypt near the Sudan border before the 8th millennium BC. Despite this, the idea of an independent bovine domestication event in Africa must be abandoned because subsequent evidence gathered over a period of thirty years has failed to corroborate this, the oldest-known domesticated cattle remains in Africa are from the Faiyum c.4400 BC. Continued desiccation forced the early ancestors of the Egyptians to settle around the Nile more permanently, however, the period from 9th to the 6th millennium BC has left very little in the way of archaeological evidence. The Nile valley of Egypt was basically uninhabitable until the work of clearing and irrigating the land along the banks was started, however it appears that this clearance and irrigation was largely under way by the 6th millennium. By that time, Nile society was already engaged in organized agriculture, at this time, Egyptians in the southwestern corner of Egypt were herding cattle and also constructing large buildings. Mortar was in use by the 4th millennium, the people of the valley and the Nile Delta were self-sufficient and were raising barley and emmer, an early variety of wheat, and stored it in pits lined with reed mats. They raised cattle, goats and pigs and they wove linen, prehistory continues through this time, variously held to begin with the Amratian culture. Between 5500 BC and the 31st century BC, small settlements flourished along the Nile, the Tasian culture was the next to appear, it existed in Upper Egypt starting about 4500 BC. This group is named for the burials found at Deir Tasa, the Tasian culture is notable for producing the earliest blacktop-ware, a type of red and brown pottery painted black on its top and interior. The Badari culture, named for the Badari site near Deir Tasa, followed the Tasian, however, the Badari culture continued to produce the kind of pottery called blacktop-ware, and was assigned the sequence dating numbers between 21 and 29. The Amratian culture is named after the site of el-Amreh, about 120 kilometres south of Badari, el-Amreh was the first site where this culture was found unmingled with the later Gerzeh culture. However, this period is attested at Nagada, and so is also referred to as the Naqada I culture. The Amratian period falls between S. D.30 and 39, newly excavated objects indicate that trade between Upper and Lower Egypt existed at this time
30.
Egyptian hieroglyphs
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Egyptian hieroglyphs were the formal writing system used in Ancient Egypt. It combined logographic, syllabic and alphabetic elements, with a total of some 1,000 distinct characters, cursive hieroglyphs were used for religious literature on papyrus and wood. The later hieratic and demotic Egyptian scripts are derived from hieroglyphic writing, the writing system continued to be used throughout the Late Period, as well as the Persian and Ptolemaic periods. Late survivals of hieroglyphic use are found well into the Roman period, with the closing of pagan temples in the 5th century, knowledge of hieroglyphic writing was lost, and the script remained undeciphered throughout the medieval and early modern period. The decipherment of hieroglyphs would only be solved in the 1820s by Jean-François Champollion, the word hieroglyph comes from the Greek adjective ἱερογλυφικός, a compound of ἱερός and γλύφω, supposedly a calque of an Egyptian phrase mdw·w-nṯr gods words. The glyphs themselves were called τὰ ἱερογλυφικὰ γράμματα the sacred engraved letters, the word hieroglyph has become a noun in English, standing for an individual hieroglyphic character. As used in the sentence, the word hieroglyphic is an adjective. Hieroglyphs emerged from the artistic traditions of Egypt. For example, symbols on Gerzean pottery from c.4000 BC have been argued to resemble hieroglyphic writing, proto-hieroglyphic symbol systems develop in the second half of the 4th millennium BC, such as the clay labels of a Predynastic ruler called Scorpion I recovered at Abydos in 1998. The first full sentence written in hieroglyphs so far discovered was found on a seal found in the tomb of Seth-Peribsen at Umm el-Qaab. There are around 800 hieroglyphs dating back to the Old Kingdom, Middle Kingdom, by the Greco-Roman period, there are more than 5,000. However, given the lack of evidence, no definitive determination has been made as to the origin of hieroglyphics in ancient Egypt. Since the 1990s, and discoveries such as the Abydos glyphs, as writing developed and became more widespread among the Egyptian people, simplified glyph forms developed, resulting in the hieratic and demotic scripts. These variants were more suited than hieroglyphs for use on papyrus. Hieroglyphic writing was not, however, eclipsed, but existed alongside the other forms, especially in monumental, the Rosetta Stone contains three parallel scripts – hieroglyphic, demotic, and Greek. Hieroglyphs continued to be used under Persian rule, and after Alexander the Greats conquest of Egypt, during the ensuing Ptolemaic and Roman periods. It appears that the quality of comments from Greek and Roman writers about hieroglyphs came about, at least in part. Some believed that hieroglyphs may have functioned as a way to distinguish true Egyptians from some of the foreign conquerors, another reason may be the refusal to tackle a foreign culture on its own terms, which characterized Greco-Roman approaches to Egyptian culture generally
31.
Karnak
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The Karnak Temple Complex, commonly known as Karnak, comprises a vast mix of decayed temples, chapels, pylons, and other buildings. Building at the complex began during the reign of Senusret I in the Middle Kingdom and continued into the Ptolemaic period, the area around Karnak was the ancient Egyptian Ipet-isut and the main place of worship of the eighteenth dynasty Theban Triad with the god Amun as its head. It is part of the city of Thebes. The Karnak complex gives its name to the nearby, and partly surrounded, the complex is a vast open-air museum, and the second largest ancient religious site in the world, after the Angkor Wat Temple of Cambodia. It is believed to be the second most visited site in Egypt. It consists of four parts, of which only the largest is currently open to the general public. The term Karnak often is understood as being the Precinct of Amun-Ra only, the three other parts, the Precinct of Mut, the Precinct of Montu, and the dismantled Temple of Amenhotep IV, are closed to the public. There also are a few temples and sanctuaries connecting the Precinct of Mut, the Precinct of Amun-Re. The Precinct of Mut is very ancient, being dedicated to an Earth and creation deity, the original temple was destroyed and partially restored by Hatshepsut, although another pharaoh built around it in order to change the focus or orientation of the sacred area. Many portions of it may have carried away for use in other buildings. The key difference between Karnak and most of the temples and sites in Egypt is the length of time over which it was developed and used. Construction of temples started in the Middle Kingdom and continued through to Ptolemaic times, approximately thirty pharaohs contributed to the buildings, enabling it to reach a size, complexity, and diversity not seen elsewhere. Few of the features of Karnak are unique, but the size. The deities represented range from some of the earliest worshiped to those worshiped much later in the history of the Ancient Egyptian culture and it also contains evidence of adaptations, using buildings of the Ancient Egyptians by later cultures for their own religious purposes. One famous aspect of Karnak is the Hypostyle Hall in the Precinct of Amun-Re,122 of these columns are 10 meters tall, and the other 12 are 21 meters tall with a diameter of over three meters. The architraves on top of columns are estimated to weigh 70 tons. These architraves may have been lifted to these heights using levers and this would be an extremely time-consuming process and also would require great balance to get to such great heights. A common alternative theory regarding how they were moved is that large ramps were constructed of sand, mud, brick or stone, if stone had been used for the ramps, they would have been able to use much less material
32.
Louvre
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The Louvre or the Louvre Museum is the worlds largest museum and a historic monument in Paris, France. A central landmark of the city, it is located on the Right Bank of the Seine in the citys 1st arrondissement, approximately 38,000 objects from prehistory to the 21st century are exhibited over an area of 72,735 square metres. The Louvre is the second most visited museum after the Palace Museum in China. The museum is housed in the Louvre Palace, originally built as a fortress in the late 12th century under Philip II, remnants of the fortress are visible in the basement of the museum. Due to the expansion of the city, the fortress eventually lost its defensive function and. The building was extended many times to form the present Louvre Palace, in 1692, the building was occupied by the Académie des Inscriptions et Belles Lettres and the Académie Royale de Peinture et de Sculpture, which in 1699 held the first of a series of salons. The Académie remained at the Louvre for 100 years, during the French Revolution, the National Assembly decreed that the Louvre should be used as a museum to display the nations masterpieces. The museum opened on 10 August 1793 with an exhibition of 537 paintings, because of structural problems with the building, the museum was closed in 1796 until 1801. The collection was increased under Napoleon and the museum renamed Musée Napoléon, the collection was further increased during the reigns of Louis XVIII and Charles X, and during the Second French Empire the museum gained 20,000 pieces. Holdings have grown steadily through donations and bequests since the Third Republic, whether this was the first building on that spot is not known, it is possible that Philip modified an existing tower. According to the authoritative Grand Larousse encyclopédique, the name derives from an association with wolf hunting den, in the 7th century, St. Fare, an abbess in Meaux, left part of her Villa called Luvra situated in the region of Paris to a monastery. This territory probably did not correspond exactly to the modern site, the Louvre Palace was altered frequently throughout the Middle Ages. In the 14th century, Charles V converted the building into a residence and in 1546, Francis acquired what would become the nucleus of the Louvres holdings, his acquisitions including Leonardo da Vincis Mona Lisa. After Louis XIV chose Versailles as his residence in 1682, constructions slowed, however, on 14 October 1750, Louis XV agreed and sanctioned a display of 96 pieces from the royal collection, mounted in the Galerie royale de peinture of the Luxembourg Palace. Under Louis XVI, the museum idea became policy. The comte dAngiviller broadened the collection and in 1776 proposed conversion of the Grande Galerie of the Louvre – which contained maps – into the French Museum, many proposals were offered for the Louvres renovation into a museum, however, none was agreed on. Hence the museum remained incomplete until the French Revolution, during the French Revolution the Louvre was transformed into a public museum. In May 1791, the Assembly declared that the Louvre would be a place for bringing together monuments of all the sciences, on 10 August 1792, Louis XVI was imprisoned and the royal collection in the Louvre became national property
33.
Babylonia
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Babylonia was an ancient Akkadian-speaking state and cultural area based in central-southern Mesopotamia. A small Amorite-ruled state emerged in 1894 BC, which contained at this time the city of Babylon. Babylon greatly expanded during the reign of Hammurabi in the first half of the 18th century BC, during the reign of Hammurabi and afterwards, Babylonia was called Māt Akkadī the country of Akkad in the Akkadian language. It was often involved in rivalry with its older fellow Akkadian-speaking state of Assyria in northern Mesopotamia and it retained the Sumerian language for religious use, but by the time Babylon was founded, this was no longer a spoken language, having been wholly subsumed by Akkadian. The earliest mention of the city of Babylon can be found in a tablet from the reign of Sargon of Akkad. During the 3rd millennium BC, a cultural symbiosis occurred between Sumerian and Akkadian-speakers, which included widespread bilingualism. The influence of Sumerian on Akkadian and vice versa is evident in all areas, from lexical borrowing on a scale, to syntactic, morphological. This has prompted scholars to refer to Sumerian and Akkadian in the millennium as a sprachbund. Traditionally, the religious center of all Mesopotamia was the city of Nippur. The empire eventually disintegrated due to decline, climate change and civil war. Sumer rose up again with the Third Dynasty of Ur in the late 22nd century BC and they also seem to have gained ascendancy over most of the territory of the Akkadian kings of Assyria in northern Mesopotamia for a time. The states of the south were unable to stem the Amorite advance, King Ilu-shuma of the Old Assyrian Empire in a known inscription describes his exploits to the south as follows, The freedom of the Akkadians and their children I established. I established their freedom from the border of the marshes and Ur and Nippur, Awal, past scholars originally extrapolated from this text that it means he defeated the invading Amorites to the south, but there is no explicit record of that. More recently, the text has been taken to mean that Asshur supplied the south with copper from Anatolia and these policies were continued by his successors Erishum I and Ikunum. During the first centuries of what is called the Amorite period and his reign was concerned with establishing statehood amongst a sea of other minor city states and kingdoms in the region. However Sumuabum appears never to have bothered to give himself the title of King of Babylon, suggesting that Babylon itself was only a minor town or city. He was followed by Sumu-la-El, Sabium, Apil-Sin, each of whom ruled in the same manner as Sumuabum. Sin-Muballit was the first of these Amorite rulers to be regarded officially as a king of Babylon, the Elamites occupied huge swathes of southern Mesopotamia, and the early Amorite rulers were largely held in vassalage to Elam
34.
Positional notation
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Positional notation or place-value notation is a method of representing or encoding numbers. Positional notation is distinguished from other notations for its use of the symbol for the different orders of magnitude. This greatly simplified arithmetic, leading to the spread of the notation across the world. With the use of a point, the notation can be extended to include fractions. The Hindu–Arabic numeral system, base-10, is the most commonly used system in the world today for most calculations, today, the base-10 system, which is likely motivated by counting with the ten fingers, is ubiquitous. Other bases have been used in the past however, and some continue to be used today, for example, the Babylonian numeral system, credited as the first positional numeral system, was base-60, but it lacked a real 0 value. Zero was indicated by a space between sexagesimal numerals, by 300 BC, a punctuation 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. Nor was it used at the end of a number, thus numbers like 2 and 120,3 and 180,4 and 240, looked the same because the larger numbers lacked a final sexagesimal placeholder. Counting rods and most abacuses have been used to represent numbers in a numeral system. This approach required no memorization of tables and could produce practical results quickly, for four centuries there was strong disagreement between those who believed in adopting the positional system in writing numbers and those who wanted to stay with the additive-system-plus-abacus. Although electronic calculators have largely replaced the abacus, the continues to be used in Japan. After the French Revolution, the new French government promoted the extension of the decimal system, some of those pro-decimal efforts—such as decimal time and the decimal calendar—were unsuccessful. Other French pro-decimal efforts—currency decimalisation and the metrication of weights and measures—spread widely out of France to almost the whole world. According to Joseph Needham and Lam Lay Yong, decimal fractions were first developed and used by the Chinese in the 1st century BC, the written Chinese decimal fractions were non-positional. However, counting rod fractions were positional, the Jewish mathematician Immanuel Bonfils used decimal fractions around 1350, anticipating Simon Stevin, but did not develop any notation to represent them. A forerunner of modern European decimal notation was introduced by Simon Stevin in the 16th century. A key argument against the system was its susceptibility to easy fraud by simply putting a number at the beginning or end of a quantity, thereby changing 100 into 5100
35.
Numerical digit
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A digit is a numeric symbol used in combinations to represent numbers in positional numeral systems. The name digit comes from the fact that the 10 digits of the hands correspond to the 10 symbols of the common base 10 numeral system, i. e. the decimal digits. In a given system, if the base is an integer. For example, the system has ten digits, whereas binary has two digits. In a basic system, a numeral is a sequence of digits. Each position in the sequence has a value, and each digit has a value. The value of the numeral is computed by multiplying each digit in the sequence by its place value, each digit in a number system represents an integer. For example, in decimal the digit 1 represents the one, and in the hexadecimal system. A positional number system must have a digit representing the integers from zero up to, but not including, thus in the positional decimal system, the numbers 0 to 9 can be expressed using their respective numerals 0 to 9 in the rightmost units position. The Hindu–Arabic numeral system uses a decimal separator, commonly a period in English, or a comma in other European languages, to denote the place or units place. Each successive place to the left of this has a value equal to the place value of the previous digit times the base. Similarly, each place to the right of the separator has a place value equal to the place value of the previous digit divided by the base. For example, in the numeral 10, the total value of the number is 1 ten,0 ones,3 tenths, and 4 hundredths. Note that the zero, which contributes no value to the number, the place value of any given digit in a numeral can be given by a simple calculation, which in itself is a compliment to the logic behind numeral systems. And to the right, the digit is multiplied by the base raised by a negative n, for example, in the number 10. This system was established by the 7th century in India, but was not yet in its modern form because the use of the digit zero had not yet widely accepted. Instead of a zero, a dot was left in the numeral as a placeholder, the first widely acknowledged use of zero was in 876. The original numerals were very similar to the ones, even down to the glyphs used to represent digits
36.
Olmec
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The Olmecs were the first major civilization in Guatemala and Mexico following a progressive development in Soconusco and modern southwestern pacific lowlands of Guatemala. They lived in the lowlands of south-central Mexico, in the present-day states of Veracruz. It has been speculated that Olmec derive in part from neighboring Mokaya and/or Mixe–Zoque, the population of the Olmecs flourished during Mesoamericas formative period, dating roughly from as early as 1500 BCE to about 400 BCE. They were the first Mesoamerican civilization, and laid many of the foundations for the civilizations that followed, among other firsts, the Olmec appeared to practice ritual bloodletting and played the Mesoamerican ballgame, hallmarks of nearly all subsequent Mesoamerican societies. The aspect of the Olmecs most familiar now is their artwork, the Olmec civilization was first defined through artifacts which collectors purchased on the pre-Columbian art market in the late 19th century and early 20th century. Olmec artworks are considered among ancient Americas most striking, the name Olmec comes from the Nahuatl word for the Olmecs, Ōlmēcatl or Ōlmēcah. This word is composed of the two words ōlli, meaning rubber, and mēcatl, meaning people, so the word means rubber people, the Olmec heartland is the area in the Gulf lowlands where it expanded after early development in Soconusco. This area is characterized by swampy lowlands punctuated by low hills, ridges, the Tuxtlas Mountains rise sharply in the north, along the Gulf of Mexicos Bay of Campeche. Here the Olmec constructed permanent city-temple complexes at San Lorenzo Tenochtitlán, La Venta, Tres Zapotes, in this region, the first Mesoamerican civilization emerged and reigned from c. The beginnings of Olmec civilization have traditionally been placed between 1400 and 1200 BCE, past finds of Olmec remains ritually deposited at El Manati shrine moved this back to at least 1600–1500 BCE. It seems that the Olmec had their roots in early farming cultures of Tabasco and these shared the same basic food crops and technologies of the later Olmec civilization. What is today called Olmec first appeared fully within the city of San Lorenzo Tenochtitlán, the rise of civilization was assisted by the local ecology of well-watered alluvial soil, as well as by the transportation network provided by the Coatzacoalcos River basin. This environment may be compared to that of other ancient centers of civilization, the Nile, Indus, and Yellow River valleys and this highly productive environment encouraged a densely concentrated population, which in turn triggered the rise of an elite class. The elite class created the demand for the production of the symbolic, the state of Guerrero, and in particular its early Mezcala culture, seem to have played an important role in the early history of Olmec culture. Olmec-style artifacts tend to appear earlier in some parts of Guerrero than in the Veracruz-Tabasco area, in particular, the relevant objects from the Amuco-Abelino site in Guerrero reveal dates as early as 1530 BC. The city of Teopantecuanitlan in Guerrero is also relevant in this regard, the first Olmec center, San Lorenzo, was all but abandoned around 900 BCE at about the same time that La Venta rose to prominence. A wholesale destruction of many San Lorenzo monuments also occurred circa 950 BCE, which may indicate an internal uprising or, less likely, an invasion. The latest thinking, however, is that changes may have been responsible for this shift in Olmec centers
37.
Maya civilization
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The Maya civilization developed in an area that encompasses southeastern Mexico, all of Guatemala and Belize, and the western portions of Honduras and El Salvador. The Archaic period, prior to 2000 BC, saw the first developments in agriculture, the first Maya cities developed around 750 BC, and by 500 BC these cities possessed monumental architecture, including large temples with elaborate stucco façades. Hieroglyphic writing was being used in the Maya region by the 3rd century BC, in the Late Preclassic a number of large cities developed in the Petén Basin, and Kaminaljuyu rose to prominence in the Guatemalan Highlands. Beginning around 250 AD, the Classic period is defined as when the Maya were raising sculpted monuments with Long Count dates. This period saw the Maya civilization develop a number of city-states linked by a complex trade network. In the Maya Lowlands two great rivals, Tikal and Calakmul, became powerful, the Classic period also saw the intrusive intervention of the central Mexican city of Teotihuacan in Maya dynastic politics. In the 9th century, there was a political collapse in the central Maya region, resulting in internecine warfare, the abandonment of cities. The Postclassic period saw the rise of Chichen Itza in the north, in the 16th century, the Spanish Empire colonized the Mesoamerican region, and a lengthy series of campaigns saw the fall of Nojpetén, the last Maya city in 1697. Classic period rule was centred on the concept of the divine king, kingship was patrilineal, and power would normally pass to the eldest son. A prospective king was expected to be a successful war leader. Maya politics was dominated by a system of patronage, although the exact political make-up of a kingdom varied from city-state to city-state. By the Late Classic, the aristocracy had greatly increased, resulting in the reduction in the exclusive power of the divine king. Maya cities tended to expand haphazardly, and the city centre would be occupied by ceremonial and administrative complexes, different parts of a city would often be linked by causeways. The principal architecture of the city consisted of palaces, pyramid-temples, ceremonial ballcourts, the Maya elite were literate, and developed a complex system of hieroglyphic writing that was the most advanced in the pre-Columbian Americas. The Maya recorded their history and ritual knowledge in screenfold books, there are also a great many examples of Maya text found on stelae and ceramics. The Maya developed a complex series of interlocking ritual calendars. As a part of their religion, the Maya practised human sacrifice, the Maya civilization developed within the Mesoamerican cultural area, which covers a region that spreads from northern Mexico southwards into Central America. Mesoamerica was one of six cradles of civilization worldwide, the Mesoamerican area gave rise to a series of cultural developments that included complex societies, agriculture, cities, monumental architecture, writing, and calendrical systems
38.
Mesoamerica
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It is one of six areas in the world where ancient civilization arose independently, and the second in the Americas along with Norte Chico in present-day northern coastal Peru. As a cultural area, Mesoamerica is defined by a mosaic of cultural traits developed and shared by its indigenous cultures, while Mesoamerican civilization did know of the wheel and basic metallurgy, neither of these technologies became culturally important. Among the earliest complex civilizations was the Olmec culture, which inhabited the Gulf coast of Mexico and extended inland, frequent contact and cultural interchange between the early Olmec and other cultures in Chiapas, Guatemala and Oaxaca laid the basis for the Mesoamerican cultural area. All this was facilitated by considerable regional communications in ancient Mesoamerica and this Formative period saw the spread of distinct religious and symbolic traditions, as well as artistic and architectural complexes. In the subsequent Preclassic period, complex urban polities began to develop among the Maya, with the rise of such as El Mirador, Calakmul and Tikal. Mesoamerica is one of three regions of the world where writing is known to have independently developed. Upon the collapse of Teotihuacán around AD600, competition between several important political centers in central Mexico, such as Xochicalco and Cholula, ensued. During the early period, Central Mexico was dominated by the Toltec culture, Oaxaca by the Mixtec. Towards the end of the period, the Aztecs of Central Mexico built a tributary empire covering most of central Mesoamerica. The distinct Mesoamerican cultural tradition ended with the Spanish conquest in the 16th century, over the next centuries, Mesoamerican indigenous cultures were gradually subjected to Spanish colonial rule. The exact geographic extent of Mesoamerica has varied through time, as the civilization extended North and South from its heartland in southern Mexico, Mesoamerica is recognized as a near-prototypical cultural area, and the term is now fully integrated in the standard terminology of pre-Columbian anthropological studies. Conversely, the sister terms Aridoamerica and Oasisamerica, which refer to northern Mexico, 10° and 22° northern latitude, Mesoamerica possesses a complex combination of ecological systems, topographic zones, and environmental contexts. A main distinction groups these different niches into two categories, the lowlands and the altiplanos, or highlands. In the low-lying regions, sub-tropical and tropical climates are most common, as is true for most of the coastline along the Pacific and Gulf of Mexico and the Caribbean Sea. The highlands show much more diversity, ranging from dry tropical to cold mountainous climates. The rainfall varies from the dry Oaxaca and north Yucatan to the humid southern Pacific, several distinct sub-regions within Mesoamerica are defined by a convergence of geographic and cultural attributes. These sub-regions are more conceptual than culturally meaningful, and the demarcation of their limits is not rigid, the Maya area, for example, can be divided into two general groups, the lowlands and highlands. The lowlands are further divided into the southern and northern Maya lowlands, the southern Maya lowlands are generally regarded as encompassing northern Guatemala, southern Campeche and Quintana Roo in Mexico, and Belize
39.
India
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India, officially the Republic of India, is a country in South Asia. It is the seventh-largest country by area, the second-most populous country, and it is bounded by the Indian Ocean on the south, the Arabian Sea on the southwest, and the Bay of Bengal on the southeast. It shares land borders with Pakistan to the west, China, Nepal, and Bhutan to the northeast, in the Indian Ocean, India is in the vicinity of Sri Lanka and the Maldives. Indias Andaman and Nicobar Islands share a border with Thailand. The Indian subcontinent was home to the urban Indus Valley Civilisation of the 3rd millennium BCE, in the following millennium, the oldest scriptures associated with Hinduism began to be composed. Social stratification, based on caste, emerged in the first millennium BCE, early political consolidations took place under the Maurya and Gupta empires, the later peninsular Middle Kingdoms influenced cultures as far as southeast Asia. In the medieval era, Judaism, Zoroastrianism, Christianity, and Islam arrived, much of the north fell to the Delhi sultanate, the south was united under the Vijayanagara Empire. The economy expanded in the 17th century in the Mughal empire, in the mid-18th century, the subcontinent came under British East India Company rule, and in the mid-19th under British crown rule. A nationalist movement emerged in the late 19th century, which later, under Mahatma Gandhi, was noted for nonviolent resistance, in 2015, the Indian economy was the worlds seventh largest by nominal GDP and third largest by purchasing power parity. Following market-based economic reforms in 1991, India became one of the major economies and is considered a newly industrialised country. However, it continues to face the challenges of poverty, corruption, malnutrition, a nuclear weapons state and regional power, it has the third largest standing army in the world and ranks sixth in military expenditure among nations. India is a constitutional republic governed under a parliamentary system. It is a pluralistic, multilingual and multi-ethnic society and is home to a diversity of wildlife in a variety of protected habitats. The name India is derived from Indus, which originates from the Old Persian word Hindu, the latter term stems from the Sanskrit word Sindhu, which was the historical local appellation for the Indus River. The ancient Greeks referred to the Indians as Indoi, which translates as The people of the Indus, the geographical term Bharat, which is recognised by the Constitution of India as an official name for the country, is used by many Indian languages in its variations. Scholars believe it to be named after the Vedic tribe of Bharatas in the second millennium B. C. E and it is also traditionally associated with the rule of the legendary emperor Bharata. Gaṇarājya is the Sanskrit/Hindi term for republic dating back to the ancient times, hindustan is a Persian name for India dating back to the 3rd century B. C. E. It was introduced into India by the Mughals and widely used since then and its meaning varied, referring to a region that encompassed northern India and Pakistan or India in its entirety
40.
Brahmagupta
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Brahmagupta was an Indian mathematician and astronomer. He is the author of two works on mathematics and astronomy, the Brāhmasphuṭasiddhānta, a theoretical treatise, and the Khaṇḍakhādyaka. According to his commentators, Brahmagupta was a native of Bhinmal, Brahmagupta was the first to give rules to compute with zero. The texts composed by Brahmagupta were composed in verse in Sanskrit. As no proofs are given, it is not known how Brahmaguptas results were derived, Brahmagupta was born in 598 CE according to his own statement. He lived in Bhillamala during the reign of the Chapa dynasty ruler Vyagrahamukha and he was the son of Jishnugupta. He was a Shaivite by religion, even though most scholars assume that Brahmagupta was born in Bhillamala, there is no conclusive evidence for it. However, he lived and worked there for a part of his life. Prithudaka Svamin, a commentator, called him Bhillamalacharya, the teacher from Bhillamala. Sociologist G. S. Ghurye believed that he might have been from the Multan region or the Abu region and it was also a center of learning for mathematics and astronomy. Brahmagupta became an astronomer of the Brahmapaksha school, in the year 628, at an age of 30, he composed Brāhmasphuṭasiddhānta which is believed to be a revised version of the received siddhanta of the Brahmapaksha school. Scholars state that he has incorported a great deal of originality to his revision, the book consists of 24 chapters with 1008 verses in the ārya meter. Later, Brahmagupta moved to Ujjain, which was also a centre for astronomy. At the mature age of 67, he composed his next well known work Khanda-khādyaka and he is believed to have died in Ujjain. Brahmagupta had a plethora of criticism directed towards the work of rival astronomers, the division was primarily about the application of mathematics to the physical world, rather than about the mathematics itself. In Brahmaguptas case, the disagreements stemmed largely from the choice of astronomical parameters, the historian of science George Sarton called him one of the greatest scientists of his race and the greatest of his time. Brahmaguptas mathematical advances were carried on to further extent by Bhāskara II, a descendant in Ujjain. Prithudaka Svamin wrote commentaries on both of his works, rendering difficult verses into simpler language and adding illustrations, lalla and Bhattotpala in the 8th and 9th centuries wrote commentaries on the Khanda-khadyaka
41.
Computus
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Computus is a calculation that determines the calendar date of Easter. The name has been used for this procedure since the early Middle Ages, for most of their history Christians have calculated Easter independently of the Jewish calendar. In principle, Easter falls on the Sunday following the moon that follows the northern spring equinox. However, the equinox and the full moon are not determined by astronomical observation. The vernal equinox is fixed to fall on 21 March, the full moon is an ecclesiastical full moon determined by reference to a lunar calendar, which again varied in different areas. The last limit arises from the fact that the crucifixion was considered to have happened on the 14th, the synodic month had already been measured to a high degree of accuracy. The schematic model that eventually was accepted is the Metonic cycle, in 1583, the Catholic Church began using 21 March under the Gregorian calendar to calculate the date of Easter, while the Eastern Churches have continued to use 21 March under the Julian calendar. The Catholic and Protestant denominations thus use a full moon that occurs four. The earliest and latest dates for Easter are 22 March and 25 April, in the Gregorian calendar those dates are as commonly understood. Easter is the most important Christian feast, and the date of its celebration has been the subject of controversy as early as the meeting of Anicetus. The rest of the Christian world at time, according to Eusebius, held to the view which still prevails. Eusebius does not say how the Sunday was decided, by the end of the 3rd century some Christians had become dissatisfied with what they perceived as the disorderly state of the Jewish calendar. The chief complaint was that the Jewish practice sometimes set the 14th of Nisan before the spring equinox, and it was explicitly stated by Peter, bishop of Alexandria that the men of the present day now celebrate before the equinox. through negligence and error. Another objection to using the Jewish computation may have been that the Jewish calendar was not unified, Jews in one city might have a method for reckoning the Week of Unleavened Bread different from that used by the Jews of another city. Because of these perceived defects in the practice, Christian computists began experimenting with systems for determining Easter that would be free of these defects. At the First Council of Nicaea in 325, it was agreed that the Christians should observe a common date, because of the divergence of tables mentioned above it was usual to negotiate a common date when discrepancies arose. It took several centuries before a common method was accepted throughout Christendom, the process of working out the details generated still further controversies. The method from Alexandria became authoritative, in its developed form it was based on the epacts of a reckoned moon according to the 19-year Metonic cycle
42.
Easter
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It is the culmination of the Passion of Jesus, preceded by Lent, a forty-day period of fasting, prayer, and penance. In Western Christianity, Eastertide, or the Easter Season, begins on Easter Sunday and lasts seven weeks, ending with the coming of the fiftieth day, Pentecost Sunday. In Eastern Christianity, the season of Pascha begins on Pascha and ends with the coming of the fortieth day, the Feast of the Ascension. The First Council of Nicaea established two rules, independence of the Jewish calendar and worldwide uniformity, which were the rules for Easter explicitly laid down by the council. No details for the computation were specified, these were worked out in practice and it has come to be the first Sunday after the ecclesiastical full moon that occurs on or soonest after 21 March, but calculations vary. Easter is linked to the Jewish Passover by much of its symbolism, in many languages, the words for Easter and Passover are identical or very similar. Easter customs vary across the Christian world, and include services, exclaiming the Paschal greeting, clipping the church. The Easter lily, a symbol of the resurrection, traditionally decorates the area of churches on this day. Additional customs that have associated with Easter and are observed by both Christians and some non-Christians include egg hunting, the Easter Bunny, and Easter parades. There are also various traditional Easter foods that vary regionally, however, it is possible that Bede was only speculating about the origin of the term since there is no firm evidence that such a goddess actually existed. In Greek and Latin, the Christian celebration was, and still is, called Πάσχα, Pascha, the word originally denoted the Jewish festival known in English as Passover, commemorating the Jewish Exodus from slavery in Egypt. In most of the non-English speaking world, the feast is known by names derived from Greek, Pascha is also a name by which Jesus himself is remembered in the Orthodox Church, especially in connection with his resurrection and with the season of its celebration. The New Testament states that the resurrection of Jesus, which Easter celebrates, is a foundation of the Christian faith, the resurrection established Jesus as the powerful Son of God and is cited as proof that God will judge the world in righteousness. For those who trust in Jesus death and resurrection, death is swallowed up in victory, any person who chooses to follow Jesus receives a new birth into a living hope through the resurrection of Jesus Christ from the dead. Through faith in the working of God those who follow Jesus are spiritually resurrected with him so that they may walk in a new way of life and receive eternal salvation. Easter is linked to the Passover and Exodus from Egypt recorded in the Old Testament through the Last Supper, sufferings and crucifixion of Jesus that preceded the resurrection. According to the New Testament, Jesus gave the Passover meal a new meaning, as in the room during the Last Supper he prepared himself. He identified the matzah and cup of wine as his soon to be sacrificed
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