1.
Bordeaux
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Bordeaux is a port city on the Garonne River in the Gironde department in southwestern France. The municipality of Bordeaux proper has a population of 243,626, together with its suburbs and satellite towns, Bordeaux is the centre of the Bordeaux Métropole. With 749,595 inhabitants and 1,178,335 in the area, it is the fifth largest in France, after Paris, Lyon, Marseille and Lille. It is the capital of the Nouvelle-Aquitaine region, as well as the prefecture of the Gironde department and its inhabitants are called Bordelais or Bordelaises. The term Bordelais may also refer to the city and its surrounding region, Bordeaux is the worlds major wine industry capital. It is home to the main wine fair, Vinexpo. Bordeaux wine has been produced in the region since the 8th century, the historic part of the city is on the UNESCO World Heritage List as an outstanding urban and architectural ensemble of the 18th century. After Paris, Bordeaux has the highest number of preserved buildings of any city in France. In historical times, around 300 BC it was the settlement of a Celtic tribe, the Bituriges Vivisci, the name Bourde is still the name of a river south of the city. In 107 BC, the Battle of Burdigala was fought by the Romans who were defending the Allobroges, a Gallic tribe allied to Rome, the Romans were defeated and their commander, the consul Lucius Cassius Longinus, was killed in the action. The city fell under Roman rule around 60 BC, its importance lying in the commerce of tin, later it became capital of Roman Aquitaine, flourishing especially during the Severan dynasty. In 276 it was sacked by the Vandals, further ravage was brought by the same Vandals in 409, the Visigoths in 414 and the Franks in 498, beginning a period of obscurity for the city. In the late 6th century, the city re-emerged as the seat of a county and an archdiocese within the Merovingian kingdom of the Franks, the city started to play a regional role as a major urban center on the fringes of the newly founded Frankish Duchy of Vasconia. Around 585, a certain Gallactorius is cited as count of Bordeaux, the city was plundered by the troops of Abd er Rahman in 732 after storming the fortified city and overwhelming the Aquitanian garrison. After Duke Eudess defeat, the Aquitanian duke could still save part of its troops, the following year, the Frankish commander descended again over Aquitaine, but clashed in battle with the Aquitanians and left to take on hostile Burgundian authorities and magnates. In 745, Aquitaine faced yet another expedition by Charles sons Pepin and Carloman against Hunald, Hunald was defeated, and his son Waifer replaced him, who in turn confirmed Bordeaux as the capital city. During the last stage of the war against Aquitaine, it was one of Waifers last important strongholds to fall to King Pepin the Shorts troops. Next to Bordeaux, Charlemagne built the fortress of Fronsac on a hill across the border with the Basques, in 778, Seguin was appointed count of Bordeaux, probably undermining the power of the Duke Lupo, and possibly leading to the Battle of Roncevaux Pass that very year
Bordeaux
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Clockwise from top:
Place de la Bourse by the
Garonne, Allees du Tourny and Maison de Vin,
Pierre Bridge on the Garonne, Meriadeck Commercial Centre, front of Palais Rohan Hotel, and
Saint-Andre Cathedral with
Bordeaux Tramway
Bordeaux
Bordeaux
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Coins of the
Bituriges Vivisci, 5th–1st century BC, derived from the coin designs of
Greeks in pre-Roman Gaul.
Cabinet des Médailles.
Bordeaux
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Merovingian tremisses minted in Bordeaux by the Church of Saint-Étienne, late 6th century.
British Museum.
2.
Operation (mathematics)
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In mathematics, an operation is a calculation from zero or more input values to an output value. The number of operands is the arity of the operation, the most commonly studied operations are binary operations of arity 2, such as addition and multiplication, and unary operations of arity 1, such as additive inverse and multiplicative inverse. An operation of arity zero, or 0-ary operation is a constant, the mixed product is an example of an operation of arity 3, or ternary operation. Generally, the arity is supposed to be finite, but infinitary operations are sometimes considered, in this context, the usual operations, of finite arity are also called finitary operations. There are two types of operations, unary and binary. Unary operations involve only one value, such as negation and trigonometric functions, binary operations, on the other hand, take two values, and include addition, subtraction, multiplication, division, and exponentiation. Operations can involve mathematical objects other than numbers, the logical values true and false can be combined using logic operations, such as and, or, and not. Vectors can be added and subtracted, rotations can be combined using the function composition operation, performing the first rotation and then the second. Operations on sets include the binary operations union and intersection and the operation of complementation. Operations on functions include composition and convolution, operations may not be defined for every possible value. For example, in the numbers one cannot divide by zero or take square roots of negative numbers. The values for which an operation is defined form a set called its domain, the set which contains the values produced is called the codomain, but the set of actual values attained by the operation is its range. For example, in the numbers, the squaring operation only produces non-negative numbers. A vector can be multiplied by a scalar to form another vector, and the inner product operation on two vectors produces a scalar. An operation may or may not have certain properties, for example it may be associative, commutative, anticommutative, idempotent, the values combined are called operands, arguments, or inputs, and the value produced is called the value, result, or output. Operations can have fewer or more than two inputs, an operation is like an operator, but the point of view is different. An operation ω is a function of the form ω, V → Y, where V ⊂ X1 × … × Xk. The sets Xk are called the domains of the operation, the set Y is called the codomain of the operation, thus a unary operation has arity one, and a binary operation has arity two
Operation (mathematics)
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– plus (addition) – minus (subtraction) – times (multiplication) – obelus (division)
3.
Minus sign
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The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning more and less, respectively, though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. In Europe in the early 15th century the letters P and M were generally used, the symbols appeared for the first time in Luca Pacioli’s mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice in 1494. The + is a simplification of the Latin et, the − may be derived from a tilde written over m when used to indicate subtraction, or it may come from a shorthand version of the letter m itself. In his 1489 treatise Johannes Widmann referred to the symbols − and + as minus and mer, was − ist, das ist minus, und das + ist das mer. They werent used for addition and subtraction here, but to indicate surplus and deficit, the plus sign is a binary operator that indicates addition, as in 2 +3 =5. It can also serve as an operator that leaves its operand unchanged. This notation may be used when it is desired to emphasize the positiveness of a number, the plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or is equivalent to and it is conventional to use the plus sign to only denote commutative operations. Subtraction is the inverse of addition, directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5, a unary operator that acts as an instruction to replace the operand by its additive inverse. For example, if x is 3, then −x is −3, similarly, − is equal to 2. The above is a case of this. All three uses can be referred to as minus in everyday speech, further, some textbooks in the United States encourage −x to be read as the opposite of x or the additive inverse of x to avoid giving the impression that −x is necessarily negative. However, in programming languages and Microsoft Excel in particular, unary operators bind strongest, so in those cases −5^2 is 25. Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers. For example, subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as 3 − −5 becomes 3 +5 =8, in grading systems, the plus sign indicates a grade one level higher and the minus sign a grade lower
Minus sign
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Plus, minus, and hyphen-minus.
4.
Negative number
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In mathematics, a negative number is a real number that is less than zero. If positive represents movement to the right, negative represents movement to the left, if positive represents above sea level, then negative represents below level. If positive represents a deposit, negative represents a withdrawal and they are often used to represent the magnitude of a loss or deficiency. A debt that is owed may be thought of as a negative asset, if a quantity may have either of two opposite senses, then one may choose to distinguish between those senses—perhaps arbitrarily—as positive and negative. In the medical context of fighting a tumor, an expansion could be thought of as a negative shrinkage, negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. The laws of arithmetic for negative numbers ensure that the common idea of an opposite is reflected in arithmetic. For example, − −3 =3 because the opposite of an opposite is the original thing, negative numbers are usually written with a minus sign in front. For example, −3 represents a quantity with a magnitude of three, and is pronounced minus three or negative three. To help tell the difference between a subtraction operation and a number, occasionally the negative sign is placed slightly higher than the minus sign. Conversely, a number that is greater than zero is called positive, the positivity of a number may be emphasized by placing a plus sign before it, e. g. +3. In general, the negativity or positivity of a number is referred to as its sign, every real number other than zero is either positive or negative. The positive whole numbers are referred to as natural numbers, while the positive and negative numbers are referred to as integers. In bookkeeping, amounts owed are often represented by red numbers, or a number in parentheses, Liu Hui established rules for adding and subtracting negative numbers. By the 7th century, Indian mathematicians such as Brahmagupta were describing the use of negative numbers, islamic mathematicians further developed the rules of subtracting and multiplying negative numbers and solved problems with negative coefficients. Western mathematicians accepted the idea of numbers by the 17th century. Prior to the concept of numbers, mathematicians such as Diophantus considered negative solutions to problems false. Negative numbers can be thought of as resulting from the subtraction of a number from a smaller. For example, negative three is the result of subtracting three from zero,0 −3 = −3, in general, the subtraction of a larger number from a smaller yields a negative result, with the magnitude of the result being the difference between the two numbers
Negative number
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This thermometer is indicating a negative
Fahrenheit temperature (−4°F).
5.
Fraction (mathematics)
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A fraction represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator displayed above a line, numerators and denominators are also used in fractions that are not common, including compound fractions, complex fractions, and mixed numerals. The numerator represents a number of parts, and the denominator. For example, in the fraction 3/4, the numerator,3, tells us that the fraction represents 3 equal parts, the picture to the right illustrates 34 or ¾ of a cake. Fractional numbers can also be written without using explicit numerators or denominators, by using decimals, percent signs, an integer such as the number 7 can be thought of as having an implicit denominator of one,7 equals 7/1. Other uses for fractions are to represent ratios and to represent division, thus the fraction ¾ is also used to represent the ratio 3,4 and the division 3 ÷4. The test for a number being a number is that it can be written in that form. In a fraction, the number of parts being described is the numerator. Informally, they may be distinguished by placement alone but in formal contexts they are separated by a fraction bar. The fraction bar may be horizontal, oblique, or diagonal and these marks are respectively known as the horizontal bar, the slash or stroke, the division slash, and the fraction slash. In typography, horizontal fractions are known as en or nut fractions and diagonal fractions as em fractions. The denominators of English fractions are expressed as ordinal numbers. When the denominator is 1, it may be expressed in terms of wholes but is commonly ignored. When the numerator is one, it may be omitted, a fraction may be expressed as a single composition, in which case it is hyphenated, or as a number of fractions with a numerator of one, in which case they are not. Fractions should always be hyphenated when used as adjectives, alternatively, a fraction may be described by reading it out as the numerator over the denominator, with the denominator expressed as a cardinal number. The term over is used even in the case of solidus fractions, Fractions with large denominators that are not powers of ten are often rendered in this fashion while those with denominators divisible by ten are typically read in the normal ordinal fashion. A simple fraction is a number written as a/b or a b
Fraction (mathematics)
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A cake with one quarter (one fourth) removed. The remaining three fourths are shown. Dotted lines indicate where the cake may be cut in order to divide it into equal parts. Each fourth of the cake is denoted by the fraction ¼.
6.
Irrational number
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In mathematics, the irrational numbers are all the real numbers, which are not rational numbers, the latter being the numbers constructed from ratios of integers. Irrational numbers may also be dealt with via non-terminating continued fractions, for example, the decimal representation of the number π starts with 3.14159265358979, but no finite number of digits can represent π exactly, nor does it repeat. Mathematicians do not generally take terminating or repeating to be the definition of the concept of rational number, as a consequence of Cantors proof that the real numbers are uncountable and the rationals countable, it follows that almost all real numbers are irrational. The first proof of the existence of numbers is usually attributed to a Pythagorean. The then-current Pythagorean method would have claimed that there must be sufficiently small. However, Hippasus, in the 5th century BC, was able to deduce that there was in no common unit of measure. His reasoning is as follows, Start with a right triangle with side lengths of integers a, b. The ratio of the hypotenuse to a leg is represented by c, b, assume a, b, and c are in the smallest possible terms. By the Pythagorean theorem, c2 = a2+b2 = b2+b2 = 2b2, since c2 = 2b2, c2 is divisible by 2, and therefore even. Since c2 is even, c must be even, since c is even, dividing c by 2 yields an integer. Squaring both sides of c = 2y yields c2 =2, or c2 = 4y2, substituting 4y2 for c2 in the first equation gives us 4y2= 2b2. Dividing by 2 yields 2y2 = b2, since y is an integer, and 2y2 = b2, b2 is divisible by 2, and therefore even. Since b2 is even, b must be even and we have just show that both b and c must be even. Hence they have a factor of 2. However this contradicts the assumption that they have no common factors and this contradiction proves that c and b cannot both be integers, and thus the existence of a number that cannot be expressed as a ratio of two integers. Greek mathematicians termed this ratio of incommensurable magnitudes alogos, or inexpressible. ”Another legend states that Hippasus was merely exiled for this revelation, the discovery of incommensurable ratios was indicative of another problem facing the Greeks, the relation of the discrete to the continuous. Brought into light by Zeno of Elea, who questioned the conception that quantities are discrete and composed of a number of units of a given size. ”However Zeno found that in fact “ in general are not discrete collections of units. That in fact, these divisions of quantity must necessarily be infinite, for example, consider a line segment, this segment can be split in half, that half split in half, the half of the half in half, and so on
Irrational number
–
The number is irrational.
7.
Euclidean vector
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In mathematics, physics, and engineering, a Euclidean vector is a geometric object that has magnitude and direction. Vectors can be added to other vectors according to vector algebra, a Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by A B →. A vector is what is needed to carry the point A to the point B and it was first used by 18th century astronomers investigating planet rotation around the Sun. The magnitude of the vector is the distance between the two points and the direction refers to the direction of displacement from A to B. These operations and associated laws qualify Euclidean vectors as an example of the more generalized concept of vectors defined simply as elements of a vector space. Vectors play an important role in physics, the velocity and acceleration of a moving object, many other physical quantities can be usefully thought of as vectors. Although most of them do not represent distances, their magnitude and direction can still be represented by the length, the mathematical representation of a physical vector depends on the coordinate system used to describe it. Other vector-like objects that describe physical quantities and transform in a similar way under changes of the system include pseudovectors and tensors. The concept of vector, as we know it today, evolved gradually over a period of more than 200 years, about a dozen people made significant contributions. Giusto Bellavitis abstracted the basic idea in 1835 when he established the concept of equipollence, working in a Euclidean plane, he made equipollent any pair of line segments of the same length and orientation. Essentially he realized an equivalence relation on the pairs of points in the plane, the term vector was introduced by William Rowan Hamilton as part of a quaternion, which is a sum q = s + v of a Real number s and a 3-dimensional vector. Like Bellavitis, Hamilton viewed vectors as representative of classes of equipollent directed segments, grassmanns work was largely neglected until the 1870s. Peter Guthrie Tait carried the standard after Hamilton. His 1867 Elementary Treatise of Quaternions included extensive treatment of the nabla or del operator ∇, in 1878 Elements of Dynamic was published by William Kingdon Clifford. Clifford simplified the quaternion study by isolating the dot product and cross product of two vectors from the complete quaternion product and this approach made vector calculations available to engineers and others working in three dimensions and skeptical of the fourth. Josiah Willard Gibbs, who was exposed to quaternions through James Clerk Maxwells Treatise on Electricity and Magnetism, the first half of Gibbss Elements of Vector Analysis, published in 1881, presents what is essentially the modern system of vector analysis. In 1901 Edwin Bidwell Wilson published Vector Analysis, adapted from Gibbs lectures, in physics and engineering, a vector is typically regarded as a geometric entity characterized by a magnitude and a direction. It is formally defined as a line segment, or arrow
Euclidean vector
8.
Commutativity
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In mathematics, a binary operation is commutative if changing the order of the operands does not change the result. It is a property of many binary operations, and many mathematical proofs depend on it. Most familiar as the name of the property that says 3 +4 =4 +3 or 2 ×5 =5 ×2, the property can also be used in more advanced settings. The name is needed there are operations, such as division and subtraction. The commutative property is a property associated with binary operations and functions. If the commutative property holds for a pair of elements under a binary operation then the two elements are said to commute under that operation. The term commutative is used in several related senses, putting on socks resembles a commutative operation since which sock is put on first is unimportant. Either way, the result, is the same, in contrast, putting on underwear and trousers is not commutative. The commutativity of addition is observed when paying for an item with cash, regardless of the order the bills are handed over in, they always give the same total. The multiplication of numbers is commutative, since y z = z y for all y, z ∈ R For example,3 ×5 =5 ×3. Some binary truth functions are also commutative, since the tables for the functions are the same when one changes the order of the operands. For example, the logical biconditional function p ↔ q is equivalent to q ↔ p and this function is also written as p IFF q, or as p ≡ q, or as Epq. Further examples of binary operations include addition and multiplication of complex numbers, addition and scalar multiplication of vectors. Concatenation, the act of joining character strings together, is a noncommutative operation, rotating a book 90° around a vertical axis then 90° around a horizontal axis produces a different orientation than when the rotations are performed in the opposite order. The twists of the Rubiks Cube are noncommutative and this can be studied using group theory. Some non-commutative binary operations, Records of the use of the commutative property go back to ancient times. The Egyptians used the property of multiplication to simplify computing products. Euclid is known to have assumed the property of multiplication in his book Elements
Commutativity
–
This image illustrates that addition is commutative.
9.
Associativity
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In mathematics, the associative property is a property of some binary operations. In propositional logic, associativity is a rule of replacement for expressions in logical proofs. That is, rearranging the parentheses in such an expression will not change its value, consider the following equations, +4 =2 + =92 × = ×4 =24. Even though the parentheses were rearranged on each line, the values of the expressions were not altered, since this holds true when performing addition and multiplication on any real numbers, it can be said that addition and multiplication of real numbers are associative operations. Associativity is not to be confused with commutativity, which addresses whether or not the order of two operands changes the result. For example, the order doesnt matter in the multiplication of numbers, that is. Associative operations are abundant in mathematics, in fact, many algebraic structures explicitly require their binary operations to be associative, however, many important and interesting operations are non-associative, some examples include subtraction, exponentiation and the vector cross product. Z = x = xyz for all x, y, z in S, the associative law can also be expressed in functional notation thus, f = f. If a binary operation is associative, repeated application of the produces the same result regardless how valid pairs of parenthesis are inserted in the expression. This is called the generalized associative law, thus the product can be written unambiguously as abcd. As the number of elements increases, the number of ways to insert parentheses grows quickly. Some examples of associative operations include the following, the two methods produce the same result, string concatenation is associative. In arithmetic, addition and multiplication of numbers are associative, i. e. + z = x + = x + y + z z = x = x y z } for all x, y, z ∈ R. x, y, z\in \mathbb. }Because of associativity. Addition and multiplication of numbers and quaternions are associative. Addition of octonions is also associative, but multiplication of octonions is non-associative, the greatest common divisor and least common multiple functions act associatively. Gcd = gcd = gcd lcm = lcm = lcm } for all x, y, z ∈ Z. x, y, z\in \mathbb. }Taking the intersection or the union of sets, ∩ C = A ∩ = A ∩ B ∩ C ∪ C = A ∪ = A ∪ B ∪ C } for all sets A, B, C. Slightly more generally, given four sets M, N, P and Q, with h, M to N, g, N to P, in short, composition of maps is always associative. Consider a set with three elements, A, B, and C, thus, for example, A=C = A
Associativity
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A binary operation ∗ on the set S is associative when
this diagram commutes. That is, when the two paths from S × S × S to S
compose to the same function from S × S × S to S.
10.
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
0 (number)
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Example of the early Greek symbol for zero (lower right corner) from a 2nd-century papyrus
0 (number)
0 (number)
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The number 605 in Khmer numerals, from the Sambor inscription (
Saka era 605 corresponds to AD 683). The earliest known material use of zero as a decimal figure.
0 (number)
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The back of Olmec stela C from
Tres Zapotes, the second oldest
Long Count date discovered. The numerals 7.16.6.16.18 translate to September, 32 BC (Julian). The glyphs surrounding the date are thought to be one of the few surviving examples of
Epi-Olmec script.
11.
Addition
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Addition is one of the four basic operations of arithmetic, with the others being subtraction, multiplication and division. The addition of two numbers is the total amount of those quantities combined. For example, in the picture on the right, there is a combination of three apples and two together, making a total of five apples. This observation is equivalent to the mathematical expression 3 +2 =5 i. e.3 add 2 is equal to 5, besides counting fruits, addition can also represent combining other physical objects. In arithmetic, rules for addition involving fractions and negative numbers have been devised amongst others, in algebra, addition is studied more abstractly. It is commutative, meaning that order does not matter, and it is associative, repeated addition of 1 is the same as counting, addition of 0 does not change a number. Addition also obeys predictable rules concerning related operations such as subtraction and multiplication, performing addition is one of the simplest numerical tasks. Addition of very small numbers is accessible to toddlers, the most basic task,1 +1, can be performed by infants as young as five months and even some members of other animal species. In primary education, students are taught to add numbers in the system, starting with single digits. Mechanical aids range from the ancient abacus to the modern computer, Addition is written using the plus sign + between the terms, that is, in infix notation. The result is expressed with an equals sign, for example, 3½ =3 + ½ =3.5. This notation can cause confusion since in most other contexts juxtaposition denotes multiplication instead, the sum of a series of related numbers can be expressed through capital sigma notation, which compactly denotes iteration. For example, ∑ k =15 k 2 =12 +22 +32 +42 +52 =55. The numbers or the objects to be added in addition are collectively referred to as the terms, the addends or the summands. This is to be distinguished from factors, which are multiplied, some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an addend at all, today, due to the commutative property of addition, augend is rarely used, and both terms are generally called addends. All of the above terminology derives from Latin, using the gerundive suffix -nd results in addend, thing to be added. Likewise from augere to increase, one gets augend, thing to be increased, sum and summand derive from the Latin noun summa the highest, the top and associated verb summare
Addition
–
Part of Charles Babbage's
Difference Engine including the addition and carry mechanisms
Addition
–
The plus sign
Addition
–
A circular slide rule
12.
Multiplication
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Multiplication is one of the four elementary, mathematical operations of arithmetic, with the others being addition, subtraction and division. Multiplication can also be visualized as counting objects arranged in a rectangle or as finding the area of a rectangle whose sides have given lengths, the area of a rectangle does not depend on which side is measured first, which illustrates the commutative property. The product of two measurements is a new type of measurement, for multiplying the lengths of the two sides of a rectangle gives its area, this is the subject of dimensional analysis. The inverse operation of multiplication is division, for example, since 4 multiplied by 3 equals 12, then 12 divided by 3 equals 4. Multiplication by 3, followed by division by 3, yields the original number, Multiplication is also defined for other types of numbers, such as complex numbers, and more abstract constructs, like matrices. For these more abstract constructs, the order that the operands are multiplied sometimes does matter, a listing of the many different kinds of products that are used in mathematics is given in the product page. In arithmetic, multiplication is often written using the sign × between the terms, that is, in infix notation, there are other mathematical notations for multiplication, Multiplication is also denoted by dot signs, usually a middle-position dot,5 ⋅2 or 5. 2 The middle dot notation, encoded in Unicode as U+22C5 ⋅ dot operator, is standard in the United States, the United Kingdom, when the dot operator character is not accessible, the interpunct is used. In other countries use a comma as a decimal mark. In algebra, multiplication involving variables is often written as a juxtaposition, the notation can also be used for quantities that are surrounded by parentheses. In matrix multiplication, there is a distinction between the cross and the dot symbols. The cross symbol generally denotes the taking a product of two vectors, yielding a vector as the result, while the dot denotes taking the dot product of two vectors, resulting in a scalar. In computer programming, the asterisk is still the most common notation and this is due to the fact that most computers historically were limited to small character sets that lacked a multiplication sign, while the asterisk appeared on every keyboard. This usage originated in the FORTRAN programming language, the numbers to be multiplied are generally called the factors. The number to be multiplied is called the multiplicand, while the number of times the multiplicand is to be multiplied comes from the multiplier. Usually the multiplier is placed first and the multiplicand is placed second, however sometimes the first factor is the multiplicand, additionally, there are some sources in which the term multiplicand is regarded as a synonym for factor. In algebra, a number that is the multiplier of a variable or expression is called a coefficient, the result of a multiplication is called a product. A product of integers is a multiple of each factor, for example,15 is the product of 3 and 5, and is both a multiple of 3 and a multiple of 5
Multiplication
–
4 × 5 = 20, the rectangle is composed of 20 squares, having dimensions of 4 by 5.
Multiplication
–
Four bags of three
marbles gives twelve marbles (4 × 3 = 12).
13.
Mathematical proof
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In mathematics, a proof is an inferential argument for a mathematical statement. In the argument, other previously established statements, such as theorems, in principle, a proof can be traced back to self-evident or assumed statements, known as axioms, along with accepted rules of inference. Axioms may be treated as conditions that must be met before the statement applies, Proofs are examples of exhaustive deductive reasoning or inductive reasoning and are distinguished from empirical arguments or non-exhaustive inductive reasoning. A proof must demonstrate that a statement is true, rather than enumerate many confirmatory cases. An unproved proposition that is believed to be true is known as a conjecture, Proofs employ logic but usually include some amount of natural language which usually admits some ambiguity. In fact, the vast majority of proofs in mathematics can be considered as applications of rigorous informal logic. Purely formal proofs, written in language instead of natural language, are considered in proof theory. The distinction between formal and informal proofs has led to examination of current and historical mathematical practice, quasi-empiricism in mathematics. The philosophy of mathematics is concerned with the role of language and logic in proofs, the word proof comes from the Latin probare meaning to test. Related modern words are the English probe, probation, and probability, the Spanish probar, Italian provare, the early use of probity was in the presentation of legal evidence. A person of authority, such as a nobleman, was said to have probity, whereby the evidence was by his relative authority, plausibility arguments using heuristic devices such as pictures and analogies preceded strict mathematical proof. It is likely that the idea of demonstrating a conclusion first arose in connection with geometry, the development of mathematical proof is primarily the product of ancient Greek mathematics, and one of the greatest achievements thereof. Thales proved some theorems in geometry, eudoxus and Theaetetus formulated theorems but did not prove them. Aristotle said definitions should describe the concept being defined in terms of other concepts already known and his book, the Elements, was read by anyone who was considered educated in the West until the middle of the 20th century. Further advances took place in medieval Islamic mathematics, while earlier Greek proofs were largely geometric demonstrations, the development of arithmetic and algebra by Islamic mathematicians allowed more general proofs that no longer depended on geometry. In the 10th century CE, the Iraqi mathematician Al-Hashimi provided general proofs for numbers as he considered multiplication, division and he used this method to provide a proof of the existence of irrational numbers. An inductive proof for arithmetic sequences was introduced in the Al-Fakhri by Al-Karaji, alhazen also developed the method of proof by contradiction, as the first attempt at proving the Euclidean parallel postulate. Modern proof theory treats proofs as inductively defined data structures, there is no longer an assumption that axioms are true in any sense, this allows for parallel mathematical theories built on alternate sets of axioms
Mathematical proof
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One of the oldest surviving fragments of Euclid's Elements, a textbook used for millennia to teach proof-writing techniques. The diagram accompanies Book II, Proposition 5.
Mathematical proof
–
Visual proof for the (3, 4, 5) triangle as in the
Chou Pei Suan Ching 500–200 BC.
14.
Integers
–
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
Integers
–
The Zahlen symbol, often used to denote the set of all integers (see
List of mathematical symbols)
15.
Real number
–
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
Real number
–
A symbol of the set of real numbers (ℝ)
16.
Abstract algebra
–
In algebra, which is a broad division of mathematics, abstract algebra is the study of algebraic structures. Algebraic structures include groups, rings, fields, modules, vector spaces, lattices, the term abstract algebra was coined in the early 20th century to distinguish this area of study from the other parts of algebra. Algebraic structures, with their homomorphisms, form mathematical categories. Category theory is a formalism that allows a way for expressing properties. Universal algebra is a subject that studies types of algebraic structures as single objects. For example, the structure of groups is an object in universal algebra. As in other parts of mathematics, concrete problems and examples have played important roles in the development of abstract algebra, through the end of the nineteenth century, many – perhaps most – of these problems were in some way related to the theory of algebraic equations. Numerous textbooks in abstract algebra start with definitions of various algebraic structures. This creates an impression that in algebra axioms had come first and then served as a motivation. The true order of development was almost exactly the opposite. For example, the numbers of the nineteenth century had kinematic and physical motivations. An archetypical example of this progressive synthesis can be seen in the history of group theory, there were several threads in the early development of group theory, in modern language loosely corresponding to number theory, theory of equations, and geometry. Leonhard Euler considered algebraic operations on numbers modulo an integer, modular arithmetic, lagranges goal was to understand why equations of third and fourth degree admit formulae for solutions, and he identified as key objects permutations of the roots. An important novel step taken by Lagrange in this paper was the view of the roots, i. e. as symbols. However, he did not consider composition of permutations, serendipitously, the first edition of Edward Warings Meditationes Algebraicae appeared in the same year, with an expanded version published in 1782. Waring proved the theorem on symmetric functions, and specially considered the relation between the roots of a quartic equation and its resolvent cubic. Kronecker claimed in 1888 that the study of modern algebra began with this first paper of Vandermonde, cauchy states quite clearly that Vandermonde had priority over Lagrange for this remarkable idea, which eventually led to the study of group theory. Paolo Ruffini was the first person to develop the theory of permutation groups and his goal was to establish the impossibility of an algebraic solution to a general algebraic equation of degree greater than four
Abstract algebra
–
The permutations of
Rubik's Cube form a group, a fundamental concept within abstract algebra.
17.
Primary education
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Primary education or elementary education is typically the first stage of compulsory education, coming between early childhood education and secondary education. Primary education usually takes place in a school or elementary school. In some countries, primary education is followed by school, an educational stage which exists in some countries. In order to achieve the goal by 2015, the United Nations estimated that all children at the entry age for primary school would have had to have been attending classes by 2009. This would depend on the duration of the level, as well as how well the schools retain students until the end of the cycle. As of 2010, the number of new teachers needed in sub-Saharan Africa alone, however, the gender gap for children not in education had also been narrowed. Between 1999 and 2008, the number of not in education worldwide had decreased from 57 percent to 53 percent, however it should also be noted that in some regions. According to the United Nations, there are things in the regions that have already been accomplished. The country doubled its enrollment ratio over the same period, other regions in Latin America such as Guatemala and Nicaragua as well as Zambia in Southern Africa broke through the 90 percent towards greater access to primary education. In Australia, students undertake preschool then 13 years of schooling before moving to vocational or higher education, Primary schooling for most children starts after they turn 5 years old. In most states, children can be enrolled earlier at the discretion of individual school principals on the basis of intellectual giftedness, in Victoria, New South Wales, Northern Territory, ACT and Tasmania students then move through Kindergarten/Preparatory School/Reception and Years 1 to 6 before starting high school. Pre-School/Kindergarten,4 to 5 years old Prep, currently, at the age of 6 children attend from the grade 1 to 4 what is called Ensino Primário, and afterwards from grade 5 to 9 the Ensino Fundamental. At the age of 15 the teenagers go to Ensino Médio, which is equivalent High School in other countries, Primary school is mandatory and consists in nine years called Ensino Fundamental, separated in Ensino Fundamental I and Ensino Fundamental II. Primary school is followed by the three years called Ensino Médio. 1st grade, 15- to 16-year-olds, 2nd grade, 16- to 17-year-olds, 3rd grade, in Canada, primary school usually begins at ages three or four, starting with either Kindergarten or Grade 1 and lasts until age 13 or 14. Many places in Canada have a split between primary and elementary schools, in Nova Scotia elementary school is the most common term. The provincial government of Nova Scotia uses the term Primary instead of Kindergarten, most children are pupils in the Danish Folkeskolen, which has the current grades, Kindergarten, 3–6 years https, //meta. wikimedia. The first three grades of school are called Algkool which can be translated as beginning school and can be confused with primary school
Primary education
–
A large elementary school in
Magome,
Japan
Primary education
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Children and teacher in a primary school classroom in
Laos
Primary education
–
An elementary school in California, United States
Primary education
–
A poster at the
United Nations Headquarters in
New York City,
New York,
USA, showing the Millennium Development Goals.
18.
Decimal
–
This article aims to be an accessible introduction. For the mathematical definition, see Decimal representation, the decimal numeral system has ten as its base, which, in decimal, is written 10, as is the base in every positional numeral system. It is the base most widely used by modern civilizations. Decimal fractions have terminating decimal representations and other fractions have repeating decimal representations, Decimal notation is the writing of numbers in a base-ten numeral system. Examples are Brahmi numerals, Greek numerals, Hebrew numerals, Roman numerals, Roman numerals have symbols for the decimal powers and secondary symbols for half these values. Brahmi numerals have symbols for the nine numbers 1–9, the nine decades 10–90, plus a symbol for 100, Chinese numerals have symbols for 1–9, and additional symbols for powers of ten, which in modern usage reach 1072. Positional decimal systems include a zero and use symbols for the ten values to represent any number, positional notation uses positions for each power of ten, units, tens, hundreds, thousands, etc. The position of each digit within a number denotes the multiplier multiplied with that position has a value ten times that of the position to its right. There were at least two independent sources of positional decimal systems in ancient civilization, the Chinese counting rod system. Ten is the number which is the count of fingers and thumbs on both hands, the English word digit as well as its translation in many languages is also the anatomical term for fingers and toes. In English, decimal means tenth, decimate means reduce by a tenth, however, the symbols used in different areas are not identical, for instance, Western Arabic numerals differ from the forms used by other Arab cultures. A decimal fraction is a fraction the denominator of which is a power of ten. g, Decimal fractions 8/10, 1489/100, 24/100000, and 58900/10000 are expressed in decimal notation as 0.8,14.89,0.00024,5.8900 respectively. In English-speaking, some Latin American and many Asian countries, a period or raised period is used as the separator, in many other countries, particularly in Europe. The integer part, or integral part of a number is the part to the left of the decimal separator. The part from the separator to the right is the fractional part. It is usual for a number that consists only of a fractional part to have a leading zero in its notation. Any rational number with a denominator whose only prime factors are 2 and/or 5 may be expressed as a decimal fraction and has a finite decimal expansion. 1/2 =0.5 1/20 =0.05 1/5 =0.2 1/50 =0.02 1/4 =0.25 1/40 =0.025 1/25 =0.04 1/8 =0.125 1/125 =0.008 1/10 =0
Decimal
–
The world's earliest decimal multiplication table was made from bamboo slips, dating from 305 BC, during the
Warring States period in China.
Decimal
–
Numeral systems
Decimal
–
Ten fingers on two hands, the possible starting point of the decimal counting.
Decimal
–
Diagram of the world's earliest decimal multiplication table (c. 305 BC) from the Warring States period
19.
Plus and minus signs
–
The plus and minus signs are mathematical symbols used to represent the notions of positive and negative as well as the operations of addition and subtraction. Their use has extended to many other meanings, more or less analogous. Plus and minus are Latin terms meaning more and less, respectively, though the signs now seem as familiar as the alphabet or the Hindu-Arabic numerals, they are not of great antiquity. In Europe in the early 15th century the letters P and M were generally used, the symbols appeared for the first time in Luca Pacioli’s mathematics compendium, Summa de arithmetica, geometria, proportioni et proportionalità, first printed and published in Venice in 1494. The + is a simplification of the Latin et, the − may be derived from a tilde written over m when used to indicate subtraction, or it may come from a shorthand version of the letter m itself. In his 1489 treatise Johannes Widmann referred to the symbols − and + as minus and mer, was − ist, das ist minus, und das + ist das mer. They werent used for addition and subtraction here, but to indicate surplus and deficit, the plus sign is a binary operator that indicates addition, as in 2 +3 =5. It can also serve as an operator that leaves its operand unchanged. This notation may be used when it is desired to emphasize the positiveness of a number, the plus sign can also indicate many other operations, depending on the mathematical system under consideration. Many algebraic structures have some operation which is called, or is equivalent to and it is conventional to use the plus sign to only denote commutative operations. Subtraction is the inverse of addition, directly in front of a number and when it is not a subtraction operator it means a negative number. For instance −5 is negative 5, a unary operator that acts as an instruction to replace the operand by its additive inverse. For example, if x is 3, then −x is −3, similarly, − is equal to 2. The above is a case of this. All three uses can be referred to as minus in everyday speech, further, some textbooks in the United States encourage −x to be read as the opposite of x or the additive inverse of x to avoid giving the impression that −x is necessarily negative. However, in programming languages and Microsoft Excel in particular, unary operators bind strongest, so in those cases −5^2 is 25. Some elementary teachers use raised plus and minus signs before numbers to show they are positive or negative numbers. For example, subtracting −5 from 3 might be read as positive three take away negative 5 and be shown as 3 − −5 becomes 3 +5 =8, in grading systems, the plus sign indicates a grade one level higher and the minus sign a grade lower
Plus and minus signs
–
Plus, minus, and hyphen-minus.
20.
Infix notation
–
Infix notation is the notation commonly used in arithmetical and logical formulae and statements. It is characterized by the placement of operators between operands – infixed operators – such as the sign in 2 +2. Infix notation is more difficult to parse by computers than prefix notation or postfix notation, however many programming languages use it due to its familiarity. It is more used in arithmetic, e. g. 2+2, in the absence of parentheses, certain precedence rules determine the order of operations. Infix notation may also be distinguished from function notation, where the name of a function suggests a particular operation, an example of such a function notation would be S in which the function S denotes addition, S = 1+3 =4. Tree traversal, Infix is also a tree traversal order and it is described in a more detailed manner on this page. A brief analysis of Reverse Polish Notation against Direct Algebraic Logic Infix to postfix convertor
Infix notation
–
Prefix notation
("Polish")
21.
Equals sign
–
The equals sign or equality sign is a mathematical symbol used to indicate equality. It was invented in 1557 by Robert Recorde, in an equation, the equals sign is placed between two expressions that have the same value. In Unicode and ASCII, it is U+003D = equals sign, the etymology of the word equal is from the Latin word æqualis as meaning uniform, identical, or equal, from aequus. The = symbol that is now accepted in mathematics for equality was first recorded by Welsh mathematician Robert Recorde in The Whetstone of Witte. The original form of the symbol was much wider than the present form.2. Thynges, can be moare equalle. … to avoid the repetition of these words, is equal to, I will set a pair of parallels, or Gemowe lines, of one length. According to Scotlands University of St Andrews History of Mathematics website, the symbol || was used by some and æ, from the Latin word aequalis meaning equal, was widely used into the 1700s. In mathematics, the sign can be used as a simple statement of fact in a specific case, or to create definitions, conditional statements. The first important computer programming language to use the sign was the original version of Fortran, FORTRAN I, designed in 1954. In Fortran, = serves as an assignment operator, X =2 sets the value of X to 2. This somewhat resembles the use of = in a definition, but with different semantics. For example, the assignment X = X +2 increases the value of X by 2, a rival programming-language usage was pioneered by the original version of ALGOL, which was designed in 1958 and implemented in 1960. ALGOL included a relational operator that tested for equality, allowing constructions like if x =2 with essentially the same meaning of = as the usage in mathematics. The equals sign was reserved for this usage, both usages have remained common in different programming languages into the early 21st century. As well as Fortran, = is used for assignment in such languages as C, Perl, Python, awk, but = is used for equality and not assignment in the Pascal family, Ada, Eiffel, APL, and other languages. A few languages, such as BASIC and PL/I, have used the sign to mean both assignment and equality, distinguished by context. However, in most languages where = has one of these meanings, a different character or, more often, following ALGOL, most languages that use = for equality use, = for assignment, although APL, with its special character set, uses a left-pointing arrow. Fortran did not have an equality operator until FORTRAN IV was released in 1962, the language B introduced the use of == with this meaning, which has been copied by its descendant C and most later languages where = means assignment
Equals sign
–
Recorde's introduction of "="
22.
Product (mathematics)
–
In mathematics, a product is the result of multiplying, or an expression that identifies factors to be multiplied. Thus, for instance,6 is the product of 2 and 3, the order in which real or complex numbers are multiplied has no bearing on the product, this is known as the commutative law of multiplication. When matrices or members of various other associative algebras are multiplied, matrix multiplication, for example, and multiplication in other algebras is in general non-commutative. There are many different kinds of products in mathematics, besides being able to multiply just numbers, polynomials or matricies, an overview of these different kinds of products is given here. Placing several stones into a pattern with r rows and s columns gives r ⋅ s = ∑ i =1 s r = ∑ j =1 r s stones. Integers allow positive and negative numbers, the product of two quaternions can be found in the article on quaternions. However, it is interesting to note that in this case, the product operator for the product of a sequence is denoted by the capital Greek letter Pi ∏. The product of a sequence consisting of one number is just that number itself. The product of no factors at all is known as the empty product, commutative rings have a product operation. Under the Fourier transform, convolution becomes point-wise function multiplication, others have very different names but convey essentially the same idea. A brief overview of these is given here, by the very definition of a vector space, one can form the product of any scalar with any vector, giving a map R × V → V. A scalar product is a map, ⋅, V × V → R with the following conditions. From the scalar product, one can define a norm by letting ∥ v ∥, = v ⋅ v, now we consider the composition of two linear mappings between finite dimensional vector spaces. Let the linear mapping f map V to W, and let the linear mapping g map W to U, then one can get g ∘ f = g = g j k f i j v i b U k. Or in matrix form, g ∘ f = G F v, in which the i-row, j-column element of F, denoted by Fij, is fji, the composition of more than two linear mappings can be similarly represented by a chain of matrix multiplication. To see this, let r = dim, s = dim, let U = be a basis of U, V = be a basis of V and W = be a basis of W. Then B ⋅ A = M W U ∈ R s × t is the matrix representing g ∘ f, U → W, in other words, the matrix product is the description in coordinates of the composition of linear functions. For inifinite-dimensional vector spaces, one also has the, Tensor product of Hilbert spaces Topological tensor product, the tensor product, outer product and Kronecker product all convey the same general idea
Product (mathematics)
–
3 by 4 is 12
23.
Division (mathematics)
–
Division is one of the four basic operations of arithmetic, the others being addition, subtraction, and multiplication. The division of two numbers is the process of calculating the number of times one number is contained within one another. For example, in the picture on the right, the 20 apples are divided into groups of five apples, Division can also be thought of as the process of evaluating a fraction, and fractional notation is commonly used to represent division. Division is the inverse of multiplication, if a × b = c, then a = c ÷ b, as long as b is not zero. Division by zero is undefined for the numbers and most other contexts, because if b =0, then a cannot be deduced from b and c. In some contexts, division by zero can be defined although to a limited extent, in division, the dividend is divided by the divisor to get a quotient. In the above example,20 is the dividend, five is the divisor, in some cases, the divisor may not be contained fully by the dividend, for example,10 ÷3 leaves a remainder of one, as 10 is not a multiple of three. Sometimes this remainder is added to the quotient as a fractional part, but in the context of integer division, where numbers have no fractional part, the remainder is kept separately or discarded. Besides dividing apples, division can be applied to other physical, Division has been defined in several contexts, such as for the real and complex numbers and for more abstract contexts such as for vector spaces and fields. Division is the most mentally difficult of the four operations of arithmetic. Teaching the objective concept of dividing integers introduces students to the arithmetic of fractions, unlike addition, subtraction, and multiplication, the set of all integers is not closed under division. Dividing two integers may result in a remainder, to complete the division of the remainder, the number system is extended to include fractions or rational numbers as they are more generally called. When students advance to algebra, the theory of division intuited from arithmetic naturally extends to algebraic division of variables, polynomials. Division is often shown in algebra and science by placing the dividend over the divisor with a line, also called a fraction bar. For example, a divided by b is written a b This can be read out loud as a divided by b, a fraction is a division expression where both dividend and divisor are integers, and there is no implication that the division must be evaluated further. A second way to show division is to use the obelus, common in arithmetic, in this manner, ISO 80000-2-9.6 states it should not be used. The obelus is also used alone to represent the operation itself. In some non-English-speaking cultures, a divided by b is written a, b and this notation was introduced in 1631 by William Oughtred in his Clavis Mathematicae and later popularized by Gottfried Wilhelm Leibniz
Division (mathematics)
–
This article is about the arithmetical operation. For other uses, see
Division (disambiguation).
24.
Modulo operation
–
In computing, the modulo operation finds the remainder after division of one number by another. Given two positive numbers, a and n, a n is the remainder of the Euclidean division of a by n. Although typically performed with a and n both being integers, many computing systems allow other types of numeric operands, the range of numbers for an integer modulo of n is 0 to n −1. See modular arithmetic for an older and related convention applied in number theory, when either a or n is negative, the naive definition breaks down and programming languages differ in how these values are defined. In mathematics, the result of the operation is the remainder of the Euclidean division. Computers and calculators have various ways of storing and representing numbers, usually, in number theory, the positive remainder is always chosen, but programming languages choose depending on the language and the signs of a or n. Standard Pascal and ALGOL68 give a positive remainder even for negative divisors, a modulo 0 is undefined in most systems, although some do define it as a. Despite its widespread use, truncated division is shown to be inferior to the other definitions, when the result of a modulo operation has the sign of the dividend, it can lead to surprising mistakes. For special cases, on some hardware, faster alternatives exist, optimizing compilers may recognize expressions of the form expression % constant where constant is a power of two and automatically implement them as expression &. This can allow writing clearer code without compromising performance and this optimization is not possible for languages in which the result of the modulo operation has the sign of the dividend, unless the dividend is of an unsigned integer type. This is because, if the dividend is negative, the modulo will be negative, some modulo operations can be factored or expanded similar to other mathematical operations. This may be useful in cryptography proofs, such as the Diffie–Hellman key exchange, identity, mod n = a mod n. nx mod n =0 for all positive integer values of x. If p is a number which is not a divisor of b, then abp−1 mod p = a mod p. B−1 mod n denotes the multiplicative inverse, which is defined if and only if b and n are relatively prime. Distributive, mod n = mod n. ab mod n = mod n, division, a/b mod n = mod n, when the right hand side is defined. Inverse multiplication, mod n = a mod n, modulo and modulo – many uses of the word modulo, all of which grew out of Carl F. Gausss introduction of modular arithmetic in 1801. Modular exponentiation ^ Perl usually uses arithmetic modulo operator that is machine-independent, for examples and exceptions, see the Perl documentation on multiplicative operators. ^ Mathematically, these two choices are but two of the number of choices available for the inequality satisfied by a remainder
Modulo operation
–
Quotient (red) and remainder (green) functions using different algorithms
25.
Exponentiation
–
Exponentiation is a mathematical operation, written as bn, involving two numbers, the base b and the exponent n. The exponent is usually shown as a superscript to the right of the base, Some common exponents have their own names, the exponent 2 is called the square of b or b squared, the exponent 3 is called the cube of b or b cubed. The exponent −1 of b, or 1 / b, is called the reciprocal of b, when n is a positive integer and b is not zero, b−n is naturally defined as 1/bn, preserving the property bn × bm = bn + m. The definition of exponentiation can be extended to any real or complex exponent. Exponentiation by integer exponents can also be defined for a variety of algebraic structures. The term power was used by the Greek mathematician Euclid for the square of a line, archimedes discovered and proved the law of exponents, 10a 10b = 10a+b, necessary to manipulate powers of 10. In the late 16th century, Jost Bürgi used Roman numerals for exponents, early in the 17th century, the first form of our modern exponential notation was introduced by Rene Descartes in his text titled La Géométrie, there, the notation is introduced in Book I. Nicolas Chuquet used a form of notation in the 15th century. The word exponent was coined in 1544 by Michael Stifel, samuel Jeake introduced the term indices in 1696. In the 16th century Robert Recorde used the square, cube, zenzizenzic, sursolid, zenzicube, second sursolid. Biquadrate has been used to refer to the power as well. Some mathematicians used exponents only for greater than two, preferring to represent squares as repeated multiplication. Thus they would write polynomials, for example, as ax + bxx + cx3 + d, another historical synonym, involution, is now rare and should not be confused with its more common meaning. In 1748 Leonhard Euler wrote consider exponentials or powers in which the exponent itself is a variable and it is clear that quantities of this kind are not algebraic functions, since in those the exponents must be constant. With this introduction of transcendental functions, Euler laid the foundation for the introduction of natural logarithm as the inverse function for y = ex. The expression b2 = b ⋅ b is called the square of b because the area of a square with side-length b is b2, the expression b3 = b ⋅ b ⋅ b is called the cube of b because the volume of a cube with side-length b is b3. The exponent indicates how many copies of the base are multiplied together, for example,35 =3 ⋅3 ⋅3 ⋅3 ⋅3 =243. The base 3 appears 5 times in the multiplication, because the exponent is 5
Exponentiation
–
Graphs of y = b x for various bases b: base 10 (green), base e (red), base 2 (blue), and base 1 / 2 (cyan). Each curve passes through the point (0, 1) because any nonzero number raised to the power of 0 is 1. At x = 1, the value of y equals the base because any number raised to the power of 1 is the number itself.
26.
Nth root
–
A root of degree 2 is called a square root and a root of degree 3, a cube root. Roots of higher degree are referred by using numbers, as in fourth root, twentieth root. For example,2 is a root of 4, since 22 =4. −2 is also a root of 4, since 2 =4. A real number or complex number has n roots of degree n. While the roots of 0 are not distinct, the n nth roots of any real or complex number are all distinct. If n is odd and x is real, one nth root is real and has the sign as x. Finally, if x is not real, then none of its nth roots is real. Roots are usually using the radical symbol or radix or √, with x or √ x denoting the square root, x 3 denoting the cube root, x 4 denoting the fourth root. In the expression x n, n is called the index, is the sign or radix. For example, −8 has three roots, −2,1 + i √3 and 1 − i √3. Out of these,1 + i √3 has the least argument,4 has two square roots,2 and −2, having arguments 0 and π respectively. So 2 is considered the root on account of having the lesser argument. An unresolved root, especially one using the symbol, is often referred to as a surd or a radical. Nth roots can also be defined for complex numbers, and the roots of 1 play an important role in higher mathematics. The origin of the root symbol √ is largely speculative, some sources imply that the symbol was first used by Arab mathematicians. One of those mathematicians was Abū al-Hasan ibn Alī al-Qalasādī, legend has it that it was taken from the Arabic letter ج, which is the first letter in the Arabic word جذر. However, many scholars, including Leonhard Euler, believe it originates from the letter r, the symbol was first seen in print without the vinculum in the year 1525 in Die Coss by Christoff Rudolff, a German mathematician
Nth root
–
Roots of integer numbers from 0 to 10. Line labels = x. x -axis = n. y -axis = n th root of x.
27.
Logarithm
–
In mathematics, the logarithm is the inverse operation to exponentiation. That means the logarithm of a number is the exponent to which another fixed number, in simple cases the logarithm counts factors in multiplication. For example, the base 10 logarithm of 1000 is 3, the logarithm of x to base b, denoted logb, is the unique real number y such that by = x. For example, log2 =6, as 64 =26, the logarithm to base 10 is called the common logarithm and has many applications in science and engineering. The natural logarithm has the e as its base, its use is widespread in mathematics and physics. The binary logarithm uses base 2 and is used in computer science. Logarithms were introduced by John Napier in the early 17th century as a means to simplify calculations and they were rapidly adopted by navigators, scientists, engineers, and others to perform computations more easily, using slide rules and logarithm tables. The present-day notion of logarithms comes from Leonhard Euler, who connected them to the function in the 18th century. Logarithmic scales reduce wide-ranging quantities to tiny scopes, for example, the decibel is a unit quantifying signal power log-ratios and amplitude log-ratios. In chemistry, pH is a measure for the acidity of an aqueous solution. Logarithms are commonplace in scientific formulae, and in measurements of the complexity of algorithms and they describe musical intervals, appear in formulas counting prime numbers, inform some models in psychophysics, and can aid in forensic accounting. In the same way as the logarithm reverses exponentiation, the logarithm is the inverse function of the exponential function applied to complex numbers. The discrete logarithm is another variant, it has uses in public-key cryptography, the idea of logarithms is to reverse the operation of exponentiation, that is, raising a number to a power. For example, the power of 2 is 8, because 8 is the product of three factors of 2,23 =2 ×2 ×2 =8. It follows that the logarithm of 8 with respect to base 2 is 3, the third power of some number b is the product of three factors equal to b. More generally, raising b to the power, where n is a natural number, is done by multiplying n factors equal to b. The n-th power of b is written bn, so that b n = b × b × ⋯ × b ⏟ n factors, exponentiation may be extended to by, where b is a positive number and the exponent y is any real number. For example, b−1 is the reciprocal of b, that is, the logarithm of a positive real number x with respect to base b, a positive real number not equal to 1, is the exponent by which b must be raised to yield x
Logarithm
–
John Napier (1550–1617), the inventor of logarithms
Logarithm
–
The
graph of the logarithm to base 2 crosses the
x axis (horizontal axis) at 1 and passes through the points with
coordinates (2, 1), (4, 2), and (8, 3). For example, log 2 (8) = 3, because 2 3 = 8. The graph gets arbitrarily close to the y axis, but
does not meet or intersect it.
Logarithm
–
The logarithm keys (lo g for base-10 and ln for base- e) on a typical scientific calculator
Logarithm
–
A
nautilus displaying a logarithmic spiral
28.
Latin
–
Latin is a classical language belonging to the Italic branch of the Indo-European languages. The Latin alphabet is derived from the Etruscan and Greek alphabets, Latin was originally spoken in Latium, in the Italian Peninsula. Through the power of the Roman Republic, it became the dominant language, Vulgar Latin developed into the Romance languages, such as Italian, Portuguese, Spanish, French, and Romanian. Latin, Italian and French have contributed many words to the English language, Latin and Ancient Greek roots are used in theology, biology, and medicine. By the late Roman Republic, Old Latin had been standardised into Classical Latin, Vulgar Latin was the colloquial form spoken during the same time and attested in inscriptions and the works of comic playwrights like Plautus and Terence. Late Latin is the language from the 3rd century. Later, Early Modern Latin and Modern Latin evolved, Latin was used as the language of international communication, scholarship, and science until well into the 18th century, when it began to be supplanted by vernaculars. Ecclesiastical Latin remains the language of the Holy See and the Roman Rite of the Catholic Church. Today, many students, scholars and members of the Catholic clergy speak Latin fluently and it is taught in primary, secondary and postsecondary educational institutions around the world. The language has been passed down through various forms, some inscriptions have been published in an internationally agreed, monumental, multivolume series, the Corpus Inscriptionum Latinarum. Authors and publishers vary, but the format is about the same, volumes detailing inscriptions with a critical apparatus stating the provenance, the reading and interpretation of these inscriptions is the subject matter of the field of epigraphy. The works of several hundred ancient authors who wrote in Latin have survived in whole or in part and they are in part the subject matter of the field of classics. The Cat in the Hat, and a book of fairy tales, additional resources include phrasebooks and resources for rendering everyday phrases and concepts into Latin, such as Meissners Latin Phrasebook. The Latin influence in English has been significant at all stages of its insular development. From the 16th to the 18th centuries, English writers cobbled together huge numbers of new words from Latin and Greek words, dubbed inkhorn terms, as if they had spilled from a pot of ink. Many of these words were used once by the author and then forgotten, many of the most common polysyllabic English words are of Latin origin through the medium of Old French. Romance words make respectively 59%, 20% and 14% of English, German and those figures can rise dramatically when only non-compound and non-derived words are included. Accordingly, Romance words make roughly 35% of the vocabulary of Dutch, Roman engineering had the same effect on scientific terminology as a whole
Latin
–
Latin inscription, in the
Colosseum
Latin
–
Julius Caesar 's
Commentarii de Bello Gallico is one of the most famous classical Latin texts of the Golden Age of Latin. The unvarnished, journalistic style of this
patrician general has long been taught as a model of the urbane Latin officially spoken and written in the floruit of the
Roman republic.
Latin
–
A multi-volume Latin dictionary in the
University Library of Graz
Latin
–
Latin and Ancient Greek Language - Culture - Linguistics at
Duke University in 2014.
29.
English language
–
English /ˈɪŋɡlɪʃ/ is a West Germanic language that was first spoken in early medieval England and is now the global lingua franca. Named after the Angles, one of the Germanic tribes that migrated to England, English is either the official language or one of the official languages in almost 60 sovereign states. It is the third most common language in the world, after Mandarin. It is the most widely learned second language and a language of the United Nations, of the European Union. It is the most widely spoken Germanic language, accounting for at least 70% of speakers of this Indo-European branch, English has developed over the course of more than 1,400 years. The earliest forms of English, a set of Anglo-Frisian dialects brought to Great Britain by Anglo-Saxon settlers in the century, are called Old English. Middle English began in the late 11th century with the Norman conquest of England, Early Modern English began in the late 15th century with the introduction of the printing press to London and the King James Bible, and the start of the Great Vowel Shift. Through the worldwide influence of the British Empire, modern English spread around the world from the 17th to mid-20th centuries, English is an Indo-European language, and belongs to the West Germanic group of the Germanic languages. Most closely related to English are the Frisian languages, and English, Old Saxon and its descendent Low German languages are also closely related, and sometimes Low German, English, and Frisian are grouped together as the Ingvaeonic or North Sea Germanic languages. Modern English descends from Middle English, which in turn descends from Old English, particular dialects of Old and Middle English also developed into a number of other English languages, including Scots and the extinct Fingallian and Forth and Bargy dialects of Ireland. English is classified as a Germanic language because it shares new language features with other Germanic languages such as Dutch, German and these shared innovations show that the languages have descended from a single common ancestor, which linguists call Proto-Germanic. Through Grimms law, the word for foot begins with /f/ in Germanic languages, English is classified as an Anglo-Frisian language because Frisian and English share other features, such as the palatalisation of consonants that were velar consonants in Proto-Germanic. The earliest form of English is called Old English or Anglo-Saxon, in the fifth century, the Anglo-Saxons settled Britain and the Romans withdrew from Britain. England and English are named after the Angles, Old English was divided into four dialects, the Anglian dialects, Mercian and Northumbrian, and the Saxon dialects, Kentish and West Saxon. Through the educational reforms of King Alfred in the century and the influence of the kingdom of Wessex. The epic poem Beowulf is written in West Saxon, and the earliest English poem, Modern English developed mainly from Mercian, but the Scots language developed from Northumbrian. A few short inscriptions from the period of Old English were written using a runic script. By the sixth century, a Latin alphabet was adopted, written with half-uncial letterforms and it included the runic letters wynn ⟨ƿ⟩ and thorn ⟨þ⟩, and the modified Latin letters eth ⟨ð⟩, and ash ⟨æ⟩
English language
–
The opening to the Old English epic poem
Beowulf,
handwritten in
half-uncial script: Hƿæt ƿē Gārde/na ingēar dagum þēod cyninga / þrym ge frunon... "Listen! We of the Spear-Danes from days of yore have heard of the glory of the folk-kings..."
English language
–
Countries of the world where English is a majority native language
English language
–
Title page of Geoffrey Chaucer's
Canterbury Tales c.1400
30.
Verb
–
A verb, from the Latin verbum meaning word, is a word that in syntax conveys an action, an occurrence, or a state of being. In the usual description of English, the form, with or without the particle to, is the infinitive. In many languages, verbs are inflected to encode tense, aspect, mood, a verb may also agree with the person, gender or number of some of its arguments, such as its subject, or object. Verbs have tenses, present, to indicate that an action is being carried out, past, to indicate that an action has been done, future, in languages where the verb is inflected, it often agrees with its primary argument in person, number or gender. With the exception of the verb to be, English shows distinctive agreements only in the person singular, present tense form of verbs. The rest of the persons are not distinguished in the verb, Latin and the Romance languages inflect verbs for tense–aspect–mood, and they agree in person and number with the subject. Verbs vary by type, and each type is determined by the kinds of words that accompany it, classified by the number of their valency arguments, usually three basic types are distinguished, intransitives, transitives, ditransitives and double transitive verbs. In addition, verbs can be nonfinite, namely, not inflected for tense, an intransitive verb is one that does not have a direct object. Intransitive verbs may be followed by an adverb or end a sentence, for example, The woman spoke softly. The athlete ran faster than the official, a transitive verb is followed by a noun or noun phrase. These noun phrases are not called predicate nouns, but are called direct objects because they refer to the object that is being acted upon. For example, My friend read the newspaper, the teenager earned a speeding ticket. A way to identify a verb is to invert the sentence. For example, The newspaper was read by my friend, a speeding ticket was earned by the teenager. Ditransitive verbs precede either two noun phrases or a phrase and then a prepositional phrase often led by to or for. For example, The players gave their teammates high fives, the players gave high fives to their teammates. When two noun phrases follow a transitive verb, the first is an object, that which is receiving something, and the second is a direct object. Indirect objects can be noun phrases or prepositional phrases, double transitive verbs are followed by a noun phrase that serves as a direct object and then a second noun phrase, adjective, or infinitive phrase
Verb
–
A single-word verb in
Spanish contains information about time (past, present, future), person and number. The process of grammatically modifying a verb to express this information is called
conjugation.