# Algebra Sentence Examples

The earliest

**algebra**consists in the solution of equations.But when I took up

**algebra**I had a harder time still.In

**algebra**he discovered the method of approximating to the real roots of an equation by means of continued fractions, and imagined a general process of solving algebraical equations of every degree.The theories of determinants and of symmetric functions and of the

**algebra**of differential operations have an important bearing upon this comparatively new branch of mathematics.But, when I took up

**Algebra**, I had a harder time still--I was terribly handicapped by my imperfect knowledge of the notation.When

**algebra**had advanced to the point where exponents were introduced, nothing would be more natural than that their utility as a means of performing multiplications and divisions should be remarked; but it is one of the surprises in the history of science that logarithms were invented as an arithmetical improvement years before their connexion with exponents was known.The medieval Arabians invented our system of numeration and developed

**algebra**.His standard work on

**algebra**, written in Arabic, and other treatises of a similar character raised him at once to the foremost rank among the mathematicians of that age, and induced Sultgn Malik-Shgh to summon him in A.H.From February to July, 1898, Mr. Keith came out to Wrentham twice a week, and taught me

**algebra**, geometry, Greek and Latin.His largest work,Trattato generale di numeri e misure, is a comprehensive mathematical treatise, including arithmetic, geometry, mensuration, and

**algebra**as far as quadratic equations (Venice, 1556, 1560).AdvertisementWe will confine ourselves here to algebraic complex numbers - that is, to complex numbers of the second order taken in connexion with that definition of multiplication which leads to ordinary

**algebra**.When 0=4 it is clear that no form, whose partition contains a part 3, can be reduced; but every form, whose partition is composed of the parts 4 and 2, is by elementary

**algebra**reducible by means of perpetuants of degree 2.He appears to have attended Dirichlet's lectures on theory of numbers, theory of definite integrals, and partial differential equations, and Jacobi's on analytical mechanics and higher

**algebra**.The above definition gives only a partial view of the scope of

**algebra**.The principal step in the modern development of

**algebra**was the recognition of the meaning of negative quantities.AdvertisementThe main work of Descartes, so far as

**algebra**was concerned, was the establishment of a relation between arithmetical and geometrical measurement.These attempts at the unification of

**algebra**, and its separation from other branches of mathematics, have usually been accompanied by an attempt to base it, as a deductive science, on certain fundamental laws or general rules; and this has tended to increase its difficulty.One of the most recent developments of

**algebra**is the algebraic theory of number, which is devised with the view of removing these difficulties.Two other developments of

**algebra**are of special importance.The theory of sequences and series is sometimes treated as a part of elementary

**algebra**; but it is more convenient to regard the simpler cases as isolated examples, leading up to the general theory.AdvertisementOne of the most difficult questions for the teacher of

**algebra**is the stage at which, and the extent to which, the ideas of a negative number and of continuity may be introduced.On the one hand, the modern developments of

**algebra**began with these ideas, and particularly with the idea of a negative number.In

**algebra**, if a be a real positive quantity and w a root of unity, then a is the amplitude of the product aw.The braille worked well enough in the languages, but when it came to geometry and

**algebra**, difficulties arose.But as yet he had only glimpses of a logical method which should invigorate the syllogism by the co-operation of ancient geometry and modern

**algebra**.AdvertisementAt the age of eight he began Latin, Euclid, and

**algebra**, and was appointed schoolmaster to the younger children of the family.In the present article, therefore, the main portions of elementary

**algebra**are treated in one section, without reference to these ideas, which are considered generally in two separate sections.As to the teaching of

**algebra**, see references under Arithmetic to works on the teaching of elementary mathematics.A special

**algebra**is one which differs from ordinary**algebra**in the laws of equivalence which its symbols obey.Ordinary

**algebra**developed very gradually as a kind of shorthand, devised to abbreviate the discussion of arithmetical problems and the statement of arithmetical facts.One of the most important questions in investigating a linear

**algebra**is to decide the necessary relations between a and b in order that this product may be zero.But when an

**algebra**is used with a particular interpretation, or even in the course of its formal development, it frequently happens that new symbols of operation are, so to speak, superposed upon the**algebra**, and are found to obey certain formal laws of combination of their own.In most cases these subsidiary algebras, as they may be called, are inseparable from the applications in which they are used; but in any attempt at a natural classification of

**algebra**(at present a hopeless task), they would have to be taken into account.Even in ordinary

**algebra**the notation for powers and roots disturbs the symmetry of the rational theory; and when a schoolboy illegitimately extends the distributive law by writing -V (a+b)a+J b, he is unconsciously emphasizing this want of complete harmony.Since, however, Geber happened to be the name of a celebrated Moorish philosopher who flourished in about the iith or 12th century, it has been supposed that he was the founder of

**algebra**, which has since perpetuated his name.There was a certain learned mathematician who sent his

**algebra**, written in the Syriac language, to Alexander the Great, and he named it almucabala, that is, the book of dark or mysterious things, which others would rather call the doctrine of**algebra**.Robert Recorde in his Whetstone of Witte (1557) uses the variant algeber, while John Dee (1527-1608) affirms that algiebar, and not

**algebra**, is the correct form, and appeals to the authority of the Arabian Avicenna.Although the term "

**algebra**" is now in universal use, various other appellations were used by the Italian mathematicians during the Renaissance.The name l'arte magiore, the greater art, is designed to distinguish it from l'arte minore, the lesser art, a term which he applied to the modern arithmetic. His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms toss or

**algebra**, cossic or algebraic, cossist or algebraist, &c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square.The principle underlying this expression is probably to be found in the fact that it measured the limits of their attainments in

**algebra**, for they were unable to solve equations of a higher degree than the quadratic or square.It was formerly the custom to assign the invention of

**algebra**to the Greeks, but since the decipherment of the Rhind papyrus by Eisenlohr this view has changed, for in this work there are distinct signs of an algebraic analysis.The particular problem - a heap (hau) and its seventh makes 19 - is solved as we should now solve a simple equation; but Ahmes varies his methods in other similar problems. This discovery carries the invention of

**algebra**back to about 1700 B.C., if not earlier.It is probable that the

**algebra**of the Egyptians was of a most rudimentary nature, for otherwise we should expect to find traces of it in the works of the Greek geometers, of whom Thales of Miletus (640-546 B.C.) was the first.Notwithstanding the prolixity of writers and the number of the writings, all attempts at extracting an algebraic analysis from their geometrical theorems and problems have been fruitless, and it is generally conceded that their analysis was geometrical and had little or no affinity to

**algebra**.The first extant work which approaches to a treatise on

**algebra**is by Diophantus, an Alexandrian mathematician, who flourished about A.D.It is more than likely that he was indebted to earlier writers, whom he omits to mention, and whose works are now lost; nevertheless, but for this work, we should be led to assume that

**algebra**was almost, if not entirely, unknown to the Greeks.We find that geometry was neglected except in so far as it was of service to astronomy; trigonometry was advanced, and

**algebra**improved far beyond the attainments of Diophantus.Of other Indian writers mention may be made of Cridhara, the author of a Ganita-sara (" Quintessence of Calculation "), and Padmanabha, the author of an

**algebra**.We refer to Bhaskara Acarya, whose work the Siddhanta-ciromani (" Diadem of an Astronomical System "), written in 1150, contains two important chapters, the Lilavati (" the beautiful [science or art] ") and Viga-ganita (" root-extraction "), which are given up to arithmetic and

**algebra**.The question as to whether the Greeks borrowed their

**algebra**from the Hindus or vice versa has been the subject of much discussion.John Wallis and Lord Brounker jointly obtained a tedious solution which was published in 1658, and afterwards in 1668 by John Pell in his

**Algebra**.Although this transition from the discontinuous to continuous is not truly scientific, yet it materially augmented the development of

**algebra**, and Hankel affirms that if we define**algebra**as the application of arithmetical operations to both rational and irrational numbers or magnitudes, then the Brahmans are the real inventors of**algebra**.His treatise on

**algebra**and arithmetic (the latter part of which is only extant in the form of a Latin translation, discovered in 1857) contains nothing that was unknown to the Greeks and Hindus; it exhibits methods allied to those of both races, with the Greek element predominating.It thus came about that while some progress was made in

**algebra**, the talents of the race were bestowed on astronomy and trigonometry.Fahri des al Karhi, who flourished about the beginning of the i 1 th century, is the author of the most important Arabian work on

**algebra**.This author questioned the possibility of solving cubics by pure

**algebra**, and biquadratics by geometry.Gabir ben Aflah of Sevilla, commonly called Geber, was a celebrated astronomer and apparently skilled in

**algebra**, for it has been supposed that the word "**algebra**" is compounded from his name.In this work, which is one of the most valuable contributions to the literature of

**algebra**, Cardan shows that he was familiar with both real positive and negative roots of equations whether rational or irrational, but of imaginary roots he was quite ignorant, and he admits his inability to resolve the so-called lation of Arabic manuscripts.The first successful attempt to revive the study of

**algebra**in Christendom was due to Leonardo of Pisa, an Italian merchant trading in the Mediterranean.His travels and mercantile experience had led E t u eopre him to conclude that the Hindu methods of computing were in advance of those then in general use, and in 1202 he published his Liber Abaci, which treats of both

**algebra**and arithmetic. In this work, which is of great historical interest, since it was published about two centuries before the art of printing was discovered, he adopts the Arabic notation for numbers, and solves many problems, both arithmetical and algebraical.This was Lucas Paciolus (Lucas de Burgo), a Minorite friar, who, having previously written works on

**algebra**, arithmetic and geometry, published, in 1494, his principal work, entitled Summa de Arithmetica, Geometria, Proportioni et Proportionalita.Contemporaneously with the remarkable discoveries of the Italian mathematicians,

**algebra**was increasing in popularity in Germany, France and England.The first treatise on

**algebra**written in English was by Robert Recorde, who published his arithmetic in 1552, and his**algebra**entitled The Whetstone of Witte, which is the second part of Arithmetik, in 1557.Of other writers who published works about the end of the 16th century, we may mention Jacques Peletier, or Jacobus Peletarius (De occulta parte Numerorum, quam Algebram vocant, 1558); Petrus Ramus (Arithmeticae Libri duo et totidem Algebrae, 1560), and Christoph Clavius, who wrote on

**algebra**in 1580, though it was not published until 1608.At this time also flourished Simon Stevinus (Stevin) of Bruges, who published an arithmetic in 1585 and an

**algebra**shortly afterwards.These works exhibit great originality and mark an important epoch in the history of

**algebra**.This writer, after having published an edition of Stevin's works in 1625, published in 1629 at Amsterdam a small tract on

**algebra**which shows a considerable advance on the work of Vieta.Passing over the invention of logarithms by John Napier, and their development by Henry Briggs and others, the next author of moment was an Englishman, Thomas Harriot, whose

**algebra**(Artis analyticae praxis) was published posthumously by Walter Warner in 1631.William Oughtred, a contemporary of Harriot, published an

**algebra**, Clavis mathematicae, simultaneously with Harriot's treatise.So far the development of

**algebra**and geometry had been mutually independent, except for a few isolated applications of geometrical constructions to the solution of algebraical problems. Certain minds had long suspected the advantages which would accrue from the unrestricted application of**algebra**to geometry, but it was not until the advent of the philosopher Rene Descartes that the co-ordination was effected.Many new fields were opened up, but there was still continual progress in pure

**algebra**.In recent times many mathematicians have formulated other kinds of algebras, in which the operators do not obey the laws of ordinary

**algebra**.This study was inaugurated by George Peacock, who was one of the earliest mathematicians to recognize the symbolic character of the fundamental principles of

**algebra**.Gregory, published a paper " on the real nature of symbolical

**algebra**."The preceding summary shows the specialized 'nature which

**algebra**has assumed since the 17th century.In 1774 he published a French translation of Leonhard Euler's Elements of

**Algebra**.Vieta is wont to be called the father of modern

**algebra**.All that is wanting in his writings, especially in his Isagoge in artem analyticam (1591), in order to make them look like a modern school

**algebra**, is merely the sign of equality - a want which is the more striking because Robert Recorde had made use of our present symbol for this purpose since 1 557, and Xylander had employed vertical parallel lines since 1575.Todhunter also published keys to the problems in his textbooks on

**algebra**and trigonometry; and a biographical work, William Whewell, account of his writings and correspondence (1876), in addition to many original papers in scientific journals.In

**algebra**, the "cube" of a quantity is the quantity multiplied by itself twice, i.e.The Mathesis universalis, a more elementary work, contains copious dissertations on fundamental points of

**algebra**, arithmetic and geometry, and critical remarks.Considered as a history of

**algebra**, this work is strongly objected to by Jean Etienne Montucla on the ground of its unfairness as against the early Italian algebraists and also Franciscus Vieta and Rene Descartes and in favour of Harriot; but Augustus De Morgan, while admitting this, attributes to it considerable merit.As a mathematician he occupied himself with many branches of his favourite science, more especially with higher

**algebra**, including the theory of determinants, with the general calculus of symbols, and with the application of analysis to geometry and mechanics.Essentially, therefore, Descartes's process is that known later as the process of isoperimeters, and often attributed wholly to Schwab.2 In 16J5 appeared the Arithmetica Infinitorum of John Wallis, where numerous problems of quadrature are dealt with, the curves being now represented in Cartesian co-ordinates, and

**algebra**playing an important part.This was the first English book on

**algebra**.Now by elementary

**algebra**we know that if the number of independent equations be equal to the number of unknown quantities all the unknown quantities can be determined, and can possess each one value only.With respect to arithmetic and

**algebra**, the science of numbers, he expresses an equally definite opinion, but unfortunately it is quite impossible to state in any satisfactory fashion the grounds for it or even its full bearing.In

**algebra**it denoted the characters which represented quantities in an equation.At Baltimore he gave an enormous impetus to the study of the higher mathematics in America, and during the time he was there he contributed to the American Journal of Mathematics, of which he was the first editor, no less than thirty papers, some of great length, dealing mainly with modern

**algebra**, the theory of numbers, theory of partitions and universal**algebra**.One geometrical interpretation of the negative sign of

**algebra**was early seen to be mere reversal of direction along a line.In his Treatise of

**Algebra**(1685) he distinctly proposes to construct the imaginary roots of a quadratic equation by going out of the line on which the roots, if real, would have been constructed.From the more purely symbolical view it was developed by Peacock, De Morgan, &c., as double

**algebra**.Hamilton, like most of the many inquirers who endeavoured to give a real interpretation to the imaginary of common

**algebra**, found that at least two kinds, orders or ranks of quantities were necessary for the purpose.But, instead of dealing with points on a line, and then wandering out at right angles to it, as Buee and Argand had done, he chose to look on

**algebra**as the science of " pure time," 1 and to investigate the properties of " sets " of time-steps.An attempt has recently been made under the name of multenions to systematize this

**algebra**.Chrystal's

**Algebra**, where also may be found details of the application of continued fractions to such interesting and important problems as the recurrence of eclipses and the rectification of the calendar.A previous writer, Rafaello Bombelli, had used them in his treatise on

**Algebra**(about 1579), and it is quite possible that Cataldi may have got his ideas from him.Nicol Saunderson (1682-1739), Euler and Lambert helped in developing the theory, and much was done by Lagrange in his additions to the French edition of Euler's

**Algebra**(1795).In addition to this, he translated various other treatises, to the number, it is said, of sixty-six; among these were the Tables of "Arzakhel," or Al Zarkala of Toledo, Al Farabi On the Sciences (De scientiis), Euclid's Geometry, Al Farghani's Elements of Astronomy, and treatises on

**algebra**, arithmetic and astrology.He was much interested, too, in universal

**algebra**, non-Euclidean geometry and elliptic functions, his papers "Preliminary Sketch of Bi-quaternions" (1873) and "On the Canonical Form and Dissection of a Riemann's Surface" (1877) ranking as classics.It was in these circumstances that he dictated to his servant, a tailor's apprentice, who was absolutely devoid of mathematical knowledge, his Anleitung zur

**Algebra**(1770), a work which, though purely elementary, displays the mathematical genius of its author, and is still reckoned one of the best works of its class.With the view of stimulating mathematicians to write annotations on this admirable work, the celebrated 's Gravesande published a tract, entitled Specimen Commentarii in Arithmeticam Universalem; and Maclaurin's

**Algebra**seems to have been drawn up in consequence of this appeal.It would therefore be better, in some ways, to retain the unit throughout, and to describe - 4A as a negative quantity, in order to avoid confusion with the " negative numbers " with which operations are performed in formal

**algebra**.The notion of imaginary intersections, thus presenting itself, through

**algebra**, in geometry, must be accepted in geometry - and it in fact plays an all-important part in modern geometry.As in

**algebra**we say that an equation of the mth order has in roots, viz.Various properties of curves in general, and of cubic curves, are established in Colin Maclaurin's memoir, "De linearum geometricarum proprietatibus generalibus Tractatus " (posthumous, say 1746, published in the 6th edition of his

**Algebra**).The work on [[Trigonometry]] and Double

**Algebra**(1849) contains in the latter part a most luminous and philosophical view of existing and possible systems of symbolic calculus.But De Morgan's influence on mathematical science in England can only be estimated by a review of his long series of publications, which commence, in 1828, with a translation of part of Bourdon's Elements of

**Algebra**, prepared for his students.She received a rather desultory education, and mastered

**algebra**and Euclid in secret after she had left school, and without any extraneous help. In 1804 she married her cousin, Captain Samuel Greig, who died in 1806; and in 1812 she married another cousin, Dr William Somerville (1771-1860), inspector of the army medical board, who encouraged and greatly aided her in the study of the physical sciences.The doctrine of geometrical continuity and the application of

**algebra**to geometry, developed in the 16th and 17th centuries mainly by Kepler and Descartes, led to the discovery of many properties which gave to the notion of infinity, as a localized space conception, a predominant importance.In the language of

**algebra**putting m l, m2, m 3, &c. for the masses of the bodies, r1.2 r1.3 r2.3, &c. for their mutual distances apart; vi, v 2, v 3, &c., for the velocities with which they are moving at any moment; these quantities will continually satisfy the equation orbit, appear as arbitrary constants, introduced by the process of integration.Michael Scott, the translator of some treatises of Aristotle and of the commentaries of Averroes, Leonardo of Pisa, who introduced Arabic numerals and

**algebra**to the West, and other scholars, Jewish and Mahommedan as well as Christian, were welcome at his court.These are mainly embodied in his three great treatises, Cours d'analyse de l'Ecole Polytechnique (1821); Le Calcul infinitesimal (1823); Lecons sur les applications du calcul infinitesimal a la g'ometrie (1826-1828); and also in his courses of mechanics (for the Ecole Polytechnique), higher

**algebra**(for the Faculte des Sciences), and of mathematical physics (for the College de France).Apply the

**algebra**of finite automata to design systems and to solve simple problems on creating acceptors for particular languages.Analysis of a multimedia stream using stochastic process

**algebra**.The vast majority of simulations of physical systems, including the movement of rigid bodies under forces, are carried out using vector

**algebra**.Have strong linkage to the Maple computer

**algebra**system.The counterexample is the quantum exterior

**algebra**in 2 variables.Just for explaining a little, a genetic

**algebra**is basically an**algebra**in which multiplication is not associative.There is no operator precedence and the

**algebra**is right associative.Boolean

**algebra**is the**algebra**that describes the simple properties of a single distinction.It enables us to extend matrix

**algebra**calculus in an easy way.Her research interests are representation theory, algebraic combinatorics and computational

**algebra**.Here, the mathematical derivation, which involves matrix

**algebra**, will be followed first, in two dimensions to make things easier.The objective is to enable students to understand more advanced econometrics most of which relies heavily on matrix

**algebra**.Its methods may assume that gens generates A and that the mapping of gens to imgs defines an

**algebra**homomorphism.The angular integrals are carried out using the methods of Racah

**algebra**.It calculates the inverse of a group

**algebra**element.These and a few further structural isomorphisms lead to tensor

**algebra**(see next combinator ).Core modules include mathematical ecology, mathematical biology and medicine and applied linear

**algebra**.Areas covered include linear

**algebra**, optimization, quadrature, differential equations, regression analysis, and time series analysis.Knowledge of Maple as a tool for doing linear

**algebra**.Here's an example describing *one* type of process associated with matrices, that I've used in teaching linear

**algebra**.In numerical linear

**algebra**he developed backward error analysis methods.Such a differential structure, expressing the local versus global nature of brain structure, is lacking in adaptive linear

**algebra**.We shall concentrate on a particularly nice class of codes called linear codes, a beautiful application of elementary linear

**algebra**.These include complete sets of curriculum material for various college and university level courses, including calculus, linear

**algebra**and engineering mathematics.These are objects studied in operator and operator

**algebra**theory.The basic toolbox also allows you to access functions in Maple's linear

**algebra**package.In 1545 he published Ars Magna the first Latin treatise on

**algebra**.Course Description This module will extend the vector

**algebra**of the first year to the calculus of three dimensional vector**algebra**of the first year to the calculus of three dimensional vectors.The arithmetical half of mathematics, which had been gradually growing into

**algebra**, and had decidedly established itself as such in the Ad logisticen speciosam notae priores of Francois Vieta (1540-1603), supplied to some extent the means of generalizing geometry.In this and other details he crowns and completes, in a form henceforth to be dominant for the language of

**algebra**, the work of numerous obscure predecessors, such as Etienne de la Roche, Michael Stifel or Stiefel (1487-1567), and_ others.He did not regard logic as a branch of mathematics, as the title of his earlier pamphlet might be taken to imply, but he pointed out such a deep analogy between the symbols of

**algebra**and those which can be made, in his opinion, to represent logical forms and syllogisms, that we can hardly help saying that logic is mathematics restricted to the two quantities, o and 1.He did good service in systematizing the operational laws of

**algebra**, and in throwing light upon the nature and use of imaginaries.He published, first in 1830, and then in an enlarged form in 1842, a Treatise on

**Algebra**, in which he applied his philosophical ideas concerning algebraical analysis to the elucidation of its elements.He published, among other mathematical works, Clavis Mathematica, in 1631, in which he introduced new signs for certain mathematical operations (see

**Algebra**); a treatise on navigation entitled Circles of Proportion, in 1632; works on trigonometry and dialling, and his Opuscula Mathematica, published posthumously in 1676.Arithmetic,

**algebra**, and the infinitesimal calculus, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and real numbers generally, which include incommensurable numbers.The importance of this

**algebra**arises from the fact that in terms of such complex numbers with this definition of multiplication the utmost generality of expression, to the exclusion of exceptional cases, can be obtained for theorems which occur in analogous forms, but complicated with exceptional cases, in the algebras of real numbers and of signed real numbers.For the subjects of this general heading see the articles

**ALGEBRA**; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.It may be regarded as based on arithmetic, or as dealing in the first instance with formal results of the laws of arithmetical number; and in this sense Sir Isaac Newton gave the title Universal Arithmetic to a work on

**algebra**.The great development of all branches of mathematics in the two centuries following Descartes has led to the term

**algebra**being used to cover a great variety of subjects, many of which are really only ramifications of arithmetic, dealt with by algebraical methods, while others, such as the theory of numbers and the general theory of series, are outgrowths of the application of**algebra**to arithmetic, which involve such special ideas that they must properly be regarded as distinct subjects.Some writers have attempted unification by treating

**algebra**as concerned with functions, and Comte accordingly defined**algebra**as the calculus of functions, arithmetic being regarded as the calculus of values.These applications are sometimes treated under arithmetic, sometimes under

**algebra**; but it is more convenient to regard graphics as a separate subject, closely allied to arithmetic,**algebra**, mensuration and analytical geometry.The association of

**algebra**with arithmetic on the one hand, and with geometry on the other, presents difficulties, in that geometrical measurement is based essentially on the idea of continuity, while arithmetical measurement is based essentially on the idea of discontinuity; both ideas being equally matters of intuition.Notation of Multiples.-The above is arithmetic. The only thing which it is necessary to import from

**algebra**is the notation by which we write 2X instead of 2 X X or 2.Miscellaneous Developments in Arithmetic. - The following are matters which really belong to arithmetic; they are usually placed under

**algebra**, since the general formulae involve the use of letters.A list of early works on

**algebra**is given in Encyclopaedia Britannica, 9th ed., vol.Although the distinction is one which cannot be ultimately maintained, it is convenient to classify the signs of

**algebra**into symbols of quantity (usually figures or letters), symbols of operation, such as +, i, and symbols of distinction, such as brackets.It was at last realized that the laws of

**algebra**do not depend for their validity upon any particular interpretation, whether arithmetical, geometrical or other; the only question is whether these laws do or do not involve any logical contradiction.They behave exactly like the corresponding symbols in arithmetic; and it follows from this that whatever " meaning " is attached to the symbols of quantity, ordinary

**algebra**includes arithmetic, or at least an image of it.From every proposition in this

**algebra**a reciprocal one may be deduced by interchanging + and X, and also the symbols o and i.This idea finds fuller expression in the

**algebra**of matrices, as to which it must suffice to say that a matrix is a symbol consisting of a rectangular array of scalars, and that matrices may be combined by a rule of addition which obeys the usual laws, and a rule of multiplication which is distributive and associative, but not, in general, commutative.The part devoted to

**algebra**has the title al-jebr wa'lmugabala, and the arithmetic begins with " Spoken has Algoritmi," the name Khwarizmi or Hovarezmi having passed into the word Algoritmi, which has been further transformed into the more modern words algorism and algorithm, signifying a method of computing.His discoveries had made him famous all over Italy, and he was earnestly solicited to publish his methods; but he abstained from doing so, saying that he intended to embody them in a treatise on

**algebra**which he was preparing.In his famous Geometria (1637), which is really a treatise on the algebraic representation of geometric theorems, he founded the modern theory of analytical geometry (see Geometry), and at the same time he rendered signal service to

**algebra**, more especially in the theory of equations.In England, multiple

**algebra**was developed by j ames Joseph Sylvester, who, in company with Arthur Cayley, expanded the theory of matrices, the germs of which are to be found in the writings of Hamilton (see above, under (B); and Quaternions).Similarly the "cube root" of a quantity is another quantity which when multiplied by itself twice gives the original quantity; thus is the cube root of a (see Arithmetic and

**Algebra**).Recorde's chief contributions to the progress of

**algebra**were in the way of systematizing its notation (see**ALGEBRA**, History).The first day I had Elementary Greek and Advanced Latin, and the second day Geometry,

**Algebra**and Advanced Greek.I was sorely perplexed, and felt discouraged wasting much precious time, especially in

**algebra**.Two days before the examinations, Mr. Vining sent me a braille copy of one of the old Harvard papers in

**algebra**.I was sorely perplexed, and felt quite discouraged, and wasted much precious time, especially in

**Algebra**.This papyrus contains the first documentary evidence of the use of

**algebra**. rhombus A quadrilateral with four sides of equal length.Cayley gave a matrix

**algebra**defining addition, multiplication, scalar multiplication and inverses.Gauss 's dissertation was a discussion of the fundamental theorem of

**algebra**.Factoring trinomials - A complete course in

**algebra**...After applying some simple

**algebra**to some trite phrases and cliches a new understanding can be reached of the secret to wealth and success.Course Description This module will extend the vector

**algebra**of the first year to the calculus of three dimensional vectors.My goal is to earn a 4.0 GPA this semester, but getting a C on my

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**algebra**homework.He also studied the first six books of Euclid and some

**algebra**, besides reading a considerable quantity of Hebrew and learning the Odes of Horace by heart.After Maclaurin's death his account of Newton's philosophical discoveries was published by Patrick Murdoch, and also his

**algebra**in 1748.For Tartaglia's discovery of the solution of cubic equations, and his contests with Antonio Marie Floridas, see

**Algebra**(History).But he seems to have been well cared for, and he was at the age of fourteen sufficiently advanced "in

**algebra**, geometry, astronomy, and even the higher mathematics," to calculate a solar eclipse within four seconds of accuracy.Each definition gives rise to a corresponding

**algebra**of higher complex numbers.For the subjects of this general heading see the articles

**ALGEBRA**, UNIVERSAL; GROUPS, THEORY OF; INFINITESIMAL CALCULUS; NUMBER; QUATERNIONS; VECTOR ANALYSIS.Under the general heading "

**Algebra**and Theory of Numbers" occur the subheadings "Elements of**Algebra**," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of Algebraic Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.During this period logarithms were invented, trigonometry and

**algebra**developed, analytical geometry invented, dynamics put upon a sound basis, and the period closed with the magnificent invention of (or at least the perfecting of) the differential calculus by Newton and Leibnitz and the discovery of gravitation.This theorem is due to Cayley, and reference may be made to Salmon's Higher

**Algebra**, 4th ed.The development of symbolic

**algebra**by the use of general symbols to denote numbers is due to Franciscus Vieta (Francois Viete, 1540-1603).This led to the idea of

**algebra**as generalized arithmetic.Although the results of ordinary

**algebra**will be taken for granted, it is convenient to give the principal rules upon which it is based.But the symbols of ordinary

**algebra**do not necessarily denote numbers; they may, for instance, be interpreted as coplanar points or vectors.Evolution and involution are usually regarded as operations of ordinary

**algebra**; this leads to a notation for powers and roots, and a theory of irrational algebraic quantities analogous to that of irrational numbers.The only known type of

**algebra**which does not contain arithmetical elements is substantially due to George Boole.Although originally suggested by formal logic, it is most simply interpreted as an

**algebra**of regions in space.The symbol e 0 behaves exactly like i in ordinary

**algebra**; Hamilton writes I, i, j, k instead of eo, el, e2, es, and in this notation all the special rules of operation may he summed up by the equalities = - I.Quaternions afford an example of a quadruple

**algebra**of this kind; ordinary**algebra**is a special case of a duplex linear**algebra**.If, in the extensive calculus of the nth category, all the units (including i and the derived units E) are taken to be homologous instead of being distributed into species, we may regard it as a (2'-I)-tuple linear

**algebra**, which, however, is not wholly associative.It should be observed that while the use of special units, or extraordinaries, in a linear

**algebra**is convenient, especially in applications, it is not indispensable.Various special algebras (for example, quaternions) may be expressed in the notation of the

**algebra**of matrices.In ordinary

**algebra**we have the disjunctive law that if ab = o, then either a = o or b= o.Under the general heading "Fundamental Notions" occur the subheadings "Foundations of Arithmetic," with the topics rational, irrational and transcendental numbers, and aggregates; "Universal

**Algebra**," with the topics complex numbers, quaternions, ausdehnungslehre, vector analysis, matrices, and**algebra**of logic; and "Theory of Groups," with the topics finite and continuous groups.They teach further the solution of problems leading to equations of the first and second degree, to determinate and indeterminate equations, not by single and double position only, but by real

**algebra**, proved by means of geometric constructions, and including the use of letters as symbols for known numbers, the unknown quantity being called res and its square census.The partition method of treating symmetrical

**algebra**is one which has been singularly successful in indicating new paths of advance in the theory of invariants; the important theorem of expressibility is, directly we exclude unity from the partitions, a theorem concerning the expressibility of covariants, and involves the theory of the reducible forms and of the syzygies.**Algebra**and geometry were the only studies that continued to defy my efforts to comprehend them.This form of

**algebra**was extensively studied in ancient Egypt; but, in accordance with the practical tendency of the Egyptian mind, the study consisted largely in the treatment of particular cases, very few general rules being obtained.For many centuries

**algebra**was confined almost entirely to the solution of equations; one of the most important steps being the enunciation by Diophantus of Alexandria of the laws governing the use of the minus sign.It is true that I was familiar with all literary braille in common use in this country--English, American, and New York Point; but the various signs and symbols in geometry and

**algebra**in the three systems are very different, and I had used only the English braille in my**algebra**.But on the night before the

**algebra**examination, while I was struggling over some very complicated examples, I could not tell the combinations of bracket, brace and radical.However, the braille worked well enough in the languages; but when it came to Geometry and

**Algebra**, it was different.In pure

**algebra**Descartes expounded and illustrated the general methods of solving equations up to those of the fourth degree (and believed that his method could go beyond), stated the law which connects the positive and negative roots of an equation with the changes of sign in the consecutive terms, and introduced the method of indeterminate coefficients for the solution of equations.'The first, second and third sections of this publication comprise respectively the papers communicated by him to the Academies of Sciences of Turin, Berlin and Paris; the fourth includes his miscellaneous contributions to other scientific collections, together with his additions to Euler's

**Algebra**, and his Lecons elementaires at the Ecole Normale in 1795.Consequently, I did not do so well as I should have done, if Teacher had been allowed to read the

**Algebra**and Geometry to me.