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arithmetic

Her progress in arithmetic has been equally remarkable.

21370It was not a special subject, like geography or arithmetic, but her way to outward things.

7850I study about the earth, and the animals, and I like arithmetic exceedingly.

5434I go to school every day I am studying reading, writing, arithmetic, geography and language.

4829She has nearly finished Colburn's mental arithmetic, her last work being in improper fractions.

3117If I suggest her leaving a problem in arithmetic until the next day, she answers, "I think it will make my mind stronger to do it now."

2418Now, to speak the truth, I had but ten cents in the world, and it surpassed my arithmetic to tell, if I was that man who had ten cents, or who had a farm, or ten dollars, or all together.

2314Arithmetic seems to have been the only study I did not like.

2315I still regarded arithmetic as a system of pitfalls.

2015Long before I learned to do a sum in arithmetic or describe the shape of the earth, Miss Sullivan had taught me to find beauty in the fragrant woods, in every blade of grass, and in the curves and dimples of my baby sister's hand.

159The arithmetic of rational numbers is now established by means of appropriate definitions, which indicate the entities meant by the operations of addition and multiplication.

1411My studies for the first year were English history, English literature, German, Latin, arithmetic, Latin composition and occasional themes.

126He taught me Latin grammar principally; but he often helped me in arithmetic, which I found as troublesome as it was uninteresting.

119I used to say I did not like arithmetic very well, but now I have changed my mind.

119The Koran, sacred and secular law, logic, poetry, arithmetic, with some medicine and geography, are the chief subjects of study.

106As to the teaching of algebra, see references under Arithmetic to works on the teaching of elementary mathematics.

106It was a long time before decimal arithmetic came into general use, and all through the 17th century exponential marks were in common use.

96The arithmetic of real numbers follows from appropriate definitions of the operations of addition and multiplication.

96The addition and multiplication of these "signed" real numbers is suitably defined, and it is proved that the usual arithmetic of such numbers follows.

96I did a little arithmetic.

31Political and ecclesiastical dissensions occupied the greatest intellects, and the only progress to be recorded is in the art of computing or arithmetic, and the trans pons asinorum of the earlier mathematicians.

32Sir Isaac Newton introduced the term Universal Arithmetic, since it is concerned with the doctrine of operations, not affected on numbers, but on general symbols.

20We 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.

20His 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.

20The Arabians more closely resembled the Hindus than the Greeks in the choice of studies; their philosophers blended speculative dissertations with the more progressive study of medicine; their mathematicians neglected the subtleties of the conic sections and Diophantine analysis, and applied themselves more particularly to perfect the system of numerals, arithmetic and astronomy.

20This 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.

20Text-books on arithmetic in general and on particular applications are numerous, and any list would soon be out of date.

20al-jebr wa'l-mugabala, transposition and removal (of terms of an equation), the name of a treatise by Mahommed ben Musa al-Khwarizmi), a branch of mathematics which may be defined as the generalization and extension of arithmetic.

21This led to the idea of algebra as generalized arithmetic.

21(iii.) By writing (A+a) 2 = A 2 + 2Aa+a 2 in the form (A+a)2= A 2 +(2A+a)a, we obtain the rule for extracting the square root in arithmetic.

21The 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.

21Franciscus Vieta (Francois Viete) named it Specious Arithmetic, on account of the species of the quantities involved, which he represented symbolically by the various letters of the alphabet.

21His 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.

21His 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).

22The 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.

10At this time also flourished Simon Stevinus (Stevin) of Bruges, who published an arithmetic in 1585 and an algebra shortly afterwards.

10Mention may also be made of his chapter on inequalities, in which he proves that the arithmetic mean is always greater than the geometric mean.

10Somewhat later he became Otto's instructor in arithmetic, and had been appointed archbishop of Ravenna before May 998.

10But reference should be made to the exhaustive studies on Hero's arithmetic by Paul Tannery," L'Arithmetique des Grecs dans Heron d'Alexandrie "(Mem.

10These theorems, which hold for the motion of a single rigid body, are true generally for a flexible system, such as considered here for a liquid, with one or more rigid bodies swimming in it; and they express the statement that the work done by an impulse is the product of the impulse and the arithmetic mean of the initial and final velocity; so that the kinetic energy is the work done by the impulse in starting the motion from rest.

10In 1764 he published his first work, The Schoolmaster's Guide, or a Complete System of Practical Arithmetic, which in 1770 was followed by his Treatise on Mensuration both in Theory and Practice.

10The parents and guardians were called upon to select whether each child should learn English or Italian next after learning reading, writing and arithmetic in Maltese.

10This does not mean, what is often alleged, that nobody before him had ever thought of choosing symbols different from numerals, such as the letters of the alphabet, to denote the quantities of arithmetic, but that he made a general custom of what until his time had been only an exceptional attempt.

10He spent a year there and a year in a school for writing and arithmetic, and then at the age of ten he was taken from school to assist his father in the business of a tallow-chandler and soapboiler.

10of arithmetic, viii.

10This applies not only to the geometrical principles but also to the arithmetical principles, and it is therefore of importance, in the earlier stages, to keep geometry, mensuration and arithmetic in close association with one another; mensuration forming, in fact, the link between arithmetic and geometry.

10The measure of the area of a rectangle is thus presented as the product of the measures of the sides, and arithmetic and mensuration are developed concurrently.

10Calculations involving feet and inches are sometimes performed by means of duodecimal arithmetic; i.e., in effect, the tables of square measure and of cubic measure are amplified by the insertion of intermediate units.

10The value of 7r for duodecimal arithmetic is 3+1/12+8/122+ 4/12 3 +8/12 4 +..

10In the latter case the two sections are taken at distances t 2H/ A l 3 = = 2887H from the middle section, where H is the total internal length; and their arithmetic mean is taken to be the mean section of the cask.

10Millis, Technical Arithmetic and Geometry (1903).

10C. Turner, Graphics applied to Arithmetic, Mensuration and Statics (1907).

10He recognizes political economy and statistics as alike sciences, and represents the distinction between them as having never been made before him, though he quotes what Smith had said of political arithmetic. While deserving the praise of honesty, sincerity and independence, he is inferior to his predecessor in breadth of view on moral and political questions.

10To Legendre is due the theorem known as the law of quadratic reciprocity, the most important general result in the science of numbers which has been discovered since the time of P. de Fermat, and which was called by Gauss the " gem of arithmetic."

10The difficulty is avoided by the use of Siacci's altitude-function A or A(u), by which y/x can be calculated without introducing sin n or tan n, but in which n occurs only in the form cos n or sec n, which varies very slowly for moderate values of n, so that n need not be calculated with any great regard for accuracy, the arithmetic mean 1(0+0) of ¢ and B being near enough for n over any arc 4)-8 of moderate extent.

10The Mathesis universalis, a more elementary work, contains copious dissertations on fundamental points of algebra, arithmetic and geometry, and critical remarks.

10(arithmetic) elementary lessons on the notation of decimal fractions.

10For the purpose of thus simplifying the operations of arithmetic, the base is taken to be Io, and use is made of tables of logarithms in which the values of x, the logarithm, corresponding to values of m, the number, are tabulated.

10With Vieta, by reason of the advance in arithmetic, the style of treatment becomes more strictly trigonometrical; indeed, the Universales Inspectiones, in which the calculation occurs, would now be called plane and spherical trigonometry, and the accompanying Canon mathematicus a table of sines, tangents and secants.'

10number in arithmetic, magnitude in geometry, stars in astronomy, a man's good in ethics; concentrates itself on the causes and appropriate principles of its subject, especially the definition of the subject and its species by their essences or formal causes; and after an inductive intelligence of those principles proceeds by a deductive demonstration from definitions to consequences: philosophy is simply a desire of this definite knowledge of causes and effects.

10(2) On the arithmetic, geometric and harmonic means between two straight lines, and the problem of representing all three in one and the same geometrical figure.

10The significance of the word "mean," i.e., middle, is seen by considering 3 instead of n quantities; these will be denoted by a, b, c. The arithmetic mean b, is seen to be such that the terms a, b, c are in arithmetical progression,.

10One while he devoted himself to the sciences, " perfecting himself in music, arithmetic, geometry and ' Life, P. 93.

10During his reign the Tibetans obtained their first knowledge of arithmetic and medicine from China; the prosperity and pastoral wealth of the country were so great that " the king built his palace with cement moistened with the milk of the cow and the yak."

10(b) How then are the primary data of mathematical cognition to be derived from an experience containing space and time relations in Hume, in regard to this problem, distinctly separates geometry from algebra and arithmetic, i.e.

10With 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.

10In the primary schools instruction in reading, writing, arithmetic, history and geography is obligatory.

10Phillips Academy, opened in 1778 (incorporated in 1780), was the first incorporated academy of the state; it was founded through the efforts of Samuel Phillips (1752-1802, president of the Massachusetts senate in 1785-1787 and in 1788-1801, and lieutenant-governor of Massachusetts in 1801-1802), by his father, Samuel Phillips (1715-1790), and his uncle, John Phillips (1719-1795), "for the purpose of instructing youth, not only in English and Latin grammar, writing, arithmetic and those sciences wherein they are commonly taught, but more especially to learn them the great end and real business of living."

10He was also the author of rhetorical exercises on hackneyed sophistical themes; of a Quadrivium (Arithmetic, Music, Geometry, Astronomy), valuable for the history of music and astronomy in the middle ages; a general sketch of Aristotelian philosophy; a paraphrase of the speeches and letters of Dionysius Areopagita; poems, including an autobiography; and a description of the Augusteum, the column erected by Justinian in the church of St Sophia to commemorate his victories over the Persians.

10All pupils were taught to recite portions of the Koran, and a proportion of the scholars learnt to read and write Arabic and a little simple arithmetic. Those pupils who succeeded in committing to memory the whole of the Koran were regarded as fiki (learned in Mahommedan law), and as such escaped liability to military conscription.

10At the same time efforts were made to stamp out all liberal culture in Andalusia, so far as it went beyond the little medicine, arithmetic and astronomy required for practical life.

103 18 E) makes Protagoras pointedly refer to sophists who, " when young men have made their escape from the arts, plunge them once more into technical study, and teach them such subjects as arithmetic, astronomy, geometry and music."

10It may be imagined further that, when he established himself at the Academy, his first care was to draw up a scheme of education, including arithmetic, geometry (plane and solid), astronomy, harmonics and dialectic, and that it was not until he had arranged for the carrying out of this programme that he devoted himself to the special functions of professor of philosophy.

10In the primary schools boys learn arithmetic, and geography and Korean history are taught, with the outlines of the governmental systems of other civilized countries.

10arithmetic of number, geometry of magnitude, astronomy of stars, politics of government, ethics of goods.

10in arithmetic, that 2+ 2 are 4, not any particular 4, and in life that all our contemporaries must die, without enumerating all their particular sorts of deaths.

10We need devices, indeed, to determine priority or superior claim to be " better known absolutely or in the order of nature," but on the whole the problem is fairly faced.4 Of science Aristotle takes for his examples sometimes celestial physics, more often geometry or arithmetic, sometimes a concrete science, e.g.

10Instruction.-Primary schools, maktab (where Persian and a little Arabic, sufficient for reading the Koran, and sometimes also a little arithmetic, are taught to boys between the ages of seven and twelve), are very numerous.

10In 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.

10Philosophy, grammar, the history and theory of language, rhetoric, law, arithmetic, astronomy, geometry, mensuration, agriculture, naval tactics, were all represented.

10Computations are made with it by means of balls of bone or ivory runp ing on slender bamboo rods, similar to the simpler board, fitted up with beads strung on wires, which is employed in teaching the rudiments of arithmetic in English schools.

10(including his earliest publication, "On the Arithmetic of Impossible Quantities," 1779, and an "Account of the Lithological Survey of Schehallion," 1811) and in the Transactions of the Royal Society of Edinburgh ("On the Causes which affect the Accuracy of Barometrical Measurements," &c.), also the articles "Aepinus" and "Physical Astronomy," and a "Dissertation on the Progress of Mathematical and Physical Science since the Revival of Learning in Europe," in the Encyclopaedia Britannica (Supplement to fourth, fifth and sixth editions).

10Playfair's contributions to pure mathematics were not considerable, his paper "On the Arithmetic of Impossible Quantities," that "On the Causes which affect the Accuracy of Barometrical Measurements," and his Elements of Geometry, all already referred to, being the most important.

10In a small commonplace book, bearing on the seventh page the date of January 1663/1664, there are several articles on angular sections, and the squaring of curves and " crooked lines that may be squared," several calculations about musical notes, geometrical propositions from Francis Vieta and Frans van Schooten, annotations out of Wallis's Arithmetic of Infinities, together with observations on refraction, on the grinding of " spherical optic glasses," on the errors of lenses and the method of rectifying them, and on the extraction of all kinds of roots, particularly those " in affected powers."

10It was his duty as professor to lecture at least once a week in term time on some portion of geometry, arithmetic, astronomy, geography, optics, statics, or some other mathematical subject, and also for two hours in the week to allow an audience to any student who might come to consult with the professor on any difficulties he had met with.

10ARITHMETIC (Gr.

10Arithmetic is usually divided into Abstract Arithmetic and Concrete Arithmetic, the former dealing with numbers and the latter with concrete objects.

10The various stages in the study of arithmetic may be arranged in different ways, and the arrangement adopted must be influenced by the purpose in view.

10The daily activities of the great mass of the adult population, in countries where commodities are sold at definite prices for definite quantities, include calculations which have often to be performed rapidly, on data orally given, and leading in general to results which can only be approximate; and almost every branch of manufacture or commerce has its own range of applications of arithmetic. Arithmetic as a school subject has been largely regarded from this point of view.

10From the educational point of view, the value of arithmetic has usually been regarded as consisting in the stress it lays on accuracy.

10This aspect of the matter, however, belongs mainly to the period when arithmetic was studied almost entirely for commercial purposes; and even then accuracy was not found always to harmonize with actuality.

10As a branch of mathematics, arithmetic may be treated logically, psychologically, or historically.

10Laws of Arithmetic

10Commercial Arithmetic 12.2.2 126.

10Arithmetic is supposed to deal with cardinal, not with ordinal numbers; but it will be found that actual numeration, beyond about three or four, is based on the ordinal aspect of number, and that a scientific treatment of the subject usually requires a return to this fundamental basis.

10If we retain the unit, the arithmetic is concrete; if we ignore it, the arithmetic is abstract.

10We can, however, denote the result of the process by a symbol, and deal with this symbol according to the laws of arithmetic. In this way we arrive at (i) negative numbers, (ii) fractional numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word).

10Hindu treatises on arithmetic show the use of fractions, containing a power of io as denominator, as early as the beginning of the 6th century A.D.

10Text-books on arithmetic usually contain explanations of the chief commercial transactions in which arithmetical calculations arise; it will be sufficient in the present article to deal with interest and discount, and to give some notes on percentages and rates in the £.

10For the latter, and for systems of notation, reference may also be made to Peacock's article " Arithmetic " in the Encyclopaedia Metropolitana, which contains a detailed account of the Greek system.

10There are very few works dealing adequately but simply with the principles of arithmetic. Homersham Cox, Principles of Arithmetic (1885), is brief and lucid, but is out of print.

10Seeley, The Grube Method of Teaching Arithmetic (1890).

10On the teaching of arithmetic, and of elementary mathematics generally, see J.

10P. Turnbull, The Teaching of Arithmetic (1903), is more elaborate.

10Lodge, Easy Mathematics, chiefly Arithmetic (1905), treats the subject broadly in its practical aspects.

10King, Report on Teaching of Arithmetic and Mathematics in the Higher Schools of Germany (1903).

10In 1830 appeared the first edition of his well-known Elements of Arithmetic, which did much to raise the character of elementary training.

10In his admirable papers upon the modes of teaching arithmetic and geometry, originally published in the Quarterly Journal of Education (reprinted in The Schoolmaster, vol ii.), he remonstrated against the neglect of logical doctrine.

10In Chinese he published books on arithmetic, geometry, algebra (De Morgan's), mechanics, astronomy (Herschel's), and The Marine Steam Engine (T.

10Wollaston's theory of moral evil as consisting in the practical contradiction of a true proposition, closely resembles the most paradoxical part of Clarke's doctrine, and was not likely to approve itself to the strong common sense of Butler; but his statement of happiness or pleasure as a " justly desirable " end at which every rational being " ought " to aim corresponds exactly to Butler's conception of self-love as a naturally governing impulse; while' the " moral arithmetic " with which he compares pleasures and pains, and endeavours to make the notion of happiness quantitatively precise, is an anticipation of Benthamism.

10It is incumbent upon the clergy to see that all children are taught reading, writing and arithmetic. The people are great readers; considering the number of the inhabitants, books and periodicals have a very extensive circulation.

10He also wrote or edited various Chinese works on geography, the celestial and terrestrial spheres, geometry and arithmetic. And the detailed history of the mission was drawn out by him, which after his death was brought home by P. Nicolas Trigault, and published at Augsburg, and later in a complete form at Lyons under the name De Expeditione Christiana apud Sinas Suscepta, ab Soc. Jesu, Ex P. Mat.

10Arithmetic >>

10In 1768 he published the Farmer's Letters to the People of England, in 1771 the Farmer's Calendar, which went through a great number of editions, and in 1774 his Political Arithmetic, which was widely translated.

10The first contains the "Arithmetic of Diophantus," with notes and additions.

10Be aware that floating point arithmetic is not exact; matrices that are theoretically equal are not always numerically equal.

10I like to perform arithmetic on such figures to see how they add up.

10Improvements were made to the boundary cases of floating-point arithmetic.

10Each of these gets a chapter, along with a final chapter devoted to floating point arithmetic.

10The correct sequence of moves to play is based on something called ' modular arithmetic ' .

10Yes your mental arithmetic is correct we had a turnout of nineteen, a record for the Mountain Biking Section.

10arithmetic expression, where a number is needed.

10arithmetic overflow occurs.

10arithmetic progression.

10arithmetic operations in Maple should now either return a number or signal a numeric event.

10arithmetic operators apply.

10arithmetic calculation took him only a few minutes.

10Keywords DOUBLE Set this keyword to force the computation to be done in double-precision arithmetic.

10integer arithmetic within the decoders does not work, due to the algorithms used.

10For the range specification, see the description of the vector arithmetic register.

10On architectures with 16-bit pointer arithmetic, only very small images can be processed.

10Arbitrary precision integer arithmetic is provided by some implementations of the language.

10The idea behind using interval arithmetic is always to ensure that the true value of a calculation is within known bounds.

10arithmetic progression of consecutive primes for any given (finite) length?

10The work which involves arithmetic progressions is Hypsicles ' On the Ascension of Stars.

10The problem is that the Spectrum ROM handles this line interpretation as an arithmetic calculation, and calls its calculating routines.

10congruence subgroup problem, in particular Serre's conjecture for arithmetic lattices in the real rank 1 simple Lie groups.

10congruence arithmetic of the great 19th century Friedrich Gauss.

10electrophysiology of normal number processing and arithmetic disabilities (dyscalculia ).

10elementary arithmetic the two-dimensional character of the paper is sometimes used.

10evaluated according to the rules given below in section 6.5 Shell Arithmetic.

10exponentiation arithmetic operator operand data types being expanded inline are as follows.

10floating point arithmetic.

10floating-point arithmetic in scripts.

10Use slate and slate pencils to practice handwriting and have a lesson in arithmetic using pounds, shillings and pence.

10interval arithmetic.

10I was quite good at maths and English, although h I hated maths once it got beyond ordinary arithmetic.

10modular arithmetic ' .

10Integer Arithmetic By default, Perl assumes that it must do most of its arithmetic in floating point.

10Create variables, use variables in assignment statements and write code to carry out simple arithmetic operations.

10larger ordinals correspond to more sophisticate descriptions of arithmetic.

10The programmer only becomes aware of the lower level when an error such as an arithmetic overflow occurs.

10Or to put it another way, Java has pointers, but no pointer arithmetic.

10precision arithmetic allows us to investigate the convergence of iterative methods, such as the Newton Raphson method.

10Are there infinitely many sets of 3 consecutive primes in arithmetic progression.

10Arithmetic manipulation of past or future sentinels will not change their values, nor cause any error.

10vector arithmetic register.

10At Potsdam the data represent the arithmetic means derived from the Fourier analysis for the individual months comprising the season.

10gives comparative results for winter (October to March) and summer at a few stations, the value for the season being the arithmetic mean from the individual months composing it.

10Summer And Winter Represent Each Six Months And The Results Are Arithmetic Means Of The Monthly Values.

10That idea of a method grew up with his study of geometry and arithmetic, - the only branches of knowledge which he would allow to be " made sciences."

104 The mathematics of which he thus speaks included the geometry of the ancients, as it had been handed down to the modern world, and arithmetic with the developments it had received in the direction of algebra.

10From a greengrocer he learnt arithmetic; and higher branches were begun under one of those wandering scholars who gained a livelihood by cures for the sick and lessons for the young.

10In an appendix of forty-one pages he gives his third method, "local arithmetic," which is performed on a chess-board, and depends, in principle, on the expression of numbers in the scale of radix 2.

10In the preface to the appendix containing the local arithmetic he states that, while devoting all his leisure to the invention of these abbreviations of calculation, and to examining by what methods the toil of calculation might be removed, in addition to the logarithms, rabdologia and promptuary, he had hit upon a certain tabular arithmetic, whereby the more troublesome operations of common arithmetic are performed on an abacus or chess-board, and which may be regarded as an amusement A facsimile of this document is given by Mark Napier in his Memoirs of John Napier (1834), p. 248.

10Napier deliberately set himself to abbreviate multiplications and divisions - operations of so fundamental a character that it might well have been thought that they were in rerum natura incapable of abbreviation; and he succeeded in devising, by the help of arithmetic and geometry alone, the one 1 The title runs as follows: Arithmetica Logarithmica, sive Logarithmorum chiliades triginta....

10The Arithmetic consists of three books, entitled-(I) De Computationibus Quantitatum omnibus Logisticae speciebus communium; (2) De Logistica Arithmetica; (3) De Logistica Geometrica.

10In the Rabdologia the rods are called "virgulae," b'ut in the passage quoted above from the manuscript on arithmetic they are referred to as "bones" (ossa).

10To Napier seems to be due the first use of the decimal point in arithmetic. Decimal fractions were first introduced by Stevinus in his tract La Disme, published in 1585, but he used cumbrous exponents (numbers enclosed in circles) to distinguish the different denominations, primes, seconds, thirds, &c. Thus, for example, he would have written 123.456 as 123@4050603.

10We see from a statement of Cassiodorus that he furnished manuals for the quadrivium of the schools of the middle ages (the " quattuor matheseos disciplinae," as Boetius calls them) on arithmetic, music, geometry and astronomy.

10In its charter this institution is described as "an academy for the purpose of promoting piety and virtue, and for the education of youth in the English, Latin and Greek languages, in writing, arithmetic, music and the art of speaking, practical geometry, logic and geography, and such other of the liberal arts and sciences or languages, as opportunity may hereafter permit."

10As circumstances allowed, she appears to have taught him reading, writing and arithmetic - acquisitions made with so little of remembered pain that " were not the error corrected by analogy," he says, " I should be tempted to conceive them as innate."

10At the age of ten Patrick was making slow progress in the study of reading, writing and arithmetic at a small country school, when his father became his tutor and taught him Latin, Greek and mathematics for five years, but with limited success.

10130), nor does Greek arithmetic as represented by these authors and by Iamblichus (end of 3rd century) show any trace of his influence, facts which can only be accounted for by his being later than those arithmeticians at least who would have been capable of understanding him fully.

10This saves a certain amount of arithmetic, but when the solution is applied in another determination additional calculations are necessary.

10Arithmetic, algebra, and the infinitesimal calculus, are sciences directly concerned with integral numbers, rational (or fractional) numbers, and real numbers generally, which include incommensurable numbers.

10With these definitions it is now possible to prove the following six premisses applying to finite cardinal numbers, from which Peano 2 has shown that all arithmetic can be deduced i.

10On the contrary, the premisses of arithmetic can be put in other forms, and, furthermore, an indefinite number of propositions of arithmetic can be proved directly from logical principles without mentioning them.

10Thus, while arithmetic may be defined as that branch of deductive reasoning concerning classes and relations which is concerned with the establishment of propositions concerning cardinal numbers, it must be added that the introduction of cardinal numbers makes no great break in this general science.

10of 1903); earlier formulations of the bases of arithmetic are given by him in the editions of 1898 and of 1901.

10Owing to the correspondence between the finite cardinals and the finite ordinals, the propositions of cardinal arithmetic and ordinal arithmetic correspond point by point.

10The arithmetic of the infinite cardinals does not correspond to that of the infinite ordinals.

10It will suffice to mention here that Peano's fourth premiss of arithmetic does not hold for infinite cardinals or for infinite ordinals.

10Under 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.

10For the subjects of this general heading see the articles ALGEBRA; ALGEBRAIC FORMS; ARITHMETIC; COMBINATORIAL ANALYSIS; DETERMINANTS; EQUATION; FRACTION, CONTINUED; INTERPOLATION; LOGARITHMS; MAGIC SQUARE; PROBABILITY.

10The state supports primary schools (352 in 1905), where reading, writing, arithmetic and history are taught; and separate instruction is given by the Orthodox, Roman Catholic, Jewish and Moslem clergy.

10He showed at an early age wellmarked mathematical powers, and his progress was so rapid in arithmetic and geometry that he was soon beyond the guidance not only of his father but of schoolmaster Schulz, who assisted in the mathematical department of his training.

10In the native schools - almost all maintained by Christian missions - Zulu and English are taught, the subjects taken being usually reading, writing, arithmetic, grammar, geography and history.

10It 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.

10The 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.

10Some 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.

10These 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.

10The 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.

10The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and geometrical measurement with the cardinal and the ordinal aspects of number respectively (see Arithmetic).

10What are called negative numbers in arithmetic, for instance, are not really negative numbers but negative quantities (Ã¯¿½ 27 (i.)); and the difficulties incident to the ideas of continuity have already been pointed out.

10But the present tendency is to regard the early association of arithmetic with linear measurement as important; and it seems to follow that we may properly (at any rate at an early stage of the subject) multiply a length by a length, and the product again by another length, the practice being dropped when it becomes necessary to give a strict definition of multiplication.

10It should be noticed that we are still dealing with the elementary processes of arithmetic, and that all the numbers contemplated in Ã¯¿½Ã¯¿½ 14-17 are supposed to be positive integers.

10Notation 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.

10In the method (a) above there is indeed a multiplication by 6; but it is a multiplication arising out of subdivision, not out of repetition (see Arithmetic), so that the total (viz.

10it cannot be said that " If A = B, then E=F " is arithmetic, while " If C = D, then E=F " is algebra.

10Algebraic treatment consists in replacing either of the terms A or B by an expression which we know from the laws of arithmetic to be equivalent to it.

10(i.) There are statements, such as A+B = B+A, which are particular cases of the laws of arithmetic, but need not be expressed as such.

10General Arithmetical Theorems. (i.) The fundamental laws of arithmetic should be constantly borne in mind, though not necessarily stated.

10(ii.) The elements of the theory of numbers belong to arithmetic. In particular, the theorem that if n is a factor of a and of b it is also a factor of pa= qb, where p and q are any integers, is important in reference to the determination of greatest common divisor and to the elementary treatment of continued fractions.

10- (i.) What are usually called " negative numbers " in arithmetic are in reality not negative numbers but negative quantities.

10Miscellaneous 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.

10The graphic method may therefore be used in arithmetic for comparing two particular magnitudes of the same kind by comparing the corresponding lengths P and Q measured along a single line OX from the same point O.

10in arithmetic; numerical coefficients of the factor as a whole being ignored (cf.

10- (i.) The results of the addition, subtraction and multiplication of multinomials (including monomials as a particular case) are subject to certain laws which correspond with the laws of arithmetic (Ã¯¿½ 26 (i.)) but differ from them in relating, not to arithmetical value, but to algebraic form.

10The commutative law in arithmetic, for instance, states that adb and b+a, or ab and ba, are equal.

10Thus to show that the arithmetic mean of n positive numbers is greater than their geometric mean (i.e.

11than the nth root of their product) we show that if any two are unequal their product may be increased, without altering their sum, by making them equal, and that if all the numbers are equal their arithmetic mean is equal to their geometric mean.

11The quadratic equation x 2 +b 2 =o, for instance, has no real root; but we may treat the roots as being +b-' - I, and - b 1, 1 - 1, if -J - i is treated as something which obeys the laws of arithmetic and emerges into reality under the condition 1 1 - I.

11When this fundamental truth had been fully grasped, mathematicians began to inquire whether algebras might not be discovered which obeyed laws different from those obtained by the generalization of arithmetic. The answer to this question has been so manifold as to be almost embarrassing.

11They 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.

11The 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.

11The Arabians more closely resembled the Hindus than the Greeks in the choice of studies; their philosophers blended speculative dissertations with the more progressive study of medicine; their mathematicians neglected the subtleties of the conic sections and Diophantine analysis, and applied themselves more particularly to perfect the system of numerals (see Numeral), arithmetic and astronomy.

11The proper statement is that, if a and b are the sides, the area is equal to the area of a rectangle whose sides are a and tb; this being, in fact, a particular case of the proposition that the area of a trapezium is equal to the area of a rectangle whose sides are its breadth and the arithmetic mean of the lengths of the two parallel sides.

11Similarly 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).

11-- Ordinary arithmetic books often contain references to local and customary weights and measures and to obsolete terms of no practical use to children.

11We may infer therefore that as early as 1594 Napier had communicated to some one, probably John Craig, his hope of being able to effect a simplification in the processes of arithmetic. Everything tends to show that the invention of logarithms 2 See Mark Napier's Memoirs of John Napier of Merchiston (1834), p. 362.

11The conclusion from these therefore was that the ratio of circumference to diameter is 34 This is a most notable piece of work; the immature condition of arithmetic at the time was the only real obstacle preventing the evaluation of the ratio to any degree of accuracy whatever.5 No advance of any importance was made upon the achievement of Archimedes until after the revival of learning.

11We can only conjecture that the lost book i., as well as book ii., was concerned with arithmetic, book iii.

11Its professors teach grammatical inflexion and syntax, rhetoric, versification, logic, theology, the exposition of the Koran, the traditions of the Prophet, the complete science of jurisprudence, or rather of religious, moral, civil and criminal law, which is chiefly founded on the Koran and the traditions, together with arithmetic as far as it is useful in matters of law.

11Such statements, however, arc capableof logical proof, and are generalizations of results obtained empirically at an elementary stage; they therefore belong more properly to the laws of arithmetic (§ 58).

11At eight I study arithmetic.

11She has made considerable progress in the study of arithmetic.

11She has also done some good work in written arithmetic.

11This is my Root Rational Fraction program, which does exact arithmetic on quantities that are square roots of rational fractions.

11Some of the functions available are measuring shapes, unit conversion, general arithmetic and different types of rates.

11Individuals in this stage are unable to perform challenging mental arithmetic, such as counting backward from 75 by 7s.

11Math. It's a subject that many people dread well into adulthood, but simple arithmetic is absolutely fundamental to proper functioning in today's society.

11That's why some of the free online preschool games are focused on learning the numbers and learning basic arithmetic equations.

11Some children were taught the basics of reading, writing and arithmetic at home while others worked on the family farm or in factories.

11Arithmetic is often the most difficult subject for a child with FAS.

11The average achievement levels for reading, spelling, and arithmetic were fourth grade, third grade, and second grade, respectively.

11An example of a criterion-referenced test is a timed arithmetic test that is scored for the number of problems answered correctly.

11Instruction for students with mathematical disorders emphasizes real-world uses of arithmetic, such as balancing a checkbook or comparing prices.

11While students are taught the basics of reading, writing and arithmetic, this teaching approach is centered on the child's interests.

11This has been one of the most popular Cool Math games for kids since 2000, and is used widely by elementary school teachers to reinforce and teach arithmetic in a different way.

11The answer came in the form of hexadecimal arithmetic.

11(See Arithmetic.)

12In elementary arithmetic the two-dimensional character of the paper is sometimes used.

12

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