Multiplication Sentence Examples

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

I think that there is never any alcoholic fermentation without there being at the same time organization, development and multiplication of globules, or the continued consecutive life of globules already formed."

When we are familiar with the treatment of quantities by equations, we may ignore the units and deal solely with numbers; and (ii.) (a) and (ii.) (b) may then, by the commutative law for multiplication, be regarded as identical.

The opportunity thus given for debate naturally stimulated the movement in favour of constitutional government, which received new impulses from the sympathetic attitude of the emperor Alexander II., his grant in 1879 of a constitution to the liberated principality of Bulgaria, and the multiplication of Nihilist outrages which pointed to the necessity of conciliating Liberal opinion in order to present a united front against revolutionary agitation.

By the multiplication of the protoplasts in these merismatic areas the substance of the plant is increased.

The most frequent are the miracle at Cana, the multiplication of the loaves and fishes, the paralytic carrying his bed, the healing of the woman with the issue of blood, the raising of Lazarus, FIG.

A characteristic feature of the calculus is that a meaning can be attached to a symbol of this kind by adopting a new rule, called that of regressive multiplication, as distinguished from the foregoing, which is progressive.

The greater the commercial and industrial prosperity of a town, the more rapid was the multiplication of craft gilds, which was a natural result of the ever-increasing division of labour.

Dean's multiplication table of 44, the number of rented apartments, wasn't perfect, but that num­ber times even a reasonable monthly rental lessened any sympathy he might have felt for the woman's financial plight.

Neper, Baron of Mercheston, near Edinburgh, and told him, among other discourses, of a new invention in Denmark (by Longomontanus, as 'tis said), to save the tedious multiplication and division in astronomical calculations.

It satisfies the condition, however, equally with logarithms, of enabling multiplication to be performed by the aid of a table of single entry; and, analytically considered, it is not so different in principle from the logarithmic method.

The life of the parasites while in the insect is characterized by an alternation of active periods, during which multiplication goes on, with resting-periods, when the Trypanosomes become attached to the epithelial cells of the host.

The Tenth Census put it beyond doubt that economic and social forces had been at work, reducing the rate of multiplication.

Later changes consist in the growth of the shell-valves over the whole area of the mantle-flaps, and in the multiplication of the gill-filaments and their 'consolidation to form gill-plates.

The multiplication of orders is attended with practical difficulties, and the distinctions between the various groups of the Linnean Neuroptera are without doubt less obvious than those between the Coleoptera (beetles) and the Diptera (two-winged flies) for example.

The insistence on the unique efficacy of the sacrifice of the altar led to the multiplication of masses, and so of altars, which were placed in the transepts or aisles or in chapels, dedicated to the saints whose relics they enshrined.

Multiplication of altars is another medieval characteristic. This also is probably a result of the edict of Pope Felix already mentioned.

It is in the 6th century that we begin to find traces of the multiplication of altars.

Darwin believed that man advanced to his present high condition through such a struggle, consequent on his rapid multiplication.

Variations in the structure and forms of leaves and leafstalks are produced by the increased development of cellular tissue, by the abortion or degeneration of parts, by the multiplication or repetition of parts and by adhesion.

With pendulums of shorter length, say metres, it is necessary to have a multiplication 80 to too fold by a double system of very light levers, in order to render the extremely slight tilting of their support perceptible.

This observation led him to further work, and he succeeded in showing that in vascular organs the presence of cells in inflammatory exudates is not the result of exudation but of multiplication of pre-existing cells.

Instead of regarding these as only ministering to the construction of the bulky portions, the living protoplasts take the first place as the essential portion of the tree, and all the other features are important mainly as ministering to their individual well-being and to their multiplication.

Of this we see evidence in the multiplication of Satans in the Book of Enoch.

Such excessive multiplication of the larger taxonomic divisions shows an imperfect sense of proportion, for if the term " class " be allowed its usual zoological value, no student can fail to recognize that the Hexapoda form a single welldefined class, from which few entomologists would wish to exclude even the Apterygogenea.

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

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

Growing knowledge of Aristotle's works and the multiplication of translations enabled students to tendency.

Here each member is a number, and the equation may, by the commutative law for multiplication, be written 2(x+I) - 4(x-2) This means that, whatever unit A we take, 2(x+ I) A Sand 5 4(x-2) A are equal.

We see first that any operation with 4a-3b can be regarded as an operation with (+)4a+(-)3b, subject to the conditions (I) that the signs (+) and (-) obey the laws (+)=(+),(+)(-)=(-)(+)= (-), (-) (-)=(+), and (2) that, when processes of multiplication are completed, a quantity is to be added or subtracted according as it has the sign (+) or (-) prefixed.

Theoretically, no limit can be assigned to the number of possible algebras; the varieties actually known use, for the most part, the same signs of operation, and differ among themselves principally by their rules of multiplication.

The unknown he terms arithmos, the number, and in solutions he marks it by the final s; he explains the generation of powers, the rules for multiplication and division of simple quantities, but he does not treat of the addition, subtraction, multiplication and division of compound quantities.

His principal discovery is concerned with equations, which he showed to be derived from the continued multiplication of as many simple factors as the highest power of the unknown, and he was thus enabled to deduce relations between the coefficients and various functions of the roots.

To the mere tradition preserved by memory and handed on by speech was then added the written record and its later multiplication by the mechanical arts of printing, by which it acquired permanence and universal distribution.

We have, in fact, a form of larval multiplication that recalls the development of digenetic Trematodes.

The rapid multiplication that takes place in the larval stage of nearly all endoparasitic forms affects the tissues of the "intermediate" host in which they live.

I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting m for the second figure or term, the rest should be produced by the continued multiplication of the terms of this seriesI X m 2 I X m - 2 This rule I therefore applied to the series to be interpolated.

This is the Dionysian or Great Paschal Period, and is formed by the multiplication of the numbers 28 and 19, that is, of the solar and lunar cycles, into each other.

The number of species included in this division is enormous, and the multiplication of families is, to some extent, a natural result of increasingly close study.

The explanation of this property of the base io is evident, for a change in the position of the decimal points amounts to multiplication or division by some power of 10, and this corresponds to the addition or subtraction of some integer in the case of the logarithm, the mantissa therefore remaining intact.

The two systems of logarithms for which extensive tables have been calculated are the Napierian, or hyperbolic, or natural system, of which the base is e, and the Briggian, or decimal, or common system, of which the base is io; and we see that the logarithms in the latter system may be deduced from those in the former by multiplication by the constant multiplier /loge io, which is called the modulus of the common system of logarithms.

The change, wrote General Walker, which produced this falling off from the traditional rate of increase of about 3% per annum, was that from the simplicity of the early times to comparative luxury; involving a rise in the standard of living, the multiplication of artificial necessities, the extension of a paid domestic service, the introduction of women into factory labor.2 In his opinion the decline in the birth-rate coincidently with the increase of immigration, and chiefly in those regions where immigration was greatest, was no mere coincidence; nor was such immigrant invasion due to a weakening native increase, or economic defence; but the decline of the natives was the effect of the increase of the foreigners, which was a shock to the principle of population among the native element.

The measures by which the government of India chiefly endeavours to reduce the liability of the country to famine are the promotion of railways; the extension of canal and well irrigation; the reclamation of waste lands, with the establishment of fuel and fodder reserves; the introduction of agricultural improvements; the multiplication of industries; emigration; and finally the improvement where necessary of the revenue and rent systems. In times of famine the function of the railways in distributing the grain is just as important as the function of the irrigation-canals in increasing the amount grown.

In common with all enlightened opinion, he complained bitterly of the excessive multiplication of exemptions, of the exaggerated extension of appeals to Rome, of the luxury of the Roman court, of the venality of the cardinals, and of the injury done to the traditional hierarchy by the very extent of the papal power, which was calculated to turn the strongest head.

Delage has distinguished as multiplication those cases in which the new individual arises from a mass of cells which remain a part of the maternal tissues during differentiation, reserving the term reproduction for those cases in which the spore or cell which is the starting-point of the new individual begins by separating from the maternal tissues; but the distinction is inconvenient in practice and does not appear to carry with it any fundamental biological significance.

The antipodal cells aid more or less in the process of nutrition of the developing embryo, and may undergo multiplication, though they ultimately disintegrate, as do also the synergidae.

She can add and subtract with great rapidity up to the sum of one hundred; and she knows the multiplication tables as far as the FIVES.

A "mall trip" turns into an exercise in percentages, while an outer space themed game tests multiplication skills in order to win.

The Greek system, however, adopted this foot as a basis for decimal multiplication, forming --

The parasites are frequently more numerous in the spleen, bone-marrow, kidneys, &c., than elsewhere, and it has been found that multiplication goes on rather more actively in the capillaries of these organs.

The definite course is not occasioned so much from the ferments which exist in the leaves themselves, but from an arrest of the digestive process which allows the rapid multiplication of the former in the intestines.

Experience, however, has shown that even when the movements of the ground are alarming the actual range of motion is so small that a satisfactory record can be obtained only by some mechanical (or optical) method of multiplication.

The prefixes for multiplication by io, io 2, io 3 and Io 4 are deca-, hecto-, kilo- and myria-, and those for division by io, io 2 and io 3 are deci-, centi- and milli-; the former being derived from Greek, and the latter from Latin.

Meanwhile the chief objection, that of "novelty," was gradually removed by the multiplication of local manifestations, the genuineness of which was proved to the satisfaction of the Roman Congregation of Rights, and in 1765 it was allowed for houses of the Visitation and certain countries.

While strongly discouraging the arbitrary multiplication of public or private fasts, the English Church seems to leave to the discretion of the individual conscience every question as to the manner in which the fasts she formally enjoins, are to be observed.

The new jurisprudence encouraged the multiplication of small properties.

This multiplication of the seed-bearing branches might at first sight be considered advantageous; but in practice the quality of the grain is found to be inferior, as if the force that should have been devoted to the maturation of the grain were, in a measure, diverted and expended in the production of additional branches to the spike.

Considering the equations ax +by +cz =d, a'x +b'y +c' z =d', a"x+b"y+cnz=d" and proceeding to solve them by the so-called method of cross multiplication, we multiply the equations by factors selected in such a manner that upon adding the results the whole coefficient of y becomes = o, and the whole coefficient of z becomes = o; the factors in question are b'c" - b"c', b"c - be", bc' - b'c (values which, as at once seen, have the desired property); we thus obtain an equation which contains on the left-hand side only a multiple of x, and on the right-hand side a constant term; the coefficient of x has the value a(b'c" - b"c') +a'(b"c - bc") +a'(bc' - b'c), and this function, represented in the form a, b,c, a' b'c', a" b" c" is said to be a determinant; or, the number of elements being 32, it is called a determinant of the third order.

To indicate the method of proof, observe that the determinant on the left-hand side, qua linear function of its columns, may be I The reason is the connexion with the corresponding theorem for the multiplication of two matrices.

We have already mentioned that in astronomical practice right ascensions are expressed in time, so that no multiplication by 15 is necessary.

For the investigation of the general scheme of sidereal structure, the multiplication of results of the kind is indispensable.

So long as Mr. Lloyd George was Minister, Dr. Addison was his right-hand man in the strenuous labours of the office, resulting in the enormous multiplication of engines of war, and in the redeeming of many vital industries, fertilizers, tungsten and potash from German control; and when Mr. Lloyd George formed a Government himself in December 1916, he placed him at the head of the department.

The very multiplication of offices, so noticeable at this time, furthered this triumph of feudalism by multiplying the links of personal dependence, and neutralizing more and more the direct action of the central authority.

The festival of the new doctrine, which turned the head of the new pontiff (June 8), the loi de Prairial, or code of legal murder (June 10), which gave the deputies themselves into his hand; and the multiplication of executions at a time when the victory of Fleurus (June 25) showed the uselessness and barbarity of this aggravation of the Reign of Terror provoked against him the victorious coalition of revenge, lassitude and fear.

Alteration in the symmetrical arrangement as well as in the completeness and regularity of flowers has been traced to suppression or the non-development of parts, degeneration or imperfect formation, cohesion or union of parts of the same whorl, adhesion or union of the parts of different whorls, multiplication of parts, and deduplication (sometimes called chorisis) or splitting of parts.

Flowers become double by the multiplication of the parts of the corolline whorl; this arises in general from a metamorphosis of the stamens.

The staminal row is more liable to multiplication of parts than the outer whorls.

Just for explaining a little, a genetic algebra is basically an algebra in which multiplication is not associative.

Recall that in Heisenberg's first paper, he defined a multiplication which was not necessarily commutative.

Then = n (f g ). In other words, the discrete Fourier transform maps convolution to pointwise multiplication.

On the set of all n × n matrices over a field F it is possible to define a multiplication in many ways.

For example, consider a floating point pipeline performing a multiplication.

This sort of multiplication is given a special name; it is scalar multiplication, because the numbers are sometimes called scalars.

Like viruses, such strategies take advantage of rapid multiplication to explode the message to thousands, to millions.

To do inner product matrix multiplication, you can use the product global function.

The matrix-vector multiplication routine must be supplied by the user.

This will depend on which comes first - part of the vector multiplication rules; do you remember?

All that is needed is the replacement of new lines for old in the seed multiplication chain.

All sexually reproducing organisms develop from a single fertilized oocyte by cell multiplication and differentiation.

Learning was done by rote and special attention was given to multiplication tables, capital cities and the islands of Scotland.

Fermentation, according to Pasteur, was caused by the growth and multiplication of unicellular organisms out of contact with free oxygen, under which circumstance they acquire the power of taking oxygen from chemical compounds in the medium in which they are growing.

It will be noticed that the rods only give the multiples of the number which is to be multiplied, or of the divisor when they are used for division, and it is evident that they would be of little use to any one who knew the multiplication table as far as 9 X9.

Thus in Cunina octonaria, the ovum develops into an actinula which buds daughteractinulae; all of them, both parent and offspring, develop into medusae, so that there is no alternation of generations, but only larval multiplication.

They are, however, in no sense evangelicals in the Western sense; for they observe rigorous fasts, reverence icons, and believe implicitly in the efficacy of the multiplication of crossings, bowings and prostrations.

Malthus is already an author whose name is probably more widely known than that of any other economist, but whose works are rarely read, and studied only by a small proportion of The old text-book and learnt like the multiplication table, it political p economy."

Marchal points out the analogy of this phenomenon to the artificial polyembryony that has been induced in Echinoderm and other eggs by separating the blastomeres, and suggests that the abundant food-supply afforded by the host-larva is favourable for this multiplication of embryos, which may be, in the first instance, incited by the abnormal osmotic pressure on the egg.

Another change has come over the condition of ornithology, as of kindred sciences, induced by the multiplication of learned societies which issue publications as well as of periodicals of greater or less scientific pretension.

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.

Therefore an increase in photosynthesis caused by the multiplication of plant microorganisms will lead to the precipitation of calcium carbonate, for carbonic acid will be withdrawn from solution to take part in carbohydrate synthesis by the plants.

By simple multiplication (al b l b2 -24a2bib2+ala2b;)xi +(aibz -ala214b2-aia2blb2+a2b2)xlx2 + (aia 2 b2 - 2a l a2b l b2 +a2/4b 2)x2; and transforming to the real form, (aob 2 - 2a1b,+a2bo)xi (aob 3 -a l b 2 - alb,+a3bo)xlx2 + (aib3 - 2a2b2+a3b1)x2, the simultaneous covariant; and now, putting b = a, we obtain twice.

The effect of these definitions is that the sum and the product of two quaternions are also quaternions; that addition is associative and commutative; and that multiplication is associative and distributive, but not commutative.

Now this is formally analogous to the distributive law of multiplication; and in fact we may look upon AFB as a particular way of multiplying A and B (not A and B).

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 deficiencies of the Greek symbolism were partially remedied; subtraction was denoted by placing a dot over the subtrahend; multiplication, by placing bha (an abbreviation of bhavita, the product ") after the factors; division, by placing the divisor under the dividend; and square root, by inserting ka (an abbreviation of karana, irrational) before the quantity.

With the constitutional changes of the 18th and 10th centuries, however, a corresponding modification took place in the character of the English episcopate; and a still further change resulted from the multiplication of colonial and missionary sees having no connexion with the state (see Anglican Communion).

In 1857 Pasteur decisively proved that fermentation was a physiological process, for he showed that the yeast which produced fermentation was no dead mass, as assumed by Liebig, but consisted of living organisms capable of growth and multiplication.

The branches of the stem arise by multiplication of the cells 01 the epidermis and cortex at a given spot, giving rise to a protuber ance, at the end of which an apical meristem is established.

They differ from other Endopterygota in the multiplication of their Malpighian tubes, and from all other Hexapoda in the union of the first abdominal segment with the thorax.

Augustus set himself against the undue multiplication of manumissions, probably considering the rapid succession of new citizens a source of social instability, and recommended a similar policy to his successor.

They protested against the multiplication of slaves from motives of vanity in the houses of the great, against the gladiatorial combats (ultimately abolished by the noble self-devotion of a monk) and against the consignment of slaves to the theatrical profession, which was often a school of corruption.

The addition and multiplication of two relation-numbers is defined by taking two relations R and S, such that (I) their fields have no Cf.

We 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.

Accordingly, the typical form for such a complex number is x+yi, and then with this notation the above-mentioned definition of multiplication is invariably adopted.

When, by practice with logarithms, we become familiar with the correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), A /5 acquires a definite meaning as the number corresponding to the extremity of a length x, on the logarithmic scale, such that 5 corresponds to the extremity of 2X.

Multiplication may or may not be commutative, and in the same way it may or may not be associative.

The commutative law for multiplication is directly illustrated; and subdivisions or groupings of the units lead to such formulae as (a+ a) (b + 0)=.

This enlargement of the bishop's parish and multiplication of the chuches under his care led to a change in the functions of the presbyterate.

From this theory, powerful in Great Britain and her colonies, supreme in the United States of America, has resulted an enormous multiplication of sects.

In France, and to a certain extent in Italy, the multiplication of chapels led to their being placed on the north and south side of the aisles, and in some cases, as at Albi in France, to the suppression of the aisles and the instalment of the chapels in their place.

Logarithms were originally invented for the sake of abbreviating arithmetical calculations, as by their means the operations of multiplication and division may be replaced by those of addition and subtraction, and the operations of raising to powers and extraction of roots by those of multiplication and division.

In all the multicellular plants of this group which have been adequately investigated, vegetative multiplication by means of what are known as hormogonia has been found to occur.

Special provision for vegetative multiplication is not common among Chlorophyceae.

In Caulerpa no other means of multiplication is as yet known.

Multiplication takes place in some cases by the endogenous formation of zoospores, the organism having come to rest; in others by longitudinal division, when the organism is still motile.

Vegetative multiplication occurs only sparingly in Rhodophyceae.

The great bulk of the code was an obstacle to the multiplication of copies of it, whilst the necessity for them was in a great degree superseded by the publication from time to time of synopses and encheiridia of its contents, composed by the most eminent jurists, of which a very full account will be found in the Histoire au droit byzantin, by the advocate Mortreuil, published in Paris in 1846.

Pellican's autobiography describes the gradual multiplication of accessible books on the subjects, and he not only studied but translated a vast mass of rabbinical and Talmudic texts, his interest in Jewish literature being mainly philological.

And by comparing one record with another, we can follow the multiplication of " holy places," and sometimes can even see them being shifted from one spot to another, as the centuries pass.

The multiplication of art periodicals, lectures, books, photographs, meetings of societies and gilds, museums, schools of arts and crafts, polytechnics, scholarships, facilities for travel, exhibitions, even those of the Royal Academy, to which objects of applied art are now admitted, not only encourages many persons to become workers and designers in the applied arts, but exposes everything to the plagiarist, who travesties the freshest idea before it has well left the hands of its originator.

For his speculations on sets had already familiarized him with the idea that multiplication might in certain cases not be commutative; so that, as the last term in the above product is made up of the two separate terms ijyz' and jizy', the term would vanish of itself when the factorlines are coplanar provided ij = - ji, for it would then assume the form ij(yz' - zy').

In particular, two species of multiplication (" inner " and " outer ") of directed lines in one plane were given.

But his claims, however great they may be, can in no way conflict with those of Hamilton, whose mode of multiplying couples (in which the " inner " and " outer " multiplication are essentially involved) was produced in 1833, and whose quaternion system was completed and published before Grassmann had elaborated for press even the rudimentary portions of his own system, in which the veritable difficulty of the whole subject, the application to angles in space, had not even been attacked.

Again a unit point p. may be regarded as by multiplication changing (a) from octonion to point-plane-scalar, (b) from point-plane-scalar to octonion, (c) from plane-scalar to linear element, (d) from linear element to plane-scalar.

As human intelligence and industry come into play the means of livelihood are proportionately extended; population multiplies, and with this multiplication production increases.

The psalms have already been dealt with, but it may be noted again how the multiplication of saints' festivals, with practically the same special psalms, tends in practice to constant repetition of about one-third of the Psalter, and correspondingly rare recital of the remaining two-thirds, whereas the Proprium de Tempore, could it be adhered to, would provide equal opportunities for every psalm.

As symbols of light and joy a candle is held on each side of the deacon when reading the Gospel at Mass; and the same symbolism underlies the multiplication of lights on festivals, their number varying with the importance of the occasion.

There is a multiplication of taxes in trade which recalls the old colonial alcabala tax, and it serves to restrict commerce and augment the cost of goods in much the same way, if not to the same degree.

It is to be noted that there is no fixed relation between toxin production and bacterial multiplication in the body, some of the organisms most active as toxin producers having comparatively little power of invading the tissues.

In the lastmentioned disease even the local multiplication depends upon the presence of other bacteria, as the tetanus bacillus has practically no power of multiplying in the healthy tissues when introduced alone.

In the 8th century the mayors of the palace are found affixing their personal seals to royal diplomas; and, once the idea was started, the multiplication of seals naturally followed.

By 1888 a new state of chaos had grown up as the result of the multiplication of bodies, and the new Redistribution Act of 1885 paved the way for a further reorganization of local matters by the Local Government Act of 1888, followed by that of 1894.

The difficulty has been increased by the confusion resulting from the multiplication of genera founded on slight characters, and from the description (in consequence of their wide distribution) of identical plants under several different genera.

The excessive multiplication of the title has tended to deprive it of much social value in itself, and under the democratic constitution of Italy it confers neither power nor precedence.

The multiplication of pilgrims after Islam soon made it necessary to clear away the nearest dwellings and enlarge the place of prayer around the Ancient House.

As an illustration, we may take the elementary processes of addition, subtraction, multiplication and division.

Or we might say that, since multiplication is a form of addition, and division a form of subtraction, there are really only two fundamental processes, viz.

The first method consists in defining the first few numbers, and forming larger numbers by groups or aggregates, formed partly by multiplication and partly by addition.

The grouping method introduces multiplication into the definition of large numbers; but this, from the teacher's point of view, is not now such a serious objection as it was in the days when children were introduced to millions and billions before they had any idea of elementary arithmetical processes.

Later, a system similar to the Hebrew was adopted, and extended by reproducing the first nine symbols of the series, preceded by accents, to denote multiplication by moo.

The application of the above principles, and of similar principles with regard to multiplication and division, to numerical quantities expressed in any of the diverse British denominations, presents no theoretical difficulty if the successive denominations are regarded as constituting a varying scale of notation (§17).

Multiplication occurs when a certain number or numerical quantity is treated as a unit (§ II), and is taken a certain number of times.

Although multiplication may arise in either of these two ways, the actual process in each case is performed by commencing with the unit and taking it the necessary number of times.

The result of the multiplication is called the product of the unit by the number of times it is taken.

Successive Multiplication.-In multiplication by repetition the unit is itself usually a multiple of some other unit, i.e.

When this new unit has been multiplied by a number, we can again take the product as a unit for the purpose of another multiplication; and so on indefinitely.

Similarly where multiplication has arisen out of the subdivision of a unit into smaller units, we can again subdivide these smaller units.

Thus we get successive multiplication; but it represents quite different operations according as it is due to repetition, in the sense of § 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of £3 by 20, 12 and 4, and B the successive reduction of £3 to shillings, pence and farthings.

Division.-From the commutative law for multiplication, which shows that 3 X 4d.

Successive Division may be performed as the converse of successive multiplication.

This gives a rule for testing multiplication, which is found in most text-books.

Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law.

The series is formed by successive multiplication, and any antilogarithm to a larger number of decimal places is formed from it in the same way by multiplication.

For practical purposes the number taken as base is so; the convenience of this being that the increase of the index by an integer means multiplication by the corresponding power of 10, i.e.

To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects.

In §§ 33 4 2 We Have Dealt With The Parallelism Of The Original Number Series With A Series Consisting Of The Corresponding Multiples Of Some Unit, Whether A Number Or A Numerical Quantity; And The Relations Arising Out Of Multiplication, Division, &C., Have Been Exhibited By Diagrams Comprising Pairs Of Corresponding Terms Of The Two Series.

For Multiplication And Division We Use A Multiplication Table, Which Is A Multiple Table, Arranged As Explained In § 36, And Giving The Successive Multiples, Up To 9 Times Or Further, Of The Numbers From 1 (Or Better, From O) To 10, 12 Or 20.

For elementary work the multiplicand may come immediately after the multiplier, as in D; the last figure of each partial product then comes immediately under the corre up to the multiplication of decimals and of approximate values of numbers, is to place the first figure of the multiplier under the first figure of the multiplicand, as in E; the first figure of each partial product will then come under the corresponding figure of the multiplier.

Contracted Multiplication.

For multiplication by a proper fraction or a decimal, it is sometimes convenient, especially when we are dealing with mixed quantities, to convert the multiplier into the sum or difference of a number of fractions, each of which has i as its numerator.

The method E of § ioi being adopted, the multiplicand and the multiplier are written with a space after as many digits (of each) as will be required in the product (on the principle explained in § 1 01); and the multiplication is performed from the left, two extra figures being kept in.

Other units are derived from these by multiplication or division by powers of io, the names being denoted by prefixes.

Multiplication, or an increase of the number of parts, gives rise to changes.

But 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.

Together, then, with the necessary multiplication of specialism, one of the chief lessons of the latter moiety of the 19th century was the unity of medicine in all its branches - a unity strengthened rather than weakened by special researches, such as those into "medical" and "surgical" pathology, which are daily making more manifest the absurdity of the distinction.

Want of cleanliness favours their multiplication in a high degree - the idea once existed, and is probably still held by the very ignorant, that they are directly engendered from dirt.

In addition, some show duplication of the gonads and of their ducts, so that we find both transverse and longitudinal repetition of these organs, without corresponding multiplication of the nervous ganglia mesenchyma, or excretory opening.

Dissolution, and finally reaches the statement of the Law of Evolution as" an integration of matter and concomitant dissipation of motion, during which the matter passes from a an indefinite incoherent homogeneity to a definite coherent heterogeneity, and during which the retained motion undergoes a parallel transformation."This process of evolution is due to" the instability of the homogeneous,"the" multiplication of effects "and their" segregation,"continuing until it ceases in complete" equilibration."Sooner or later, however, the reverse process of Dissolution, with its absorption of motion and disintegration of matter, which indeed has always been going on to some extent, must prevail, and these oscillations of the cosmic process will continue without end.

The improved cultural conditions become apparent in the multiplication of the varieties of tools, weapons and ornaments made possible by the more adaptable qualities of the new material; and that the development of the Bronze age culture in the lake dwellings followed the same course as in the surrounding regions where the people dwelt on the dry land is evident from the correspondence of the types of implements, weapons, ornaments and utensils common to both these conditions of life.

Since Metchnikoff's introduction (see Longevity) of the use of soured milk for dietetic purposes-the lactic acid bacillus destroying pathogenic bacteria in the intestine-a great impetus has been given to the multiplication of laboratory preparations containing 'cultures of the bacillus; and in recent years much benefit to health has, in certain cases, been derived from the discovery.

Nor is motility by means of cilia known in the group. In the unicellular forms, cell-division involves multiplication of the plant.

The coefficient of correlation is 0.5699, which indicates that the standard deviation of an array is equal to that of the leaves in general multiplied by 1 / I - (0.5699) 1; and performing this multiplication, we find 1.426 as the standard deviation of an array.

Cayley gave a matrix algebra defining addition, multiplication, scalar multiplication and inverses.

Math, another available topic, includes several games to help children practice their multiplication tables.

It's also easy to do a few simple multiplication and division steps to figure out how much mulch you need.

You are shown a mathematical equation that is missing the sign (addition, subtraction, multiplication and division).

The virus is believed to enter the body through the mouth with primary multiplication occurring in the lymphoid tissues in the throat, where it can persist for about one week.

In the first stage, four to 20 weeks gestation, rapid cell division and multiplication (hyperplasia) occurs as the embryo grows into a fetus.

Hypertrophy-An increase in the size of a tissue or organ brought about by the enlargement of its cells rather than cell multiplication.

Jumping rope while reciting multiplication facts helps your child to remember them as well as get some good exercise.

Teaching multiplication facts should always come before teaching division facts just like teaching addition comes before teaching subtraction.

If you employ a variety of strategies and techniques to help your child memorize their multiplication facts, learning division will be much easier.

Start by teaching the concept that repeated addition is the same as multiplication.

It's also helpful to teach children to read the multiplication sign as "groups of".

Doing this helps the child break down the concept of multiplication and makes it easier to grasp.

Another helpful thing to do is to have your child write down an entire family set of multiplication problems before having him memorize them.

Therefore, this is the next logical place to start when teaching multiplication facts.

It is small enough still that most kids are able to count by 3's and while the goal is to get them to know the multiplication problem automatically, baby steps are better than no steps towards progress!

Let your child use a multiplication chart.

School House Rock's Multiplication Songs will help your child memorize all of the times tables in record time.

You can play something like "Multiplication Go Fish" where instead of asking for a certain number of card, you ask for a multiplication problem that would yield that number.

Learning how to teach multiplication to your child effectively is an important step in your child's mastery of mathematics.

Multiplication is essential in understanding patterns in math and learning how to divide.

One of the most common mistakes in teaching multiplication is pushing students too far ahead before they are developmentally ready to grasp the material at hand.

Understand that it is common to introduce multiplication in some form by second grade but that does not mean that your child will be ready for it.

Once you break down the process of multiplication into smaller steps, it becomes easier to teach.

Rather than thinking of teaching mastery of multiplication, think instead of teaching one small portion that will lead to mastery.

In this way, you will be teaching multiplication mastery incrementally.

The first step in teaching about multiplication is helping your child to see numbers in fixed arrays.

The next step in how to teach kids multiplication is teaching them how to count by 2's, 5's and 10's.

Teaching kids that multiplication and repeated addition are the same is an important step for your child to learn how to multiply. 2 + 2 + 2 is the same 3 x 2 and both answers are six.

This is the very essence of what multiplication is.

Once your child understands that multiplication is a ways of understanding groups of numbers, you can start having her memorize multiplication facts.

Longer multiplication and division problems get very complicated when you have to use your fingers!

Graphs reinforce multiplication while bar models reinforce algebraic concepts.

Fractions do not always seem as tangible a concept to young students as the mathematical fundamentals of addition, subtraction, multiplication, and division.

The link seems to be in the fact that vitamin B12 assists with cell multiplication and division, which is also the mechanism by which tumors spread.

For example, a picture of a penny in one square, a picture of two pennies on another square, the $ sign on a third square, etc. The game can also be played for multiplication facts.

Sum Time (ages 8 and up) - This math-based board game emphasizes addition, subtraction, multiplication, and division.

Factor Frenzy - This fast-paced, get-up-and-move game focuses on multiplication facts.

You can use simple number recognition for young children, or multiplication and division problems for older children.

If you have ever had to force your child to sit and memorize their multiplication tables or other difficult concepts, you know that a fun way will inspire learning with fewer tears and headaches.

All games are reasonably priced ranging from $5.99 for a Sudoko puzzle book to $45.95 for an electronic multiplication game.

A Plus Math offers an assortment of games teaching addition, subtraction, multiplication, division and even some algebra.

Still, this simple multiplication should give you a general idea of what you will have to pay per child.

Provide a greater challenge once your child masters these math skills by asking him to give you the multiplication family for a division problem and vice versa.

Play Kids Games is an interactive site that offers math games in addition, subtraction, multiplication and division.

Some American math teachers have experimented with having their students recite multiplication tables while jumping on a mini trampoline.

We therefore define algebraical division by means of algebraical multiplication, and say that, if P and M are multinomials, the statement " P/M = Q " means that Q is a multinomial such that MQ (or QM) and P are identical.

If A 1 = X a i e i, B i = /if i e i, the distributive law of multiplication is preserved by assuming A1B1=E(a0 i 3)eiej; it follows that A 1 B 1 = - B 1 A 1, and that A l 2 = o.

The multiplication or division here involves the omission of the unit " boy," and the operation is incomplete.

The conversion of the unit may be represented as multiplication by a factor 4 boy apples, so that the operation is 4 apples X 5 boysboy 4 apples r boy Y 5 X I X I boy = 5 X 4 apples= 20 apples.

The multiplication of thongs for purposes of flogging is found in the old Roman flagellum, a scourge, which had sometimes three thongs with bone or bronze knots fastened to them.

Brooks, on the other hand, as stated above, regards the medusa as the older type and looks upon both polyp and medusa, in the Hydromedusae, as derived from a free-swimming or floating actinula, the polyp being thus merely a fixed nutritive stage, possessing secondarily acquired powers of multiplication by budding.

For multiplication, for instance, we have the statement that, if P and Q are two quantities, containing respectively p and q of a particular unit, then p X Q = q X P; or the more abstract statement that p X q= q X p.

The sum and product of two quaternions are defined by the formulae mi ase + F+lases = (a s + 133) es 2arer X ZO,es = Fiarfseres, where the products e,e, are further reduced according to the following multiplication table, in which, for example, the eo e1 e2 e3 second line is to be read eieo = e1, e 1 2 = - eo, e i e 2 = es, eie3 = - e2.

The use of the slips for the purpose of multiplication is now evident; thus to multiply 2085 by 736 we take out in this manner the multiples corresponding to 6, 3, 7, and set down the digits as they are obtained, from right to left, shifting them back one place and adding up the columns as in ordinary multiplication, viz.

The formation and gradually increasing thickness of its bark are explained by the continually increasing need of adequate protection to the living cortex, under the strain of the increasing framework which the enormous multiplication of its living protoplasts demands, and the development of which leads to continual rupture of the exterior.

Persian influence is also responsible for the vast multiplication of good spirits or angels, Gabriel, Raphael, Michael, &c., who play their part in apocalyptic works, such as the Book of Daniel and the Book of Enoch.