The operations of addition and multiplication of two given cardinal numbers can be defined by taking two classes a and 13, satisfying the conditions (1) that their cardinal numbers are respectively the given numbers, and (2) that they contain no member in common, and then by defining by reference to a and (3 two other suitable classes whose cardinal numbers are defined to be respectively the required sum and product of the cardinal numbers in question.
(iii.) The multiplication and division of monomials is effected by means of the law of indices.
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."
The addition and multiplication of these "signed" real numbers is suitably defined, and it is proved that the usual arithmetic of such numbers follows.
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.
By the multiplication of the protoplasts in these merismatic areas the substance of the plant is increased.
Has been brought about rather by extermination than specialization, and their distinctive facies by the development and multiplication of the surviving types.
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.
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 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.
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.
= (1 +Plain) (1 = er,nam; and, by multiplication, II (1 +ala+a2a2+...) = II (1-}-biP+b2P 2 +...
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.
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.
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.
Growing knowledge of Aristotle's works and the multiplication of translations enabled students to tendency.
The multiplication by 3 and the reversal of direction.
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.
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.
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.
He forms n equations from f by separate multiplication by x, ...x, I, in succession, and similarly treats 4) with m multipliers I.
= exp,udl where exp denotes (by the rule over exp) that the multiplication of operators is symbolic as in Taylor's theorem.
(vii.) The only exception that may be made to the above rule is that an expression involving multiplication-dots only, or a simple fraction written with the solidus, may have the brackets omitted for additions or subtractions, provided the figures are so spaced as to prevent misunderstanding.
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.
(iii.) The general statement of the laws of operation of fractions is perhaps best deferred until we come to fractional numbers, when letters can be used to express the laws of multiplication and division of such numbers.
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.
This latter process, however, is itself based on a definition of division in terms of multiplication (ï¿½ï¿½ 15, 16).
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.
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.
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 binary products e i e j, however, are expressible as linear functions of the units e i by means of a " multiplication table " which defines the special characteristics of the algebra in question.
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.
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.
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.
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.
Of this we see evidence in the multiplication of Satans in the Book of Enoch.
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.
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.
If we form the product A.D by the theorem for the multiplication of determinants we find that the element in the i th row and k th column of the product is akiAtil+ak2A12 +ï¿½ï¿½ï¿½ +aknAin, the value of which is zero when k is different from i, whilst it has the value A when k=i.
Application to Symmetric Function Multiplication.-An example will explain this.
(multiplication symbolic) ?r1! ?2,ï¿½..
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 idea of (-5) as a number with which we can perform such operations as multiplication comes later (ï¿½ 49)ï¿½ (ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the subdivision of a unit.
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.
All this is analogous to the corresponding formulae in the barycentric calculus and in quaternions; it remains to consider the multiplication of two or more extensive quantities The binary products of the units i are taken to satisfy the equalities e, 2 =o, i ej = - eeei; this reduces them to.
By assuming the truth of the associative law of multiplication, and taking account of the reducing formulae for binary products, - 'el ' 'e2 ' 'e3 we may construct derived units of the third, fourth ...
For instance, if n= 4, E r = e l e 3, Es= e 2 e 3 e 4, we have IErIE8 = (-e2e4) (- e 1) = ele2e4 = l e3, consequently, by the rule of regressive multiplication, eie3ï¿½e2e3e4 = e3.
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).
Multiplication may or may not be commutative, and in the same way it may or may not be associative.