See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

equation], the name of a See also:

• TREATISE

treatise by Mahommed See also:

• BEN (from Old Eng. bennan, within)

ben Musa al-Khwarizmi)  , a See also:

• BRANCH (from the Fr. branche, late Lat. branca, an animal's paw)

branch of See also:

• MATHEMATICS (Gr. /saBnµar1Kil, Sc. vOcvn or E7rlQTt'µn; from p iN.La, "learning" or "science ")

mathematics which may be defined as the generalization and See also:

• EXTENSION (Lat. ex, out ; tendere, to stretch)

extension of See also:

• ARITHMETIC (Gr. apeOµ7run7, sc. TEXVn, the art of counting, from (ipLBµos, number)

arithmetic . The subject-See also:

• MATTER

matter of See also:

• ALGEBRA (from the Arab. al-jebr wa'l-mugabala, transposition and removal [of terms of an equation], the name of a treatise by Mahommed ben Musa al-Khwarizmi)

algebra will be treated in the following See also:

• ARTICLE (from Lat. articulus, a joint)

article under three divisions:—A . Principles of See also:

• ORDINARY (med. Lat. ordinarius, Fr. ordinaire)

ordinary algebra; B . See also:

• SPECIAL

Special kinds of algebra; C . See also:

• HISTORY

History . Special phases of the subject are treated under their own headings, e.g . ALGEBRAIC FORMS; See also:

• BINOMIAL (from the Lat. bi-, bis, twice, and nomen, a name or term)

BINOMIAL; COMBINATORIAL See also:

• ANALYSIS (Gr. avci and Meta, to break up into parts)

ANALYSIS; DETERMINANTS; See also:

• EQUATION (from Lat. aequatio, aequare, to equalize)

EQUATION; CONTINUED FRACTION; See also:

• FUNCTION

FUNCTION; See also:

• GROUPS
• GROUPS, THEORY

GROUPS, THEORY OF; See also:

• LOGARITHM (from Gr. Abyos, word, ratio, and apx0µos, number)

LOGARITHM; NUMBER; See also:

• PROBABILITY (Lat. probabilis, probable or credible)

PROBABILITY; See also:

• SERIES (a Latin word from serere, to join)

SERIES . A . PRINCIPLES OF ORDINARY ALGEBRA 1 . The above See also:

• DEFINITION (Lat. definitio, from de-finire, to set limits to, describe)

definition gives only a partial view of the See also:

• SCOPE (through Ital. scopo, aim, purpose, intent, from Gr. o'KOaos, mark to shoot at, aim, o ic07reiv, to see, whence the termination in telescope, microscope, &c.)

scope of algebra . It may be regarded as based on arithmetic, or as dealing in the first instance with formal results of the See also:

• LAWS

laws of arithmetical number; and in this sense See also:

• SIR

Sir See also:

• ISAAC (Hebrew for " he laughs," on explanatory references to the name, see ABRAHAM)

Isaac See also:

• NEWTON
• NEWTON, ALFRED (1829–1907)
• NEWTON, JOHN (1725-1807)
• NEWTON, JOHN (1823-1895)
• NEWTON, SIR CHARLES THOMAS (1816-1894)
• NEWTON, SIR ISAAC (1642-1727)

Newton gave the See also:

• TITLE (0. Fr. title, mod. titre, from Lat. titulus)

title Universal Arithmetic to a See also:

• WORK

work on algebra . Any definition, however, must have reference to the See also:

• STATE
• STATE, GREAT OFFICERS OF

state of development of the subject at the See also:

• TIME (0. Eng. Lima, cf. Icel. timi, Swed. timme, hour, Dan. time; from the root also seen in " tide," properly the time of between the flow and ebb of the sea, cf. O. Eng. getidan, to happen, " even-tide," &c.; it is not directly related to Lat. tempus)
• TIME, MEASUREMENT OF
• TIME, STANDARD

time when the definition is given .

2 . The earliest algebra consists in the See also:

• SOLUTION (from Lat. solvere, to loosen, dissolve)

solution of equations . The distinction between algebraical and arithmetical reasoning then lies mainly in the fact that the former is in a more condensed See also:

• FORM (Lat. forma)

form than the latter; an unknown quantity being represented by a special See also:

• SYMBOL (Gr. o-uµ(3oXov, a sign)

symbol, and other symbols being used as a See also:

• KIND (O. E. ge-cynde, from the same root as is seen in " kin," supra)

kind of shorthand for verbal expressions . This form of algebra was extensively studied in See also:

• ANCIENT
• ANCIENT (also spelt ANTIENT; derived, through the Fr. ancien, old, from the late Lat. antianum, from ante, before)

ancient See also:

• EGYPT

Egypt; but, in accordance with the See also:

• PRACTICAL

practical tendency of the See also:

• EGYPTIAN

Egyptian mind, the study consisted largely in the treatment of particular cases, very few See also:

• GENERAL
• GENERAL (Lat. generalis, of or relating to a genus, kind or class)

general rules being obtained . 3 . For many centuries algebra was confined almost entirely to the solution of equations; one of the most important steps being the enunciation by See also:

• DIOPHANTUS

Diophantus of See also:

• ALEXANDRIA
• ALEXANDRIA (Arab. Iskenderia)

Alexandria of the laws governing the use of the minus sign . The knowledge of these laws, however, does not imply the existence of a conception of negative quantities . The development of symbolic algebra by the use of general symbols to denote See also:

• NUMBERS, BOOK OF
• NUMBERS, PARTITION OF

numbers is due to Franciscus See also:

• VIETA (or VIETE), FRANCOIS, SEIGNEUR DE LA BIGOTIERE (1540-1603)

Vieta (See also:

• FRANCOIS (1558-1614)

Francois Viete, 1J40-1603) . This led to the See also:

• IDEA (Gr. Ibia, connected with i&eiv, to see; cf. Lat. species from specere, to look at)

idea of algebra as generalized arithmetic . 4 . The See also:

• PRINCIPAL

principal step in the See also:

• MODERN

modern development of algebra was the recognition of the meaning of negative quantities . This appears to have been due in the first instance to See also:

• ALBERT
• ALBERT (1490'1545)
• ALBERT (149o-1568)
• ALBERT (1522–1557)
• ALBERT (FRANCIS CHARLES AUGUSTUS ALBERT EMMANUEL) (1819—1861)
• ALBERT (FRIEDRICH RUDOLF ALBRECHT)
• ALBERT, FREDERICK AUGUSTUS
• ALBERT, MADAME (c. 2805—2846)

Albert See also:

• GIRARD, JEAN BAPTISTE [known as " Le Pere Girard " or" Le Pere Gregoire "] (1765-185o)
• GIRARD, PHILIPPE HENRI DE (1775-1845)
• GIRARD, STEPHEN (1750-1831)

Girard (1595-1632), who extended Vieta's results in various branches of mathematics .

His work, however, was little known at the time, and later was overshadowed by the greater work of See also:

• DESCARTES, RENE (1596-1650)

Descartes (1596-1650) . 5 . The See also:

• MAIN (from the Aryan root which appears in " may " and " might," and Lat. magnus, great)
• MAIN (Lat. Moenus)

main work of Descartes, so far as algebra was concerned, was the See also:

• ESTABLISHMENT (0. Fr. establissement, Fr. etablissement, late Norm. Fr. establishement, from O. Fr. establir, Fr. etablir, Lat. stabilire, to make stable)

establishment of a relation between arithmetical and geometrical measurement . This involved not only the geometrical See also:

• INTERPRETATION (from Lat. interpretari, to expound, explain, inter pres, an agent, go-between, interpreter; inter, between, and the root pret-, possibly connected with that seen either in Greek 4 p4'ew, to speak, or irpa-rrecv, to do)

interpretation of negative quantities, but also the idea of continuity; this latter, which is the basis of modern analysis, leading to two See also:

• SEPARATE

separate but allied developments, viz° the theory of the function and the theory of limits . 6 . The See also:

• GREAT

great development of all branches of mathematics in the two centuries following Descartes has led to the See also:

• TERM

term algebra being used to See also:

• COVER (from the Fr. convert, from couvrir, to cover, Lat. cooperire)

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

• COMTE, AUGUSTE [ISIDORE AUGUSTE MARIE FRANCOIS XAVIER] (1798-1857)

Comte accordingly defined algebra as the calculus of functions, arithmetic being regarded as the calculus of values . 7 . These attempts at the unification of algebra, and its separation from other branches of mathematics, have usually been accompanied by an See also:

• ATTEMPT (Lat. adtemptare, attentare, to try)

attempt to See also:

• BASE

base it, as a deductive See also:

• SCIENCE (Lat. scientia, from scire; to learn, know)

science, on certain fundamental laws or general rules; and this has tended to increase its difficulty . In reality, the variety of algebra corresponds to the variety of phenomena . Neither mathematics itself, nor any branch or set of branches of mathematics, can be regarded as an isolated science . While, therefore, the logical development of algebraic reasoning must depend on certain fundamental relations, it is important that in the See also:

• EARLY
• EARLY, JUBAL ANDERSON (1816-1894)

early study of the subject these relations should be introduced gradually, and not until there is some empirical acquaintance with the phenomena with which they are concerned .

8 . The extension of the range of subjects to which mathematical methods can be applied, accompanied as it is by an extension of the range of study which is useful to the ordinary worker, has led in the latter See also:

• PART

part of the 19th See also:

• CENTURY (from Lat. centuria, a division of a hundred men)

century to an important reaction against the specialization mentioned in the preceding See also:

• PARAGRAPH

paragraph . This reaction has taken the form of a return to the affiance between algebra and See also:

• GEOMETRY

geometry (§5), on which modern See also:

• ANALYTICAL

analytical geometry is based; the See also:

• ALLIANCE

alliance, however, being concerned with the application of graphical methods to particular cases rather than to general expressions . These applications are sometimes treated under arithmetic, sometimes under algebra; but it is more convenient to regard graphics as a separate subject, closely allied to arithmetic, algebra, See also:

• MENSURATION (Lat. mensura, a measure)

mensuration and analytical geometry . 9 . The association of algebra with arithmetic on the one See also:

• HAND
• HAND (a word common to Teutonic languages; cf. Ger. Hand, Goth. handus)
• HAND, FERDINAND GOTTHELF (1786-185r)

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 See also:

• INTUITION (from Lat. intueri, to look at)

intuition . The difficulty first arises in elementary mensuration, where it is partly met by associating arithmetical and geometrical measurement with the See also:

• CARDINAL (Lat. cardinalis)

cardinal and the ordinal aspects of number respectively (see ARITHMETIC) . Later, the difficulty recurs in an acute form in reference to the continuous variation of a function . Reference to a geometrical interpretation seems at first sight to throw See also:

• LIGHT

light on the meaning of a See also:

• DIFFERENTIAL

differential coefficient; but closer analysis reveals new difficulties, due to the geometrical interpretation itself . One of the most See also:

• RECENT

recent developments of algebra is the algebraic theory of number, which is devised with the view of removing these difficulties . The See also:

• HARMONY (Gr. dpµovia, a concord of musical sounds, dpA'ew to join; &plan midi (sc. TeXvil)

harmony between arithmetical and geometrical measurement, which was disturbed by the See also:

• GREEK
• GREEK, ETRUSCAN AND

Greek geometers on the See also:

• DISCOVERY

discovery of irrational numbers, is restored by an unlimited See also:

• SUPPLY (through Fr. from Lat. supplere, to fill up)

supply of the causes of disturbance . 1o .

Two other developments of algebra are of special importance . The theory of sequences and series is sometimes treated as a part of elementary algebra; but it is more convenient to regard the simpler cases as isolated examples, leading up to the general theory . The treatment of equations of the second and higher degrees introduces imaginary and complex numbers, the theory of which is a special subject . I r . One of the most difficult questions for the teacher of algebra is the See also:

• STAGE (Fr. 6/age; from Lat. stare, to stand)

stage at which, and the extent to which, the ideas of a negative number and of continuity may be introduced . On the one hand, the modern developments of algebra began with these ideas, and particularly with the idea of a negative number . On the other hand, the lateness of occurrence of any particular mathematical idea is usually closely correlated with its See also:

• INTRINSIC (through Fr. intrinsique, from Lat. intrinsecus, inwardly; inter, within, secus, following, from root of sequi, to follow)

intrinsic difficulty . Moreover, the ideas which are usually formed on these points at an early stage are incomplete; and, if the incompleteness of an idea is not realized, operations in which it is implied are See also:

• APT

apt to be purely formal and See also:

• MECHANICAL

mechanical . What are called negative numbers in arithmetic, for instance, are not really negative numbers but negative quantities (§ 27 (i.)); and the difficulties incident to the ideas of continuity have already been pointed out . 12 . In the See also:

• PRESENT

present article, therefore, the main portions of elementary algebra are treated in one See also:

• SECTION
• SECTION (Lat. sectio, cutting, secare, to cut)

section, without reference to these ideas, which are considered generally in two separate sections . These three sections may therefore be regarded as to a certain extent concurrent .

They are preceded by two sections dealing with the introduction to algebra from the arithmetical and the graphical sides, and are followed by a section dealing briefly with the developments mentioned in §§ 9 and 10 above . I . Arithmetical Introduction to Algebra . 13 . See also:

• ORDER
• ORDER (through Fr. ordre, for earlier ordene, from Lat. ordo, ordinis, rank, service, arrangement; the ultimate source is generally taken to be the root seen in Lat. oriri, rise, arise, begin; cf. " origin ")
• ORDER, HOLY

Order of Arithmetical Operations.—It is important, before beginning the study of algebra, to have a dear idea as to the meanings of the symbols used to denote arithmetical operations . (i.) Additions and subtractions are performed from See also:

• LEFT

left to right . Thus 3 lb + 5 lb— 7 lb + 2 lb means that 5 lb is to be added to 3 lb, 7 lb subtracted from the result, and 2 lb added to the new result . (ii.) The above operation is performed with r lb as the unit of counting, and the See also:

• PROCESS

process would be the same with any other unit; e.g. we should perform the same process to find 3s.+5s.—7s.+2s . Hence we can separate the numbers from the See also:

• COMMON

common unit, and replace 3 lb+5lb - 7lb+2lb by (3+5—7+2) lb, the additions and subtractions being then performed by means of an addition-table . (iii.) Multiplications, represented by X, are performed from right to left . Thus 5 X3 X7 X I lb means 5 times 3 times 7 times 1 lb; i.e. it means that r lb is to be multiplied by 7, the result by 3, and the new result by 5 . We may regard this as meaning the same as 5X3 X7 lb, since 7 lb itself means 7X 1 lb, and the lb is the unit in each See also:

• CASE
• CASE, JOHN (d. 1600)

case .

But it does not mean the same as 5X 21 lb, though the two are equal, i.e. give the same. result (see § 23) . This See also:

• RULE

rule as to the meaning of X is important . If it is intended that the first number is to be multiplied by the second, a special sign such as x should be used . (iv.) The sign means that the quantity or number preceding it is to be divided by the quantity or number following it . (v.) The use of the solidus / separating two numbers is for convenience of See also:

• PRINTING (from Lat. imprimere, O. Fr. empreindre)

printing fractions or fractional numbers . Thus 16/4 does not mean 16+4, but Y . (vi.) Any See also:

• COMPOUND
• COMPOUND (from Lat. componere, to combine or put together)

compound operation not coming under the above descriptions is to have its meaning made clear by brackets; the use of a pair of brackets indicating that the expression between them is to be treated as a whole . Thus we should not write 8X7+6, but (8X7)+6, or 8X(7+6) . The sign X coming immediately before, or immediately after, a See also:

• BRACKET

bracket may be omitted; e.g . 8X(7+6) may be written 8(7+6) . This rule as to using brackets is not always observed, the See also:

• CONVENTION (Lat. conventio, an assembly or agreement, from convenire, to come together)
• CONVENTION, THE NATIONAL

convention sometimes adopted being that multiplications or divisions are to be performed before additions or subtractions . The convention is even pushed to such an extent as to make " 42+33 of 7+5 " mean 42+(33 of 7)+5 "; though it is not clear what "Find the value of 42+31 times 7+5" would then mean .

There are See also:

• GRAVE

grave objections to an arbitrary rule of this kind, the See also:

• CHIEF (from Fr. chef, head, Lat. caput)

chief being the useless See also:

• WASTE (O. Fr. wast, guast, gast, gaste; Lat. vastus, vast, desolate)

waste of See also:

• MENTAL

mental See also:

• ENERGY (from the Gr. ivipyela; iv, in, ipyov, work)

energy in remembering it . (vii.) The only exception that may be made to the above rule is that an expression involving multiplication-dots only, or a See also:

• SIMPLE

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 . Thus 8+(7X6)+3 may be written 8+7.6+3, and 8+3+3 may be written 8+7/6+3 . But should be written (3.5)/(2.4), not 3.5/2.4 . 2 .4 14 . Latent Equations.—The equation exists, without being shown as an equation, in all those elementary arithmetical processes which come under the See also:

• HEAD (in 0. Eng. heafad; the word is common to Teutonic languages; cf. Dutch hoofd, Ger. Haupt, generally taken to be in origin connected with Lat. caput, Gr. KerbOvi7)
• HEAD, SIR EDMUND WALKER, BART
• HEAD, SIR FRANCIS BOND, BART

head of inverse operations; i.e. processes which consist in obtaining an See also:

• ANSWER (derived from and, against, and the same root as swear)

answer to the question " Upon what has a given operation to be performed in order to produce a given result?" or to the question " What operation of a given kind has to be performed on a given quantity or number in order to produce a given result?" (i.) In the case of subtraction the second of these two questions is perhaps the simpler . Suppose, for instance,,that we wish to know how much will be left out of 1os. after spending 3s., or how much has been spent out of See also:

• ROS, or DE
• ROS, SIR RICHARD (b. 1429)

ros. if 3s. is left . In either case we may put the question in two ways: (a) What must be added to 3s. in order to produce 1os., or (b) To what must 3s. be added in order to produce 1os . If the answer to the question is X, we have either (a) ros . = 3s . +X, .. X =1os .

-3s . or (b) Ios.=X+3s., ..X=1os.—3s . (ii.) In the above case the two different kinds of statement See also:

• LEAD
• LEAD (pronounced iced)

lead to arithmetical formulae of the same kind . In the case of See also:

• DIVISION (from Lat. dividere, to break up into parts, separate)

division we get two kinds of arithmetical See also:

• FORMULA
• FORMULA (Lat. diminutive of forma, shape, pattern, &c., especially used of rules of judicial procedure)

formula, which, however, may be regarded as requiring a single kind of numerical process in order to determine the final result . (a) If 24d. is divided into 4 equal portions, how much will each portion be ? Let the answer be X; then 24d . =4XX, .. X=; of 24d . (b) Into how many equal portions of 6d. each may 24d. be divided ? Let the answer be x; then 24d . =xX6d., .'. x=24d . 6d .

(iii.) Where the See also:

• DIRECT

direct operation is See also:

• EVOLUTION

evolution, for which there is no commutative See also:

• LAW
• LAW (0. Eng. lagu, M. Eng. lawe; from an old Teutonic root lag, " lie," what lies fixed or evenly; cf. Lat. lex, Fr. loi)
• LAW, JOHN (1671—1729)
• LAW, WILLIAM (1686-1761)

law, the two inverse operations are different in kind . (a) What would be the dimensions of a cubical See also:

• VESSEL (O. Fr. vaissel, from a rare Lat. vascellum, dim. of vas, vase, urn)

vessel which would exactly hold 125 litres; a litre being a cubic decimetre ? Let the answer be X; then 125 c.dm . = X3, .. X = 125 c.dm . = ;I 125 dm . (b) To what See also:

• POWER [WILLIAM GRATTAN] TYRONE (1797-1841)

power must 5 be raised to produce 125 ? Let the answer be x; then See also:

• I25

I25=5,.. x =log6 125 . 15 . With regard to the above, the following paints should be noted . (i) When what we require to know is a quantity, it is simplest to See also:

• DEAL

deal with this quantity as a whole . In (i.), for instance, we want to find the amount by which Ios. exceeds 3s., not the number of shillings in this amount .

It is true that we obtain this result by subtracting 3 from Io by means of a subtraction-table (See also:

• CONCRETE
• CONCRETE (Lat. concretes, participle of concrescere, to grow together)

concrete or ideal); but this table merely gives the generalized results of a number of operations of addition or subtraction performed with concrete See also:

• UNITS, DIMENSIONS OF
• UNITS, PHYSICAL

units . We must See also:

• COUNT (Lat. comes, gen. comitis, Fr. comte, Ital. conte, Span. conde)

count with something; and the successive somethings obtained by the addition of successive units are in fact numerical quantities, not numbers . Whether this principle may legitimately be extended to the notation adopted in (iii.) (a) of § 14 is a See also:

• MOOT

moot point . 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 See also:

• RATE

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 . (2) The results may be stated briefly as follows, the more usual form being adopted under (iii.) (a): (i.) If A=B+X, or=X+B, then X=A-B . (ii.) (a) If A=m times X, then X= m of A . (b) If A=x times M, then x=A=M . (iii.) (a) If n=xP, then xn . (b) If n=as, then x=See also:

• LOG (a word of uncertain etymological origin,possibly onomatopoeic; the New English Dictionary rejects the derivation from Norwegian lag, a fallen tree)

log. n . The important thing to See also:

• NOTICE

notice is that where, in any of these five cases, one statement is followed by another, the second is not to be regarded as obtained from the first by logical reasoning involving such general axioms as that " if equals are taken from equals the remainders are equal "; the fact being that the two statements are merely different ways of expressing the same relation . To say, for instance, that X is equal to A - B, is the same thing as to say that X is a quantity such that X and B, when added, make up A; and the above five statements of necessary connexion between two statements of equality are in fact nothing more than See also:

• DEFINITIONS

definitions of the symbols - , I mm of,=, '/, and log .. An apparent difficulty is that we use a single symbol - to denote the result of the two different statements in (i.) (a) and (i.) (b) of § 14 .

This is due to the fact that there are really two kinds of subtraction, respectively involving counting forwards (complementary addition) and counting backwards (ordinary subtraction); and it suggests that it may be See also:

• WISE, HENRY ALEXANDER (1806-1876)
• WISE, ISAAC MAYER (1819-1900)

wise not to use the one symbol - to represent the result of both operations until the commutative law for addition has been fully grasped . .16 . In the same way, a statement as to the result of an inverse operation is really, by the definition of the operation, a statement as to the result of a direct operation . If, for instance, we state that A= X - B, this is really a statement that X= A+B . Thus, corresponding to the results under § 15 (2), we have the following: (1) Where the inverse operation is performed on the unknown quantity or number: (i.) If A=X-B, then X=A+B . (ii.) (a) If M=m of X, then X=m times M . (b) If m=X=M, then X=m times M . (iii.) (a) If a=Vx, then x=aP . (b) If p = log.x, then x =a' . (2) Where the inverse operation is performed with the unknown quantity or number: (i.) If B=A-X, then A=B+X . (ii.) (a) If m=A=X, then A=m times X . (b) If M=z of A, then A=x times M .

(iii.) (a) Ifp=login, then n=xP . (b) If a ='/ n, then n =a' . In each of these cases, however, the reasoning which enables us to replace one statement by another is of a different kind from the reasoning in the corresponding cases of § 15 . There we proceeded from the direct to the inverse operations; i.e. so far as the nature of arithmetical operations is concerned, we launched out on the unknown . In the present section, however, we return from the inverse operation to the direct; i.e. we rearrange our statement in its simplest form . The statement, for instance, that 32- X=25, is really a statement that 32 is the sum of x and 25 . 17 . The five equalities which stand first in the five pairs of equalities in § 15 (2) may therefore be taken as the main types of a simple statement of equality . When we are See also:

• FAMILIAR (through the Fr. familier, from Lat. familiaris, of or belonging to the familia, family)

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 five processes of See also:

• DEDUCTION (from Lat. deducere, to take or lead from or out of, derive)

deduction then reduce to four, which may be described as (i.) subtraction, (ii.) division, (iii.) (a) taking a See also:

• ROOT (late O.E. rot, adopted from Scand., cf. Norw. and Swed. rot, Dan. rod; the true O.E. word was wyrt, plant, represented in Ger. Wurz or Wurzel; the ultimate root is the same in both words, and is seen in Lat. radix)
• ROOT, ELIHU (1845– )

root, (iii.) (b) taking logarithms . It will be found that these (and particularly the first three) cover practically all the processes legitimately adopted in the elementary theory of the solution of equations; other processes being sometimes liable to introduce roots which do not satisfy the See also:

• ORIGINAL

original equation . 18 .

It should be noticed that we are still dealing with the elementary processes of arithmetic, and that all the numbers contemplated in §§ 14-17 are supposed to be See also:

• POSITIVE (or PORTABLE) ORGAN

positive integers . If, for instance, we are told that 15= i of (x - 2), what is meant is that (I) there is a number u such that X=U-{-2, (2) there is a number v such that u=4 times v, and (3) 15=3 times v . From these statements, working backwards, we find successively that v=5, U=20, x= 22 . The deductions follow directly from the definitions, and such mechanical processes as "clearing of fractions " find no See also:

• PLACE (through Fr. from Lat. platea, street; Gr. IrAar6s, wide)

place (§ 21 (ii.)) . The extension of the methods to fractional numbers is part of the establishment of the laws governing these numbers (§ 27 (ii.)) . Iq . Expressed Equations.—The simplest forms of arithmetical equation arise out of abbreviated solutions of particular problems . In accordance with § 15, it is desirable that our statements should be statements of equality of quantities rather than of numbers; and it is convenient in the early stages to have a distinctive notation, e.g. to represent the former by See also:

• CAPITAL (i.e. capital stock or fund)
• CAPITAL (Lat. caput, head)

capital letters and the latter by small letters . As an example, take the following . I buy 2 lb of See also:

• TEA (Chinese cha, Amoy dialect t€)

tea, and have 6s . 8d. left out of Icis.; how much per lb did tea cost ? (I) In ordinary See also:

• LANGUAGE (adapted from the Fr. langage, from langue, tongue, Lat. lingua)

language we should say: Since 6s .

8d. was left, the amount spent was Ios . - 6s . 8d., i.e. was 3s . 4d . There-fore 2 lb of tea cost 3S . 4d . Therefore 1 lb of tea cost Is . 8d . (2) The first step towards arithmetical reasoning in such a case is the introduction of the sign of equality . Thus we say: Cost of 2 lb tea+6s . 8d . =1os .

. . Cost of 2 lb tea =Ios . -6s . 8d . =3s . 4d . : . Cost of 1 lb tea = Is . 8d (3) The next step is to show more distinctly the unit we are dealing with (in addition to the See also:

• MONEY

money unit), viz. the cost of r lb tea . We write: (2 Xcost of i lb tea) +6s . 8d . = ros .

2 Xcost of r lb tea =10S . -6s . 8d . =3s . 4d . Cost of I lb tea = Is . 8d . (4) The stage which is See also:

• INTRODUCTORY

introductory to algebra consists merely in replacing the unit " cost of i lb tea " by a symbol, which may be a See also:

• LETTER (through Fr. lettre from Lat. littera or litera, letter of the alphabet; the origin of the Latin word is obscure; it has probably no connexion with the root of linere, to smear, i.e. with wax, for an inscription with a stilus)

letter or a See also:

• MARK
• MARK, CARL (1858– )
• MARK, GOSPEL OF ST
• MARK, ST

mark such as the mark of interrogation, the See also:

• ASTERISK (from Gr. oo-repicreos, a little star)

asterisk, &c . If we denote this unit by X, we have (2XX) +6s . 8d . = ros . 2XX = IOS .

– 6s . 8d . =3S . 4d . X=Is.8d . 20 . Notation of Multiples.—The above is arithmetic . The only thing which it is necessary to import from algebra is the notation by which we write 2X instead of 2 X X or 2 . X . This is rendered possible by the fact that we can use a single letter to represent a single number or numerical quantity, however many digits are contained in the number . It must be remembered that, if a is a number, 3a means 3 times a, not a times 3; the latter must be represented by aX3 or a .3 . The number by which an algebraical expression is to, be multiplied is called its coefficient .

Thus in 3a the coefficient of a is 3 . But in 3 .4a the coefficient of 4a is 3, while the coefficient of a is 3.4 . 21 . Equations with Fractional Coefficients.—As an example of a special form of equation we may take ix+ix = to . (i.) There are two ways of proceeding . (a) The statement is that (I) there is a number u such that x= 2u,(2) there is a number v such that x=3v, and (3) u+v= 10 . We may therefore conveniently take as our unit, in place of x, a number y such that x=6y . We then have 3Y+2y= 10, whence 5y=1o, y=2, x=6y=12 . (b) We can collect coefficients, i.e. combine the separate quantities or numbers expressed in terms of x as unit into a single quantity or number so expressed, obtaining ix= to . By successive stages we obtain (§ r8) X=2, x=12; or we may write at once x=5~ of ro=5 of ro=12 . The latter is the more advanced process, implying some knowledge of the laws of fractional numbers, as well as an application of the associative law (§ 26 (i.)) . (ii.) Perhaps the worst thing we can do, from the point of view of intelligibility, is to " clear of fractions " by multiplying both sides by 6 .

It is no doubt true that, if Ix+ix=to, then 3x+ 2x=6o (and similarly if 2x+ax+sx=ro, then 3x+2x+x=6o); but the fact, however interesting it may be, is of no importance for our present purpose . In 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), sQ that the See also:

• TOTAL

total (viz. to) is unaltered . 22 . Arithmetical and Algebraical Treatment of Equations.—The following will illustrate the passage from arithmetical to algebraical reasoning . " See also:

• COAL

Coal See also:

• COSTS

costs 3S. a ton more this See also:

• YEAR

year than last year . If 4 tons last year cost I04s., how much does a ton cost this year ? " If we write X for the cost per ton this year, we have 4(X–3s.) = 104s . From this we can deduce successively X - 3s . = 26s., X = 2qs . But, if we transform the equation into 4X– I2s . = 104s., we make an essential alteration . The original statement was with regard to X-3s. as the unit; and from this, by the application of the distributive law (§ 26 (i.)), we have passed to a statement with regard to X as the unit .

This is an algebraical process . In the same way, the transition from (x2+4x+4)-4=21 to X2+4X+4= 25, or from (x+2)2=25 to x+2=s/25, is arithmetical; but the transition from x2+4x+4= 25, to (x+2)2= 25 is algebraical, since it involves a See also:

• CHANGE (derived through the Fr. from the Late Lat. cambium, cambiare, to barter; the ultimate derivation is probably from the root which appears in the Gr. «a urmu', to bend)

change of the number we are thinking about . Generally, we may say that algebraic reasoning in reference to equations consists in the alteration of the form of a statement rather than in the deduction of a new statement; i.e. it cannot be said that " If A=B, then E = F " is arithmetic, while " If C= D, then E = F " is algebra . Algebraic treatment consists in replacing either of the terms A or B by an expression which we know from the laws of arithmetic to be See also:

• EQUIVALENT

equivalent to it . The subsequent reasoning is arithmetical . 23 . Sign of Equality.—The various meanings of the sign of equality (=) must be distinguished . (i.) 4X3lb=12 lb . This states that the result of the operation of multiplying 3 lb by 4 is 12 lb . (ii.) 4X3 lb=3X4 lb . This states that the two operations give the same result; i.e that they are equivalent . (iii.) A's See also:

• SHARE (O. Eng. scearu, chiefly in compounds, e.g. land-scearu, a share of land, from sceran to cut; cf. " shear" )

share =5s., or 3 times A's share = 15s .

Either of these is a statement of fact with regard to a particular quantity; it is usually called an equation, but sometimes a conditional equation, the term " equation " being then extended to cover (i.) and (ii.) . (iv.) x3=xXxXx . This is a definition of x3; the sign = is in such cases usually replaced by = .. (v.) .24d . =2s . This is usually regarded as being, like (ii.), a statement of equivalence . It is, however, only true if is. is equivalent to 12d., and the correct statement is then IS . X24d . =2S . 12d . If the operator j 5 X is omitted, the statement is really an equation, giving Is. in terms of Id. or See also:

• VICE

vice versa . The following statements should be compared: X=A's share =1 of Io=3X£5=£15- X =A's share= of £io= of 30 =415 .

In each case, the first sign of equality comes under (iv.) above, the second under (iii.), and the See also:

• FOURTH

fourth under (i.); but the third sign comes under (i.) in the first case (the statement being that i of £ro=5) and under (ii.) in the second . It will be seen from § 22 that the application of algebra to equations consists in the interchange of equivalent expressions, and therefore comes under (i.) and (ii.) . We replace 4(x-3), for instance, by 4x-4.3i because we know that, whatever the value of x may be, the result of subtracting 3 from it and multiplying the See also:

• REMAINDER, REVERSION

remainder by 4 is the same as the result of finding 4x and 4.3 separately and subtracting the latter from the former . A statement such as (i.) or (ii.) is sometimes called an identity . The two expressions whose equality is stated by an equation or an identity are its members . 24 . Use of Letters in General Reasoning.—It may be assumed that the use of letters to denote quantities or numbers will first arise in dealing with equations, so that the letter used will in each case represent a definite quantity or number; such general statements as those of §§ 15 and 16 being deferred to a later stage . In addition to these, there are cases in which letters can use-fully, be employed for general arithmetical reasoning . (i.) There are statements, such as A+B=B+A, which are particular cases of the laws of arithmetic, but,need not be ex-pressed as such . 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 pXQ=qXP; or the more abstract statement that pXq=qXp . (ii.) The general theory of ratio and proportion requires the. use of general symbols . (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 See also:

• EXPRESS (through the French from the past participle of the Lat. exprimere, to press out, by transference used of representing objects in painting or sculpture, or of thoughts, &c. in words)

express the laws of multiplication and division of such numbers .

(iv.) Variation is generally included in See also:

• TEXT (Lat. textum, woven fabric, from texere, to weave)

text-books on algebra, but apparently only because the reasoning is general . It is part of the general theory of quantitative relation; and in its elementary stages is a suitable subject for graphical treatment (§ 31) . 25 . Preparation for Algebra.—The calculation of the values of simple algebraical expressions for particular values of letters involved is a useful exercise, but its tediousness is apt to make the subject repulsive . What is more important is to verify particular examples of general formulae . These formulae are of two kinds:—(a) the general properties, such as m(a+b) = ma+mb, on which algebra is based, and (b) particular formulae such as (x–a)(x+a) = x'–a' . Such verifications are of value for two reasons . In the first place, they lead to an understanding of what is meant by the use of brackets and by such a statement as 3(7+2) =3.7+3 .2 . This does not mean (cf . § 23) that the algebraic result of performing the operation 3(7+2) is 3 • 7+3 • 2; it means that if we convert 7+2 into the single number 9 and then multiply by 3 we get the same result as if we converted 3 . 7 and 3 . 2 into 21 and 6 respectively and added the results .

In the second place, particular cases See also:

• LAY

lay the See also:

• FOUNDATION (Lat. fundatio, from fundare, to found)

foundation for the general formula . Exercises in the collection of coefficients of various letters occurring in a complicated expression are usually performed mechanically, and are probably of very little value . 26 . General Arithmetical Theorems . (i.) The fundamental laws of arithmetic (q.v.) should be constantly See also:

• BORNE, KARL LUDWIG (1786–1837)

borne in mind, though not necessarily stated . The following are some special points . (a) The commutative law and the associative law are closely related, and it is best to establish each law for the case of two numbers before proceeding to the general case . In the case of addition, for instance, suppose that we are satisfied that in a+b+c+d+e we may take any two, as b and c, together (association) and interchange them (See also:

• COMMUTATION (from Lat. commutare, to change)

commutation) . Then we have a+b+c+d+e=a+c+b+d+e . Thus any pair of adjoining numbers can be interchanged, so that the numbers can be arranged in any order . (b) The important form of the distributive law is m(A+B) = mA+mB . The form (m+n)A=mA+nA follows at once from the fact that A is the unit with which we are dealing .

(c) The fundamental properties of subtraction and of division are that A–B +B = A and m X m of A= A, since in each case the second operation restores the original quantity with which we started . (ii.) The elements of the theory of numbers belong to arithmetic . In particular, the theorem that if n is a See also:

• FACTOR (from Lat. facere, to make or do)

factor of a and of See also:

• BIT (from the verb " to bite," either in the sense of a piece bitten off, or an act of biting, or a thing that bites or is bitten)

bit is also a factor of pa~gb, 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 . Graphic methods are useful here (§ 34 (iv.)) . The law of relation of successive convergents to a continued fraction involves more advanced methods (see § 42 (iii.) and CONTINUED FRACTION) . (iii.) There are important theorems as to the relative value of fractions; e.g . (a) If a =d, then each =Paqc b qd' is nearer to I than b is; and, generally, if E,a (b) , then pa+qc lies between the two . (All the numbers are, of course, pb+qd supposed to be positive.) 27 . Negative Quantities and Fractional Numbers.—(i.) What are usually called " negative numbers " in arithmetic are in reality not negative numbers but negative quantities . If a See also:

• PERSON, OFFENCES AGAINST THE

person has to receive 7s. and pay 5s., with a See also:

• NET

net result of +2s., the order of the operations is immaterial . If he pays first, he then has -5s . This is sometimes treated as a See also:

• DEBT (Lat. debitum, a thing owed)

debt of 5s.; an alternative method is to recognize that our zero is really arbitrary, and that in fact we shift it with every operation of addition or subtraction .

But when we say " -5s." we mean "–(5s.)," not " (–5)s."; the idea of (–5) as a number with which we can, perform such operations as multiplication comes later (§ 49) . (ii.) On the other hand, the conception of a fractional number follows directly from the use of fractions, involving the sub-division of a unit . We find that fractions follow certain lawscorresponding exactly with those of integral multipliers, and we are therefore able to deal with the fractional numbers as if they were integers . 28 . See also:

• MISCELLANEOUS

Miscellaneous Developments in Arithmetic.—The following are matters which really belong to arithmetic; they are usually placed under algebra, since the general formulae involve the use of letters . (i.) Arithmetical Progressions such as 2, 5, 8, .—The formula for the rth term is easily obtained . The problem of finding the sum of r terms is aided by graphic See also:

• REPRESENTATION

representation, which shows that the terms may be taken in pairs, working from the outside to the See also:

• MIDDLE

middle; the two cases of an See also:

• ODD (in middle English odde, from old Norwegian oddi, an angle of a triangle; the old Norwegian oddamann is used of the third man who gives a casting vote in a dispute)

odd number of terms and an even number of terms may be treated separately at first, and then combined by the ordinary method, viz. See also:

• WRITING
• WRITING (the verbal noun of " to write," O. Eng. writan, to inscribe)

writing the series backwards . In this, as in almost all other cases, particular examples should be worked before obtaining a general formula . (ii.) The law of indices (positive integral indices only) follows at once from the definition of a', a', a', . . . as abbreviations of a.a, a.a.a, a.a.a.a, . . ., or (by See also:

• ANALOGY (Gr. avaXo-yLa, proportion)

analogy with the definitions of 2, 3, 4, . . . themselves) of a.a, a.a', a.a', .

. . successively . The treatment of roots and of logarithms (all being positive integers) belongs to this subject; a={/n and p=log,,n being the inverses of n=aP (cf . §§ 15, 16) . The theory may be extended to the cases of p= i and p = o; so that a' means a.a.a.1, a' means a.a.r, a' means a.I, and a° means 1 (there being then none of the multipliers a) . The terminology is sometimes confused . In n=aP, a is the root or base, p is, the See also:

• INDEX

index or logarithm, and n is the power or antilogarithm . Thus a, a', a', . . . are the first, second, third, . . . See also:

• POWERS, HIRAM (1805-1873)

powers of a . But aP is sometimes incorrectly described as " a to the power p "; the power being thus confused with the index or logarithm . (iii.) Scales of Notation lead, by considering, e.g., how to express in the See also:

• SCALE
• SCALE (1) A

scale of lo a number whose expression in the scale of 8 is 2222222, to (iv.) Geometrical Progressions.—It should be observed that the radix of the scale is exactly the same thing as the root mentioned under (ii.) above; and it is better to use the term " root " throughout . Denoting the root by a, and the number 2222222 in this scale by N, we have N = 2222222 .

aN=22222220 . Thus by adding 2 to aN we can subtract N from aN+ 2, obtaining 20000000, which is = 2 . a'; and from this we easily pass to the general formula for the sum of a geometrical progression having a given number of terms . (v.) Permutations and Combinations may be regarded as arithmetical recreations; they become important algebraically in reference to the binomial theorem (§§ 41,44)• (vi.) Surds and Approximate Logarithms.—From the arithmetical point of view, surds present a greater difficulty than negative quantities and fractional numbers . We cannot solve the equation 7s.+X=4s.; but we are accustomed to transactions of lending and borrowing, and we can therefore invent a negative quantity -3s. such that -3s.+3s.=0 . We cannot solve the equation 7X=4s.; but we are accustomed to subdivision of units, and we can therefore give a meaning to X by inventing a unit +s. such that 7 X s = Is., and can thence pass to the idea of fractional numbers . When, however, we come to the equation x2 = 5, where we are dealing with numbers, not with quantities, we have no concrete facts to assist us . We can, however, find a number whose square shall be as nearly equal to 5 as we please, and it is this number that we treat arithmetically as 15 . We may take it to (say) 4 places of decimals; or we may suppose it to be taken to See also:

• I000

I000 places . In actual practice, surds mainly arise out of mensuration; and we can then give an exact definition by graphical methods . When, by practice with logarithms, we become familiar with the See also:

• CORRESPONDENCE
• CORRESPONDENCE (from med. scholastic Lat. correspondentia, corrcspondere, compounded of Lat. cum, with, and respond ere, to answer; cf. Fr. correspondance)

correspondence between additions of length on the logarithmic scale (on a slide-rule) and multiplication of numbers in the natural scale (including fractional numbers), 115 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 . Thus the concrete fact required to enable us to pass arithmetically from the conception of a fractional number to the conception of a surd is the fact of performing calculations by means of logarithms . In the same way we regard log,oa, not as a new kind of number, but as an approximation .

(vii.) The use of fractional indices follows directly from this See also:

• PARALLELISM, PSYCHOPHYSICAL

parallelism . We find that the product a'" X X am is equal to aim; and, by definition, the product ;la X;la X Ja is equal to a, which is al . This suggests that we should write ;la as al/3; and we find that the use of fractional indices in this way satisfies the laws of integral indices . It should be observed that, by analogy with the definition of a fraction, aP/q mean (al/q)P, not (aP)hJ . II . Graphical Introduction to Algebra . 29 . The science of graphics is closely related to that of mensuration . While mensuration is concerned with the representation of geometrical magnitudes by numbers, graphics is concerned with the representation of numerical quantities by geometrical figures, and particularly by lengths . An important development, covering such diverse matters as the See also:

• EQUILIBRIUM (from the Lat. aequus, equal, and libra, a balance)

equilibrium of forces and the algebraic theory of complex numbers (§ 66), has relation to cases where the numerical quantity has direction as well as magnitude . There are also cases in which graphics and mensuration are used jointly; a variable numerical quantity is represented by a graph, and the principles of men§uration are then applied to determine related numerical quantities . General aspects of the subject are considered under MENSURATION; VECTOR ANALYSIS; INFINITESIMAL CALCULUS .

30 . The elementary use of graphic methods is qualitative rather than quantitative; i.e. it is for purposes of See also:

• ILLUSTRATION

illustration and See also:

• SUGGESTION

suggestion rather than for purposes of deduction and exact calculation . We start with related facts, and adopt a particular method of visualizing the relation . One of the relations most commonly illustrated in this way is the time-relation; the passage of time being associated with the passage of a point along a straight See also:

• LINE

line, so that equal intervals of time are represented by equal lengths . 31 . It is important to begin the study of graphics with concrete cases rather than with tracing values of an algebraic function . Simple examples of the time-relation are—the number of scholars present in a class, the height of the See also:

• BAROMETER (from Gr. (3apoc, pressure, and µETpov, measure)

barometer, and the See also:

• READING

reading of the thermometer, on successive days . Another useful set of graphs comprises those which give the relation between the expressions of a length, See also:

• VOLUME

volume, &c., on different systems of measurement . Mechanical, commercial, economic and statistical facts (the latter usually involving the time-relation) afford numerous examples . 32 . The ordinary method of representation is as follows . Let X and Y be the related quantities, their expressions in terms of selected units A and B being x and y, so that X=x.A, Y = y .

B . For graphical representation we select units of length L and M, not necessarily identical . We take a fixed line OX, usually See also:

• DRAWN

drawn horizontally; for each value of X we measure a length or See also:

• ABSCISSA (from the Lat. abscissus, cut off)

abscissa ON equal to x.L, and draw an See also:

• ORDINATE

ordinate NP at right angles to OX and equal to the corresponding value of y . M . The assemblage of ordinates NP is then the graph of Y . The series of values of X will in general be discontinuous, and the graph will then be made up of a See also:

• SUCCESSION (Lat. successio, from succedere, to follow after)

succession of parallel and (usually) equidistant ordinates . When the series is theoretically continuous, the theoretical graph will be a continuous figure of which the lines actually drawn are ordinates . The upper boundary of this figure will be a line of some sort; it is this line, rather than the figure, that is sometimes called the " graph." It is better, however, to treat this as a secondary meaning . In particular, the equality or inequality of values of two functions is more readily grasped by comparison of the lengths of the ordinates of the graphs than by inspection of the relative positions of their bounding lines . 33• The importance of the bounding line of the graph lies in the fact that we can keep it unaltered while we alter the graph as a whole by moving OX up or down . We might, for instance, read temperature from 6o° instead of from o° . Thus we form the conception, not only of a zero, but also of the arbitrariness of position of this zero (cf .

§ 27 (i.)); and we are assisted to the conception of negative quantities . On the other hand. the alteration in the direction of the bounding line, due to alteration in the unit of measurement of Y, is useful in relation to geometrical See also:

• PROJECTION

projection . This, however, applies mainly to the representation of values of Y . Y is represented by the length of the ordinate NP, so that the representation is cardinal; but this ordinate really corresponds to the point N, so that the representation of X is ordinal . It is therefore only in certain special cases, such as those of simple time-relations (e.g . " J is aged 40, and K is aged 26; when will J be twice as old as K?"), that the graphic method leads without arithmetical reasoning to the properties of negative values . In other cases the continuation of the graph may constitute a dangerous extrapolation . 34 . Graphic representation thus rests on the principle that equal numerical quantities may be represented by equal lengths, and that a quantity mA may be represented by a length mL, where A and L are the respective units; and the science of graphics rests on the converse See also:

• PROPERTY

property that the quantity represented by pL is pA, i.e. that pA is determined by finding the number of times that L is contained in pL . The 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 . (i.) To See also:

• DIVIDE

divide P by Q, we cut off from P successive portions each equal to Q, till we have a piece R left which is less than Q . Thus P=kQ+R, where k is an integer .

(ii.) To continue the division we may take as our new unit a submultiple of Q, such as Q/r, where r is an integer, and repeat the process . We thus get P=kQ+m.Q/r+S=(k+m/r)Q+S, where S is less than Qfr . Proceeding in this way, we may be able to express P4 -Q as the sum of a finite number of terms k+m/r+n/See also:

• R2(rt-s2)

r2+ . . . ; or, if r is not suitably chosen, we may not . If, e.g. r= ro, we get the ordinary expression of P/Q as an integer and a decimal; but, if P/Q were equal to 1/3, we could not express it as a decimal with a finite number of figures . (iii.) In the above method the choice of r is arbitrary . We can avoid this arbitrariness by a different See also:

• PROCEDURE (Fr. procedure, from Lat. procedere, to go for-ward)

procedure . Having obtained R, which is less than Q, we now repeat with Q and R the process that we adopted with P and Q; i.e. we cut off from Q successive portions each equal to R . Suppose we find Q=sR+T, then we repeat the process with R and T; and so on . We thus express P+ Q in the form of a continued fraction, k + s +I I ,which isusually written, forconciseness, k+s t-r{+&c., t-I &c . or k+s+t+&c .

(iv.) If P and Q can be expressed in the forms pL and qL, where p and q are integers, R will be equal to (p–kq)L, which is both less than pL and less than qL . Hence the successive remainders are successively smaller multiples of L, but still integral multiples, so that the series of quotients k, s, t, .. . will ultimately come to an end . Moreover, if the last divisor is uL, then it follows from the theory of numbers (§ 26 (ii.)) that (a) u is a factor of p and of q, and (b) any number which is a factor of p and q is also a factor of u . Hence u is the greatest common measure of p and q . 35• In relation to algebra, the graphic method is mainly useful in connexion with the theory of limits (§§ 58, 61) and the functional treatment of equations (§ 6o) . As regards the latter, there are two classes of cases . In the first class come equations in a single unknown; here the function which is equated to zero is the Y whose values for different values of X are traced, and the solution of the equation is the determination of the points where the ordinates of the graph are zero . The second class of cases comprises equations involving two unknowns; here we have to deal with two graphs, and the solution of the equation is the determination of their common ordinates . Graphic methods also enter into the See also:

• CONSIDERATION (from Lat. considerare, to look at closely, examine, generally taken to be from con-, and the base seen in sidus, sideris, a star, the word being supposed to be originally an astrological or astronomical term)

consideration of irrational numbers (§ 65) . 36 . Monomials.—(i.) An expression such as a.z.a.a.b.c.3.a.a.c, denoting that a series of multiplications is to be performed, is called a monomial; the numbers (arithmetical or algebraical) which are multiplied together being its factors .

An expression denoting that two or more monomials are to be added or subtracted is a multinomial or polynomial, each of the monomials being a term of it . A multinomial consisting of two or of three terms is a binomial or a trinomial . (ii.) By means of the commutative law we can collect like terms of a monomial, numbers being regarded as like terms . Thus the above expression is equal to 6a5bc2, which is, of course, equal to other expressions, such as 6ba5c2 . The numerical factor 6 is called the coefficient of a5bc2 (§ 20); and, generally, the coefficient of any factor or of the product of any factors is the product of the remaining factors . (iii.) The multiplication and division of monomials is effected by means of the law of indices . Thus 6a5bc2=5a2bc=Ia3c, since b°= 1 . It must, of course, be remembered (§ 23) that this is a statement of arithmetical equality; we See also:

• CALL (from Anglo-Saxon ceallian, a common Teutonic word, cf. Dutch kallen, to talk or chatter)

call the statement an " identity," but we do not mean that the expressions are the same, but that, whatever the numerical values of a, b and c may be, the expressions give the same numerical result . In order that a monomial containing am as a factor may be divisible by a monomial containing aP as a factor, it is necessary that p should be not greater than m . (iv.) In algebra we have a theory of highest common factor and lowest common multiple, but it is different from the arithmetical theory of greatest common divisor and least common multiple . We disregard. numerical coefficients, so that by the H.C.F. or L.C.M. of 6a5bc2 and I2a4b2cd we mean the H.C.F. or L.C.M. of a5bc2 and a4b2cd . The H.C.F. is then an expression of the form aPbec'd", where p, q, r, s have the greatest possible values consistent with the See also:

• CONDITION (Lat. condicio, from condicere, to agree upon, arrange; not connected with conditio, from condere, conditum, to put together)

condition that each of the given expressions shall be divisible by aPbgC'd' .

Similarly the L.C.M. is of the form aPbac'd', where p, q, r, s have the least possible values consistent with the condition that aPb°c'd' shall be divisible by each of the given expressions . In the particular case it is clear that the H.C.F. is a4bc and the L.C.M. is a5b2c2d . The extension to multinomials forms part of the theory of factors (§ 51) . 37 . Products of T/! ultinomials.—(i.) Special arithmetical results may often be used to lead up to algebraical formulae . Thus a comparison of numbers occurring in a table of squares I2= 1I2=I21 22=4 122=144 32=9 132=169 suggests the formula (A+a)2=See also:

• A2(z)

A2+zAa+a2 . Similarly the equalities 99 X 101 = 9999 =10000 - I 98X102=9996=10000-4 97X103=9991 =10000-9 lead up to (A-a) (A+a)=A2-a2 . These, with (A-a)2= A2-zAa+a2, are the most important in elementary work . (ii.) These algebraical formulae involve not only the distributive law and the law of signs, but also the commutative law . Thus (A+a)2=(A+a)(A+a)=A(A+a) +a(A+a)=AA+Aa+aA+aa; and the grouping of the second and third terms as zAa involves treating Aa and aA as identical . This is important when we come to the binomial theorem (§ 41, and cf . § 54 (i.)) .

(iii.) By writing (A+a)2=A2+zAa+a2 in the form (A+a)2= A2+(2A+a)a, we obtain the rule for extracting the square root in arithmetic . (iv.) When the terms of a multinomial contain various powers of x, and we are specially concerned with x, the terms are usually arranged in descending (or ascending) order of the indices; terms which contain the same power being grouped so as to give a single coefficient . Thus 2bx-4x2+6ab+3ax would be written -4x2+(3a+2b)x+6ab . It is not necessary to regard -4 here as a negative number; all that is meant is that 4x2 has to be subtracted . (v.) When we have to multiply two multinomials arranged according to powers of x, the method of detached coefficients enables us to omit the powers of x during the multiplication . If any power is absent, we treat it as present, but with coefficient o . Thus, to multiply x3-2x+r by 2x2+4, we write the process +1+0—2+1 +2+0+4 +2+0—4+2 +0+0—0+0 +4+o-8+4 +2+0+0+2-8+4 giving 2x5+2x2-8x+4 as the result . 38 . Construction and Transformation of Equations.—(i.) The statement of problems in equational form should precede the solution of equations . (ii.) The solution of equations is effected by transformation, which may be either arithmetical or algebraical . The principles of arithmetical transformation follow from those stated in §§ 15-18 by replacing X, A, B, m, M, x, n, a and p by any expressions involving or not involving the unknown quantity or number and representing positive numbers or (in the case of X, A, B and M) positive quantities . The principle of algebraic transformation has been stated in § 22; it is that, if A=B is an equation (i.e. if either or both of the expressions A and B involves x, and A is arithmetically equal to B for the particular value of x which we require), and if B = C is an identity (i.e. if B and C are expressions involving x which are different in form but are arithmetically equal for all values of x), then the statement A= C is an equation which is true for the same value of x for which A=B is true .

(iii.) A special rule of transformation is that any expression may be transposed from one See also:

• SIDE
• SIDE (mod. Eski Adalia)

side of an equation to the other, provided its sign is changed . This is the rule of transposition . Suppose, for instance, that P+Q - R+S = T . This may be written (P+Q-R)+S=T; and this statement, by definition of the sign -, is the same as the statement that (P+Q- R) = T-S . Similarly the statements P+Q-R-S=T and P+ Q-R=T+S are the same . These transpositions are purely arithmetical . To transpose a term which is not the last term on either side we must first use the commutative law, which involves an algebraical transformation . Thus from the equation P+Q-R+S=T and the identity P+Q-R+S=P-R+S+Q we have the equation P-R+S+Q=T, which is the same statement as P-R+S=T-Q . (iv.) The procedure is sometimes stated differently, the transposition being regarded as a corollary 'from a general theorem that the roots of an equation are not altered if the same expression is added to or subtracted from both members of the equation . The objection to this (cf . § 21 (ii.)) is that we do not need the general theorem, and that it is unwise to cultivate the See also:

• HABIT (through the French from Lat. habitus, from habere, to have, hold, or, in a reflective sense, to be in a certain condition; in many of the English senses the French use habitude, not habit)

habit of laying down a general law as a See also:

• JUSTIFICATION

justification for an isolated See also:

• ACTION

action . (v.) An alternative method of obtaining the rule of trans-position is to change the zero from which we measure .

Thus from P+Q-R+S=T we deduce P+(Q-R+S)=P+(T-P) . If instead of measuring from zero we measure from P, we find Q-R+S=T-P . The difference between this and (iii.) is that we transpose the first term instead of the last; the two methods corresponding to the two cases under (i.) of § 15 (2) . (vi.) In the same way, we do not lay down a general rule that an equation is not altered by multiplying both members by the same number . Suppose, for instance, that -Ex+I) =4(x-2) . Here each member is a number, and the equation may, by the commutative law for multiplication, be written 2(x+I) =4(X–2) S This means that, whatever unit A we take, 2 (x5 I A and 4(x3 2) A are equal . We therefore take A to be 15, and find that 6(x+1)=2o(x-2) . Thus, if we have an equation P=Q, where P and Q are numbers involving fractions, we can clear of fractions, not by multiplying P and Q by a number m, but by applying the equal multiples P and Q to a number m as unit . If the P and Q of our equation were quantities expressed in terms of a unit A, we should restate the equation in terms of a unit Alm, as explained in §§ 18 and 21 (i.) (a) . (vii.) One result of the rule of transposition is that .we can transpose all the terms in x to one side of equation, and all the terms not containing x to the other . An equation of the form ax= b, where a and b do not contain x, is the See also:

• STANDARD
• STANDARD, BATTLE OF THE

standard form of simple equation . (viii.) The quadratic equation is the equation of two expressions, monomial or multinomial, none of the terms involving any power of x except x and x2 .

The standard form is usually taken to be See also:

• AXE (0. Eng. aex; a word common, in different forms, in the Teutonic languages, and akin to the Greek 41v17; the New English Dictionary prefers the spelling " ax ")

axe+bx+c = o, from which we find, by transformation, - (2 ax+b)2=b2-4ac, and thence x=See also:

• TII

tii(b2-4ac}-b 2a This only gives one root . As to the other root, see § 47 (iii.) . 39 . Fractional Expressions.—An equation may involve a fraction of the form Q, where Q involves x . (i.) If P and Q can (algebraically) be written in the forms RA and SA respectively, where A may or may not involve x, then Q = RA - R provided A is not =o . SA - S I 2 I 3 3 1 4 6 4 1 5 10 TO 5 6 15 20 15 6 &c., where the first line stands for (A+a)°=r . A°a°, and the successive numbers in the (n---i)th line are the coefficients of A"a° A"-'a1, ...Ma" in the n+1 terms of the multinomial equivalent to (A+a)" In the same way we have (A-¢)2=A2-2Aa+a2, (A-a)3 =A3-3A2a+3Aa2-a3, ... , so that the multinomial equivalent to (A-al" has the same coefficients as the multinomial equivalent to (A+a)", but with signs alternately + and - . The multinomial which is equivalent to (At a) n, and has its terms arranged in ascending powers of a, is called the expansion of (Ara)" . 41 . The binomial theorem gives a formula for writing down the coefficient of any stated term in the expansion of any stated power of a given binomial . (i.) For the general formula, we need only consider (A+a)" .

It is clear that, since the numerical coefficients of A and of a are each 1, the coefficients in the expansions arise from the grouping and addition of like terms (§ 37 (ii.)) . We therefore determine the coefficients by counting the grouped terms individually, instead of adding them . To individualize the terms, we replace (A+a) (A+a) (A+a) . . . by (A+a) (B+b) (C+c) . . ., so that no two terms are the same; the " like " -ness which determines the placing of two terms in one See also:

• GROUP

group being the fact that they become equal (by the commutative law) when B, C, . and b, c, .... are each replaced by A and a respectively . Suppose, for instance, that n=5, so that we take five factors (A+a) (B+b) (C+c) (D+d) (E+e) and find their product . The coefficient of Alas in the expansion of (A+a)5 is then the number of terms such as ABcde, AbcDe, AbCde, . . . , in each of which there are two large and three small letters . The first term is ABCDE, in which all the letters are large; and the coefficient of A2a3 is therefore the number of terms which can be obtained from ABCDE by changing three, and three only, of the large letters into small ones . We can begin with any one of the 5 letters, so that the first change can be made in 5 ways . There are then 4 letters left, and we can change any one of these .

Then 3 letters are left, and we can change any one of these . Hence the change can be made in 3.4.5 ways . If, however, the 3.4.5 results of making changes like this are written down, it will be seen that any one term in the required product is written down several times . Consider, for instance, the term AbcDe, in which the small letters are bce . Any one of these 3 might have appeared first, any one of the remaining second, and the remaining 1 last . The term therefore occurs I . 2 . 3 times . This applies to each of the terms in which there are two large and three small letters . The total number of such terms in the multinomial equivalent to (A+a) (B+b) (C+c) (D+d) (E+e) is therefore (3.4 . 5) - (1 . 2.3) ; and this is therefore the coefficient of A2¢3 in the expansion of (A+a)5 The reasoning is quite general; and, in the same way, the coefficient of A"-'a' in the expansion of (A+a)" is {(n-r+I) (n-r+2) .

. . (n-r)n) + {1.2.3 . . . r} . It is usual to write this as a fraction, inverting the order of the factors in the numerator . Then, if we denote it by 11(r), so that -n (n-I)...(n-r+I) n(r)_ I . 2 . 3 ... r we have (A+a)"=nsiA"+n(I)A^-'-a+...+nf)A°-''a'+...+nvaa" (2), where n(o), introduced for consistency of notation, is defined by See also:

• NCO

nco^ (3)• This is the binomial theorem for a positive integral index . (ii.) To verify this, let us denote the true coefficient of A"-'ar by (,), so that we have to prove that (;) =n(r0, where no.) is defined by (I); and let us inspect the actual process of multiplying the expansion of (A+ a) ' by A+a in order to obtain that of (A+a) " . Using detached coefficients (§ 37 (V.)), the multiplication is represented by the following:— 1 I+ (n I) + (n 2 1) +...+ (n -r 1) +...+ (n- I) n–2 +I I+ (1) + (2) +...+ (y) 1+...+ ~nn I~ +I, so that (r>(nrI)+(r-I) . Now suppose that the formula (2) has been established for every power of A+a up to the (n-r)th inclusive, so that (n -r I)=(n-I)(r),(r-I) (n - 1) (r_1) . Then (r) ,the coefficient of A"-'a' in the expansion of (A+a)", is equal to (n-1)(r)+ (n-1)(r_I) . But it may be shown that (r being >o) (ii.) In an equation of the form Q = V, the expressions P, Q, U, V are usuaily numerical .

We then have Q . QV . = V . QV, or PV = UQ, as in § 38 (vi.) . This is the rule of See also:

• CROSS

cross-multiplication . (iii.) The restriction in (i.) is important . Thus 2x2- x +x–2 (x-1) (x+1) is equal to x+1, except when x= 1 . For this (x – I) (x+2) x+2 latter value it becomes which has no direct meaning, and requires interpretation (§ 6r) . 40 . Powers of a Binomial.—We know that (A+a)2=A2+ 2Aa+a2 . Continuing to develop the successive powers of A+a into multinomials, we find that (A+a)3=A3+3A2a+3Aa2+a3 &c.; each power containing one more term than the preceding power, and the coefficients, when the terms are arranged in descending powers of A, being given by the following table: n(r)=(n-1)(r)+(n-I)(r-I) (4), and therefore (r) =n(,) . Hence the formula (2) is also true for the nth power of A+a .

But it is true for the 1st and the and powers; therefore it is true for the 3rd; therefore for the 4th; and so on . Hence it is true for all positive integral powers of n . (iii.) The product I . 2 . 3 . . . r is denoted by L r or r!, and is called factorial r . The form r! is better for printing, but the form Jr is more convenient for ordinary use . If we denote n(n-I) . . . (n-r+I) (r factors) by n(r), then n(r)---n'"/r !. (iv.) We can write n(:) in the more symmetrical form n ! n(r)(n-r) r ! which shows that n(r) =n(,,-,) (6) .

We should have arrived at this form in (i.) by considering the selection of terms in which there are to be two large and three small letters, the large letters being written down first . The terms can be built up in 5! ways; but each will appear 2 ! 3! times . (v.) Since n(,) is an integer, n(r) is divisible by r!; i.e. the product of any r consecutive integers is divisible by r ! (see § 42 (ii.)) . (vi.) The product r! arose in (i.) by the successive multiplication of r, r-1, r- 2, . . . I . In practice the successive factorials I!, 2!, 3 ! . . . are supposed to be obtained successively by introduction of new factors, so that r!=r . (r-I) !

(7) . Thus in defining r! as 1 . 2 . 3 . . . r we regard the multiplications as taking place from left to right; and similarly in n(r) . A product in which multiplications are taken in this order is called a continued product . (vii.) In order to make the formula (5) hold for the extreme values n(o) and no) we must adopt the convention that o ! =1 (8) . This is consistent with (7), which gives 1!= 1.0 !. It should be observed that, for r=o, (4) is replaced by n(o) _ (n - I)(o) (9), and similarly, for the final terms, we should See also:

• NOTE
• NOTE (Lat. nota, mark, sign, from noscere, to know)

note that p(Q)=o if 4>p (to) . (viii.) If u, denotes the term involving ar in the expansion of (A+a)n, then ur/ur_1= {(n-r+r)/r}.a/A . This decreases as r increases; its value ranging from na/A to a/(nA) .

If na<A, the terms will decrease from the beginning; if n A< a, the terms will increase up to the end; if na > A and nA > a, the terms will first increase up to a greatest term (or two consecutive equal greatest terms) and then decrease . (ix.) The position of the greatest term will depend on the relative values of A and a; if a/A is small, it will be near the beginning . See also:

• ADVANTAGE

Advantage can be taken of this, when n is large, to make approximate calculations, by omitting terms that are negligible . (a) Let S, denote the sum u,)+ul+ . . . +u,,this sum being taken so as to include the greatest term (or terms); and let ur+1/ur=8, so that B< I . Then the sum of the remaining terms u,+1+ur+2+ . . .+u„ is less than (1+6+62+ . . . +e"-r-1)ur+1, which is less than ur+1/(1-B); and therefore (A+a)" lies between S, and Sr+ur+i/(I-B) . We can therefore stop as soon as ur+1l (1-8) becomes negligible . (b) In the same way, for the expansion of (A-- a) n, let a,. denote uo-u1+ .

. . u, . Then, provided v, includes the greatest term, it will be found that (A-a)" lies between Or and Qr}1 . For actual calculation it is most convenient to write the theorem in the form nn=r (Ata)^=A^(Itx)^=A^t Ix . An+ 2 x.Ix.A^t ... where x----a/A; thus the successive terms are obtained by successive multiplication . To apply the method to the calculation of N", it is necessary that we should be able to express N in the form A+a or A-a, where a is small in comparison with A, An is easy to calculate and a/A is convenient as a multiplier . 42 . The reasoning adopted in § 41 (ii.) illustrates two generalmethods of procedure . We know that (A+a) n is equal to a multinomial of n+r terms with unknown coefficients, and we require to find these coefficients . We therefore represent them by separate symbols, in the same way that we represent the unknown quantity in an equation by a symbol . This is the method of undetermined coefficients . We then obtain a set of equations, and by means of these equations we establish the required result by a process known as mathematical See also:

• INDUCTION (from Lat. inducere, to lead into; cf. Gr. ilrayu yli)

induction .

This process consists in proving that a property involving p is true when p is any positive integer by proving (I) that it is true when p=1, and (2) that if it is true when p=n, where n is any positive integer, then it is true when p=n+r . The following are some further examples of mathematical induction . (i.) By adding successively 1, 3, 5 . . . we obtain 1, 4, 9, .. . This suggests that, if u^ is the sum of the first n odd numbers, then u„=n2 . Assume this true for u1, u2, . . ., u,, . Then u,,+1=un,+(2n+I)=n2+(2n+I)=(n+i)2, so that it is true for u„+I . But it is true for u1 . Therefore it is true generally . (ii.) We can prove the theorem of § 41 (v.) by a See also:

• DOUBLE
• DOUBLE (from the Mid. Eng. duble, the form which gives the present pronunciation, through the Old Er. duble, from Lat. duplus, twice as much)

double application of the method . (a) It is clear that every integer is divisible by 1 !.

(b) Let us assume that the product of every set of p consecutive integers is divisible by p!, and let us try to prove that the product of every set of p+1 consecutive integers is divisible by,(p+r) . Denote the product n(n+I) . . . (n+r- r) by n[r) . Then the See also:

• ASSUMPTION, FEAST OF

assumption is that, whatever positive integral value n may have, n(PI is divisible by p!. h (I) n(P+I]—(n—1)[P+1] =n(n+I) ... (n+p—1)(n+p)-(n—I)n . . . (n+p-I)=(p+r). n(P) . But, by See also:

• HYPOTHESIS (from Gr. inrorcOivat, to put under; cf. Lat. suppositio, from sub-ponere)

hypothesis, n[P) is divisible by p !. Therefore n)P+1)-(n-I))P+1) is divisible by p !. Therefore, if (n-I)(P+I] is divisible by (p+1)!, n(P+17 is divisible by (p+1)!• (2) But 1(P+1)=(p+i)!, which is divisible by (p+I) !. (3) Therefore n)P+11 is divisible by (p+I)!, whatever positive integral value n may have .

(c) Thus, if the theorem of § 41 (v.