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offs, would be 271-530, but expressed in terms of 100 it would be - 27 1 530.

Fractions other than decimal fractions are usually called vulgar fractions.

75. Decimal N umbers.-—Instead of regarding the -r 53 in 27153 as meaning 111,535, we may regard the difierent figures in the expression as denoting numbers in the successive orders of submultiples of 1 on a denary scale. Thus, on the grouping system, 27-153 will mean 2.10+7+1/ro+5/10#+3/ro', while on the counting system it will mean the result of counting through the tens to 2, then through the ones to 7, then through tenths to I, and so on. A number made up in this way may be called a decimal number, or, more briefly, a decimal. It will be seen that the definition includes integral numbers.

76. Sum: and Difl'crcnces of Decimals.——To add or subtract decimals, we must reduce them to the same denomination, i.c. if one has more figures after the decimal point than the other, we must add sufficient o’s to the latter to make the numbers of figures equal. Thus, to add 5-413 to 3-8, we must write the latter as 3-800. Or we may treat the former as the sum of 5-4 and -or3, and recombine the -013 with the sum of 3-8 and 5'477. Product of Decimals.—To multiply two decimals exactly, we multiply them as if the point were absent, and then insert it so that the number of figures after the point in the product shall be equal to the sum of the numbers of figures after the points in the original decimals.

In actual practice, however, decimals only represent approximations, and the process has to be modified (§ III).

78. Division by Decimal.-—To divide one decimal by another, we must reduce them to the same denomination, as explained in § 76, and then omit the decimal points. Thus 5-4r3+3-8= Hil+i8~8$=5413+38m

79. Historical Development of Fractions and Decimals—The fractions used in ancient times were mainly of two kinds: unitfractions, i.e. fractions representing aliquot parts (Q 103), and fractions with a definite denominator.

The Egyptians as a rule used only unit-fractions, other fractions being expressed as the sum of unit-fractions. The only known exception was the use of Q as a single fraction. Except in the case of Q and s, the fraction was expressed by the denominator, with a special symbol above it.

The Babylonians expressed numbers less than 1 by the numerator of a fraction with denominator 60; the numerator only being written. The choice of 60 appears to have been connected with the reckoning of the year as 360 days; it is perpetuated in the present subdivision of angles.

The Greeks originally used unit‘fractions, like the Egyptians; later they introduced the sexagesimal fractions of the Babylonians, extending the system to four or more successive subdivisions of the unit representing a degree. They also, but apparently still later and only occasionally, used fractions of the modern kind. In the sexagesimal system the numerators of the successive fractions (the denominators of which were the successive powers of 60) were followed by ' , ' , ", ", the denominator not being written. This notation survives in reference to the minute (') and second (") of angular measurement, and has been extended, by analogy, to the foot (’) and inch ("). Since 2 represented 60, and 0 was the next letter, the latter appears to have been used to denote absence of one of the fractions; but it is not clear that our present sign for zero was actually derived from this. In the case of fractions of the more general kind, the numerator was written first with ', and then the denominator, followed by ”, was written twice. A different method was used by Diophantus, accents being omitted, and the denominator being written above and to the right of the numerator.

The Romans commonly used fractions with denominator 12; these Were described as unciae (ounces), being twelfths of the a: (pound).

The modern system of placing the numerator above the denominator is due to the Hindus; but the dividing line is a later invention. Various systems were tried before the present

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notation came to be generally accepted. Under one system, for

instance, the continued sumg-l-fi-l-gygTswould be denoted by g—;—:l5; this is somewhat similar in principle to a decimal notation, but with digits taken in the reverse order.

Hindu treatises on arithmetic show the use of fractions, containing a power of 10 as denominator, as early as the beginning of the 6th century AD. There was, however, no development in the direction of decimals in the modern sense, and the Arabs, by whom the Hindu notation of integers was brought to Europe, mainly used the sexagesimal division in the ' ' " notation. Even where the decimal notation would seem to arise naturally, as in the case of approximate extraction of a square root, the portion which might have been expressed as a decimal was converted into scxagesimal fractions. It was not until A.D. 1585 that a decimal notation was published by Simon Stevinus of Bruges. It is worthy of notice that the invention of this notation appears to have been due to practical needs, being required for the purpose of computation of compound interest. The present decimal notation, which is a development of that of Stevinus, was first used in r617 by H. Briggs, the computer of logarithms.

8o. Fraclions of Concrete Quantities.—The British systems of coinage, weights, lengths, &c., afford many examples of the use of fractions. These may be divided into three classes, as follows:—

(i) The fraction of a concrete quantity may itself not exist as a concrete quantity, but be represented by a token. Thus, if we take a shilling as a unit, we may divide it into 12 or 48 smaller units; but corresponding coins are not really portions of a shilling, but objects which help us in counting. Similarly we may take the farthing as a unit, and invent smaller units, represented either by tokens or by no material objects at all. Ten marks, for instance, might be taken as equivalent to a farthing; but 13 marks are not equivalent to anything except one farthing and three out of the ten acts of counting required to arrive at another farthing.

(ii) In the second class of cases the fraction of the unit quantity is a quantity of the same kind, but cannot be determined with absolute exactness. Weights come in this class. The ounce, for instance, is one-sixteenth of the pound, but it is impossible to find 16 objects such that their weights shall be exactly equal and that the sum of their weights shall be exactly equal to the weight of the standard pound.

(iii) Finally, there are the cases of linear measurement, where it is theoretically possible to find, by geometrical methods, an exact submultiple of a given unit, but both the unit and the submultiple are not really concrete objects, but are spatial relations embodied in objects.

Of these three classes, the first is the least abstract and the last the most abstract. The first only involves number and counting. The second involves the idea of equalin as a necessary characteristic of the units or subunits that are used. The third involves also the idea of continuity and therefore of unlimited subdivision. In weighing an object with ounce-weights the fact that it weighs more than 1 lb 3 oz. but less than 1 lb 4 02. does not of itself suggest the necessity or possibility of subdivision of the ounce for purposes of greater accuracy. But in measuring a distance we may find that it is “ between” two distances differing by a unit of the lowest denomination used, and a subdivision of this unit follows naturally.

VII. Arraoxnmnon

81. Approximate Character of N umbers.—The numbers (integral or decimal) by which we represent the results of arithmetical operations are often only approximately correct. All numbers, for instance,which represent physical measurements,are limited in their aCcuracy not only by our powers of measurement but also by the accuracy of the measure we use as our unit. Also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations Even where numbers are supposed to be exact, calculations based on them can often only be approximate. We might, for instance, calculate the exact cost of 3 lb 5 oz. of meat at 9% . a lb, but there are no coins in which we could pay this exact amount.

When the result of any arithmetical operation or operations is represented approximately but not exactly by a number, the excess (positive or negative) of this number over the number which would express the result exactly is called the error.

82. Degree of Accuracy—There are three principal ways of expressing the degree of accuracy of any number, i.e. the extent to which it is equal to the number it is intended to represent.

(i) A number can be correct to so many places of decimals. This means (cf. 5 71) that the number difiers from the true value by less than one~half of the unit represented by r in the last place of decimals. For instance, 143 represents 1} correct to 3 places of decimals, since it differs from it by less than -0005. The final figure, in a case like this, is said to be corrected.

This method is not good for comparative purposes. Thus - X43 and r4-286 represent respectively 1} and 12'“ to the same number of places of decimals, but the latter is obviously more exact than the former.

(ii) A number can be correct to so many significant figures. The significant figures of a number are those which commence with the first figure other than zero in the number; thus the significant figures of 13-027 and of 00013027 are the same.

This is the usual method; but the relative accuracy of two numbers expressed to the same number of significant figures depends to a certain extent on the magnitude of the first figure. Thus -r4286 and ~857r4 represent »} and § correct to 5 significant figures; but the latter is relatively more accurate than the former. For the former sh0ws only that l lies between -142855 and -142865, or, as it is better expressed, between -r4285§ and Q4286}; but the latter shows that 1?, lies between -85713§ and ~857r4}, and therefore that 4} lies between 1428511; and -r4285,97.

In either of the above cases, and generally in any case where a number is known to be within a certain limit on each side of the stated value, the limit of error is expressed by the sign *. Thus the former of the above two statements would give #= -r4286*-ooooos. It should be observed that the numerical value of the error is to be subtracted from or added to the stated value according as the error is positive or negative.

(iii) The limit of error can be expressed as a fraction of the number as stated. Thus §=~r43=l=-0005 can be written +=-14s(1*rh)

83. Accuracy after Arithmetical Operations—If the numbers which are the subject of operations are not all exact, the accuracy of the result requires special investigation in each case.

Additions and subtractions are simple. If, for instance, the values of a and b, correct to two places of decimals, are 3-58 and 1-34, then 2-24, as the value of a—b, is not necessarily correct to two places. The limit of error of each being =h -oo5, the limit of error of their sum or difference is =4= -or.

For multiplication we make use of the formula (§ 60 (i)) (a'=|=a) (b'=*= B) = a’b'+aB=h (a’B-l-b’a). If a’ and b’ are the stated values, and =*= a. and * B the respective limits of error, we ought strictly to take a'b’+o.;3 as the product. with a limit of error=l= (a'fl+b’a). In practice, however, both a5 and a certain portion of a'b' are small in comparison with 0’6 and b’a, and we therefore replace (fl/+0.13 by an approximate valueI and increase the limit of error so as to cover the further error thus introduced. In the case of the two numbers given in the last paragraph, the product lies between 3-575Xr-335=4-772625 and 3-585 Xi-345=4-821825. We might take the product as (3-58X 1'34)+(-oo5)'==4-7o7225, the limits of error being *-oos(_3-58+r-34)===-0246; but it is more convenient to write it in such a form as 4‘7o7*~025 or 4~80* '03.

If the number of decimal places to which a result is to be accurate is determined beforehand. it is usually not necessary in the actual working to go to more than two or three places

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find any number whose cube is 2000.

It is, however, possible to find a number whose cube shall approximate as closely as we please to 2000. Thus the cubes of 12-5 and of 12-6 are respectively r953-125 and 2000376, so that the number whose cube difiers as little as possible from 2000 is somewhere between r2-5 and 12-6. Again the cube of 12-59 is 1995-616979, so that the number lies between 12-59 and 12-60. We may therefore consider that there is some number a: whose cube is 2000, and we can find this number to any degree of accuracy that we please.

A number of this kind is called a surd; the surd which is the pth root of N is written UN, but if the index is a it is usually omitted, so that the square root of N is written y/N.

85. Surd as a P0‘wer.-—We have seen (§§ 43,44) that, if we take the successive powers of a number N, commencing with r, they may be written N“, N‘, N“, N‘, . . . , the series of indices being the standard series; and we have also seen (§ 44) that multiplication of any two of these numbers corresponds to addition of their indices. Hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law. The number denoted by N 4 will therefore be such that NlXNiXNl=Nf*l*l=-=N; i.e. it will be the cube root of N. By analogy with the notation of fractional numbers, Nl will be N§*i=NlXN§; and, generally, N5 will mean the product of 9 numbers, the product of q of which is equal to N. Thus Ni will not mean the some as N5, but will mean the square of Nl; but this will be equal to N5, Le. ('JN)*=\‘/N.

86. Multiplication and of Sunk—To add or subtract fractional numbers, we must reduce them to a common denominator ; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index. Thus {2Xd5= 25X5l=2'X53=4l>(125l=5ool=§/5oo.

87. Antilogarithms.—-If we take a fixed number, e.g. a, as base, and take as indices the successive decimal numbers to any particular number of places of decimals, we get a series of antilogan'llmu of‘ the indices to this base. Thus, if we go to two places of decimals, we have as the integral series the numbers 1, 2, 4, 8,

. which are the values of 2°, 2', 2', . . . and we insert within this series the successive powers of x, where x is such that x'°° = 2. We thus get the numbers 2-°', 24", 2-°', . . . ,which are the anti— logarithms of -01, -02, ~03, . . . to base 2; the first antilogarithm being 2-°°= r, which is thus the antilogarithm of o to this (or any other) base. 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. If, for instance, we have found 2-", then the value of 2-'" is found from it by multiplying by the 6th power of the loooth root of 2.

For practical purposes the number taken as base is IO; the convenience of this being that the increase of the index by an integer means multiplication by the corresponding power of re, i.e. it means a shifting of the decimal point. In the same way. by dividing by powers of to we may get negative indices.

88. L0gari¢hms.—If N is the antilogarithm of p to the base a, i.e. if N—a’, then p is called the logarithm of N to the base a. and is written log, N. As the table of antilogarithms is formed by successive multiplications, so the logarithm of any given number is in theory found by successive divisions. Thus, to find the logarithm of a number to base 2, the number being greater than 1, we first divide repeatedly by 2 until we get a number between 1 and 2; then divide repeatedly by 1°42 until we get anumber between 1 and “W2; then divide repeatedly by loo~12; and so on. If, for instance, we find that the number is approximately equal to 23 X (“la/2)5 X (“W/'2)7 X (‘°°°\/2)4, it may be written 2'-"“, and its logarithm to base 2 is 3-574.

For a further explanation of logarithm, and for an explanation of the treatment of cases in which an antilogarithm is less' than 1, see LOGARITHM.

For practical purposes logarithms are usually calculated to base 10, so that log", 1o= 1, log“, 1oo= 2, &c.

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89. Change of Denomination of a numerical quantity is usually called reduction, so that this term covers, e.g., the expression of £153, 75. 4d. as shillings and pence and also the expression of 3067s. 4d. as £, s. and d.

The usual statement is that to express £153, 7s. as shillings we multiply 155 by 20 and add 7. This, as already explained (§37), is incorrect. £153 denotes 153 units, each of which is {1 or 20s.; and therefore we must multiply 20s. by 153 and add 75., i.e. multiply 20 by 153 (the unit being now 15.) and add 7. This is the expression of the process on the grouping method. On the

counting method we have

A a scale with every 20th

shilling marked as a £;

there are 153 of these 20’s, and 7 over.

The simplest case, in which the quantity can be expressed as an integral number of the largest units [3 involved, has already been considered (§§ 37, 42). The same method can be applied in other cases ,5. by regarding a quantity expressed in several denominations as a fractional number of units of the largest denomination mentioned; thus 75. 4d. is to be taken as meaning 71415., but

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and of 36808d. to £, s. d., on this principle, is shown in diagrams A and B above.

For reduction of pounds to shillings, or shillings to pounds, we must consider that we have a multiple-table (§ 36) in which the multiples of £1 and of zos. are arranged in parallel columns; and similarly for shillings and pence.

90. Change of Unit—The statement “ £153=3o60s.” is not a statement of equality of the same kind as the statement “ 153X:o=3060,” but only a statement of equivalence for certain purposes; in other words, it does not convey an absolute truth. It is therefore of interest to see whether we cannot replace it by an absolute truth.

To do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects. If we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5. We cannot multiply 4 apples by 5 boys, for then we should get 20 “ boy-apples,” an expression which has no meaning. Or, again, to distribute 20 apples amongst 5 boys, we are not regarded as dividing 2o apples by 5 boys, but as dividing 20 apples by the number 5. The multiplication or division here involves the omission of the unit “ boy,” and the operation is incomplete. The complete operation, in each case, is as follows.

(i) In the case of multiplication we commence with the conception of the number “ 5 " and the unit “ boy "; and we then convert this unit into 4 apples, and thus obtain the result,

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so many times 4 lb by the same number of times 7 lb, while “ 1X ” symbolizes the replacing of 4 times something by 7 times that something. X. Annmrrcar. Rsxsonmo 91. Correspondence of Series of Numbers.— In §§ 33-42 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 outof multiplication, division, &c., have been exhibited by diagrams comprising pairsof corresponding terms of the two series. This, however, is only a particular case of the correspondence of two series. In considering addition, for instance, we have introduced two parallel series, each being the original numberSeries, but the two being placed in diflerent positions. If we add 1,2,3, . . . to 6, we obtain a series 7,8,9, . . . , the terms of which correspond with those of the original series 1,2,3, . . . ' Again, in §§61-7 5 and 84-88 we have considered various kinds of numbers other than those in the original number-series. In general, these have involved two of the original numbers, e.g. 53 involves 5 and 3, and log, 8 involves 2 and 8. In some cases, however, e.g. in the case of negative numbers and reciprocals, only one is involved; and there might be three or more, as in the case of a number expressed by (a+b)'*. If all but one of these constituent elements are settled beforehand, e.g. if we take the numbers 5, 52,53, . . . , ortbenumbers‘41,°\/2,'\/3, ...or logo 1-001, log", 1-002, log" 1-003 . . . we obtain a series in which each term corresponds with a term of the original number-series. This correspondence is usually shown by tabulation, Le. by the formation of a table in which the original series is shown in one A B C column, and each term of the second series is placed

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in a second column op

6+" " 4" n 4" posite the corresponding term of the first series,

0 6 o o o -ooo each column being headed 1 7 1 4 1 1-000 by a description of its 2 8 2 8 2 1-414 contents. It is sometimes 3 9 3 12 3 1-732 convenient to begin the . . . . first series with o, and even

to give the series of nega

tive numbers; in most

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cases, however, these latter are regarded as belonging to a difierent series, and they need not be considered here. The diagrams, A, B, C are simple forms of tables; A giving a sum-series, B a multiple-series, and C a series of square roots, calculated approximately.

92. Correspondence of Numerical Quantities—Again, in § 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is equivalent to the corresponding quantity in the other series. We might extend this principle to cases in which the terms of two series, whether of numbers or

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Length of Volume cube. This relation is not one of pro?dge in 0‘ portion; but it may nevertheless be "‘Ches' cube' expressed by tabulation, as shown at D.

- 93. Interpolation.—In most cases the 0 Nu. . . I l cub in‘ quantity in the second column may - be regarded as increasmg or decreasing 2 8 cub. in. . . 27 cub in continuously as the number in the first 3 ' ' column increases, and it has inter' ' mediate values corresponding to inter' mediate (Le. fractional or decimal) numbers not shown in the table. The

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table in such cases is not, and cannot

be, complete, even up to the number to which it goes. For instance, a cube whose edge is 1%- in. has a definite volume, viz.) 3i cub. in. The determination of any such intermediate value is performed by Interpolation (q.v.).

In treating a fractional number, or the corresponding value of the quantity in the second column, as intermediate, we are in effect regarding the numbers 1, 2, 3, . . . , and the corresponding numbers in the second column, as denoting points between which other numbers lie, is. we are regarding the numbers as ordinal, not cardinal. The transition is similar to that which arises in the case of geometrical measurement (§ 26), and it is an essential feature of all reasoning with regard to continuous quantity, such as we have to deal with in real life.

94. Nature of Arithmetical Reasoning—The simplest form of arithmetical reasoning consists in the determination of the term in one series corresponding to a given term in another series, when the relation between the two series is given; and it implies, though it does not necessarily involve, the establishment of each series as a whole by determination of its unit. A method

_ involving the determination of the unit is called a unitary method. When the unit is not determined, the reasoning is algebraical rather than arithmetical. If, for instance, three terms of a proportion are given, the fourth can be obtained by the relation given at the end of § 57, this relation being then called the Rule of Three; but this is equivalent to the use of an algebraical formula.

More complicated forms of arithmetical reasoning involve the use of series, each term in which corresponds to particular terms in two or more series jointly; and cases of this kind are usually dealt with by special methods, or by means of algebraical formulae. The old-fashioned problems about the amount of work done by particular numbers of men, women and boys, are of this kind, and really involve the solution of simultaneous equations. They are not suitable for elementary purposes, as~the arithmetical relations involved are complicated and difficult to grasp.

XI. Mernons or CALCULATION (i.) Exact Calculation.

95. ll’arking from Left—It is desirable, wherever possible, to perform operations on numbers or numerical quantities from the left, rather than from the right. There are several reasons for this. In the first place, an operation then corresponds more closely, at an elementary stage, with the concrete process which it represents. If, for instance, we had one sum of £3, :55. 9d. and another of £2, 65. 5d., we should add them by putting the coins of each denomination together and commencing the addition with the £. In the second place, this method fixes the attention at once on the larger. and therefore more important, parts of the quantities concerned, and thus prevents arithmetical processes from becoming too abstract in character. In the third place, it is a better preparation for dealing with approximate calculations. Finally, experience shows that certain operations in which the result is written down at once—cg. addition or subtraction of two numbers or quantities, and multiplication by some small numbers-are with a little practice performed more quickly and more accurately from left to right.

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96. Addition.—There is no difference in principle between addition (or subtraction) of numbers and addition (or subtraction) of numerical quantities. In each case the grouping system involves rearrangement, which implies the commutative law, while the counting system requires the expression of a quantity in different denominations to be regarded as a notation in a varying scale (§§ i7, 32). We need therefore consider numerical quantities only, our results being applicable to numbers by regarding the digits as representing multiples of units in different denominations.

When the result of addition in one denomination can be partly expressed in another denomination, the process is technically called carrying. The name is a bad one, since it does not correspond with any ordinary meaning of the verb. It would be better described as exchanging, by analogy with the “changing” of subtraction. When, e.g., we find that the sum of 17s. and 18s. is 355., we take out 20 of the 35 shillings, and exchange them for £1.

To add from the left, we have to look ahead to see whether the next addition will require an exchange. Thus, in adding £3, 175. ed. to £2, 18s. od., we write down the sum of £3 and £2 as £6, not as £5, and the sum of 17s. and 18s. as 155., not as 355.

When three or more numbers or quantities are added together, the result sh0uld always be checked by adding both upwards and downwards. It is also useful to look out for pairs of numbers or quantities which make 1 of the next denomination, e.g. 7 and 3, or 8d. and 4d.

97. Subtraction.—To subtract £3, 55. 4d. from £9, 7s. 8d., on the grouping system, we split up each quantity into its denominations, perform the subtractions independently, and then regroup the results as the “ remainder ” £6, 25. 4d. On the counting system we can count either forwards or backwards, and we can work either from the left or from the right. If we count forwards we find that to convert £3, 55. 4d. into £9, 75. 8d. we must successively add £6, 25. and 4d. if we work from the left, or 4d., 25. and £6 if we work from the right. The intermediate values obtained by the successive additions are different according as we work from the left or from the right, being £9, 55. 4d. and £9, 75. 4d. in the one case, and £3, 55. 8d. and £3, 75. 8d. in the other. If we count backwards, the intermediate values are £3, 75. 8d. and £3, 55. 8d. in the one case, and £9, 75. 4d. and £9, 55. 4d. in the other.

The determination of each element in the remainder involves reference to an addition-table. Thus to subtract 5s. from 75. we refer to an addition-table giving the sum of any two quantities, each of which is one of the series 05., 15., . . . 195.

Subtraction by counting forward is called complementary addition.

To subtract £3, 55. 8d. from £9, 108. 4d., on the grouping system, we must change 15. out of the 105. into 12d., 50 that we subtract £3, 5s. 8d. from £9, 95. 16d. On the counting system it will be found that, in determining the number of shillings in the remainder, we subtract 55. from 95. if we count forwards, working from the left, or backwards, working from the right; while, if we count backwards, working from the left, or forwards, working from the right, the subtraction is of 65. from 105. In the first two cases the successive values (in direct or reverse order) are £3, 55. 8d., £9, 55. 8d., £9, 95. 8d. and £9, 105. 4d.; while in the last two cases they are £9, 108. 4d., £3, 105.4d., £3, 65. 4d. and £3, 55. 8d.

In subtracting from the left, we look ahead to see whether a r in any denomination must be reserved for changing; thus in subtracting 274 from 637 we should put down 2 from 6 as 3, not as 4, and 7 from 3 as 6.

98. M ulliplicatian- Table.—F or 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 to, 12 or 20. The column (vertical) headed 3 will give the multiples of 3, while the row (horizontal) commencing with 3 will give the values of 3 X t. 3 X 2, . . . To multiply by 3 we use the row. To divide by 3, in the sense of partition, we also use the row; but to divide by 3 as a unit we use the column.

99. Multiplication by a Small Number.—-The idea of a large multiple of a small number is simpler than that of a small multiple of a large number, but the calculation of the latter is easier. It is therefore convenient, in finding the product of two numbers, to take the smaller as the multiplier.

To find 3 times 427, we apply the distributive law (§ 58 (vi) ) that3.427 =3(400+20+7)= 3.4oo+3.2o+3.7. This, ifwe regard 3.427 as 427+427+427, is a direct consequence of the commutative law for addition (§ 58 (iii) ), which enables us to add separately the hundreds, the tens and the ones. To find 3.400, we treat 100 as the unit (as in addition), so that 3.400=3.4.1oo= 1 2.1oo= 1 200; and similarly for 3.20. These are examples of the associative law for multiplication (§ 58 (iv) ).

100. Special Cases.—The following are some special rules:—

(i) To multiply by 5, multiply by 10 and divide by 2. (And conversely, to divide by 5, we multiply by 2 and divide by 10.)

(ii) In multiplying by 2, from the left, add 1 if the next figure of the multiplicand is 5, 6, 7, 8 or 9.

(iii) In multiplying by 3, from the left, add 1 when the next figures are not less than 33 . . . 334 and not greater than 66 . . . 666, and 2 when they are 66 . . . 667 and upwards.

(iv) To multiply by 7, 8, 9, 11 or 12, treat the multiplier as 10-3, 10-2,1o-1, 10+1 or Io+2;and similarly for 13, 17, 18,19,810

(v) To multiply by 4 or 6, we can either multiply from the left by 2 and then by_ 2 or 3, or multiply from the right by 4 or 6; or we can treat the multiplier as 5—1 or 5+1.

101. M ultiplication by a Large Number.—When both the numbers are large, we split up one of them, preferably the multiplier. into separate portions. Thus 231.4273= (2004-3041) 4273= 200.4273+3o.4273+1.4273. This gives the partial products, the sum of which is the complete product. The process is shown fully in A below,—

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and more concisely in B. To multiply 4273 by 200, we use the commutative law, which gives 200.4273=2X1oo><4273= 2x4273X100=8546X100=8546oo; and similarly for 30.4273. In B the terminal o’s of the partial products are omitted. It is usually convenient to make out a preliminary table of multiples up to 10 times; the table being checked at 5 times (§ 100) and at 10 times.

The main difficulty is in the correct placing of the curtailed partial products. The first step is to regard the product of two numbers as containing as many digits as the two numbers put together. The table of multiples will then be as in C. The next step is to arrange the multiplier and‘ the multiplicand above the partial products. 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 corresponding figure of the multiplier. A better method, which leads

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The principle is that 162.427=100.427+6o.427+2.427= 1.427oo+6.427o+2.427; but, instead of

1 427 writing down the separate products. we

6 427 (in effect) write 42700, 4270 and 427 in

2 427 separate rows, with the multipliers 1, 6, 2 in the margin, and then multiply each

m number in each column by the corre

sponding multiplier in the margin, making allowance for any figures to be “ carried.” Thus the second figure (from the right) is given by 1+2.2+6.7 =47, the 1 being carried.

103. Aliquot Parts.—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 1 as its numerator. Such fractions are called aliquot parts (from Lat. aliquot, some, several). This can usually be done in a good many ways. Thus§=1-—§,andalso=§+§.and 15%=-15=~1lo+1}0>== {—316=l+-410. The fractions should generally be chosen so that each part of the product may be obtained from an earlier part by a comparatively simple division. Thus l-i-fqIlg is a simpler expression for 155 than

The process may sometimes be applied two or three times in succession; thus 135=§-§ = (1 —§) (1—3), and =§~H = (I —i) (1+116)

104. Practice.—The above is a particular case of the method called practice, but the nomenclature of the method is confusing. There are two kinds of practice, simple practice and comPound practice, but the latter is the simpler of the two. To find the cost of 2 lb 8 oz. of butter at rs. 2d. a ll), we multiply 15. 2d. by 2-1-5;=2%. This straightforward process is called “compound ” practice. “ Simple ” practice involves an application of the commutative law. To find the cost of u articles at {a, bs. cd. each, we express £0, bs. cd. in the form {,(a-i-j'), where f is a fraction (or the sum of several fractions); we then say that the cost, being nX£(a+f), is equal to (a+f)><£n, and apply the method of compound practice, i.e. the method of aliquot parts.

105. Multiplication of a Mixed Number.—When a mixed quantity or a mixed number has to be multiplied by a large number, it is sometimes convenient to express the former in terms of one only of its denominations. Thus, to multiply {7, 13s. 6d. by 460, we may express the former in any of the ways £7 -675, "4°07 of £1, 15355., 153-55., 307 sixpences, or 1842 pence. Expression in {I and decimals of {1 is usually recommended, but it depends on circumstances whether some other method may not be simpler.

A sum of money cannot be expressed exactly as a decimal of {1 unless it is a multiple of id. A rule for approximate conversion is that 1s. = -05 of£1, and that 25d. = '01 of {1. For accurate conversion we write ~1£ for each 25., and -001£ for each farthing beyond 25.,their number being firstincrcased by onetwenty-fourth.

106. Division.'—Of the two kinds of division, although the idea of partition is perhaps the more elementary, the process of measuring is the easier to perform, since it is equivalent to a

F series of subtractions. Starting from the dividend, we in theory keep on

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4273 subtracting the unit, and count the

number of subtractions that have to

x 0987063 be performed until nothing is left. In 200 0854600 actual practice, of course, we subtract large multiples at a time. Thus, to

x ‘200 132463 divide 987063 by 427, we reverse the procedure of § 101, but with inter

30 128190 mediate stages. We first construct the x—230 04273 multiple-table C, and then subtract successively 200 times, 30 times and

I 04273 1 times; these numbers being the parx_;_;, 0000 tial quotients. The theory of the process is shown fully in F. Treating x

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as the unknown quotient corresponding to the original dividend,

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