Again, in §§ 61-75 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.53involves 5 and 3, and log28 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)n. If all but one of these constituent elements are settled beforehand,e.g.if we take the numbers 5, 52, 53, ..., or the numbers3√1,3√2,3√3, ... or log101.001, log101.002, log101.003 ... we obtain a series in which each term corresponds with a term of the original number-series.
This correspondence is usually shown bytabulation, i.e. by the formation of a table in which the original series is shown in one column, and each term of the second series is placed in a second column opposite the corresponding term of the first series, each column being headed by a description of its contents. It is sometimes convenient to begin the first series with 0, and even to give the series of negative numbers; in most cases, however, these latter are regarded as belonging to a different 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 isequivalentto the corresponding quantity in the other series. We might extend this principle to cases in which the terms of two series, whether of numbers orof numerical quantities, merelycorrespondwith each other, the correspondence being the result of some relation. The volume of a cube, for instance, bears a certain relation to the length of an edge of the cube. This relation is not one of proportion; but it may nevertheless be expressed by tabulation, as shown at D.
93.Interpolation.—In most cases the quantity in the second column may be regarded as increasing or decreasing continuously as the number in the first column increases, and it has intermediate values corresponding to intermediate (i.e.fractional or decimal) numbers not shown in the table. The 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 1½ in. has a definite volume, viz. 33⁄8cub. in. The determination of any such intermediate value is performed byInterpolation(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,i.e.we are regarding the numbers asordinal, 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 aunitarymethod. 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 theRule 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. Methods of Calculation
(i.)Exact Calculation.
95.Working 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, 15s. 9d. and another of £2, 6s. 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—e.g.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.
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 (§§ 17, 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 calledcarrying. The name is a bad one, since it does not correspond with any ordinary meaning of the verb. It would be better described asexchanging, by analogy with the “changing” of subtraction. When,e.g., we find that the sum of 17s. and 18s. is 35s., 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, 17s. 0d. to £2, 18s. 0d., we write down the sum of £3 and £2 as £6, not as £5, and the sum of 17s. and 18s. as 15s., not as 35s.
When three or more numbers or quantities are added together, the result should 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, 5s. 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, 2s. 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, 5s. 4d. into £9, 7s. 8d. we must successively add £6, 2s. and 4d. if we work from the left, or 4d., 2s. 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, 5s. 4d. and £9, 7s. 4d. in the one case, and £3, 5s. 8d. and £3, 7s. 8d. in the other. If we count backwards, the intermediate values are £3, 7s. 8d. and £3, 5s. 8d. in the one case, and £9, 7s. 4d. and £9, 5s. 4d. in the other.
The determination of each element in the remainder involves reference to an addition-table. Thus to subtract 5s. from 7s. we refer to an addition-table giving the sum of any two quantities, each of which is one of the series 0s., 1s., ... 19s.
Subtraction by counting forward is calledcomplementary addition.
To subtract £3, 5s. 8d. from £9, 10s. 4d., on the grouping system, we mustchange1s. out of the 10s. into 12d., so that we subtract £3, 5s. 8d. from £9, 9s. 16d. On the counting system it will be found that, in determining the number of shillings in the remainder, we subtract 5s. from 9s. 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 6s. from 10s. In the first two cases the successive values (in direct or reverse order) are £3, 5s. 8d., £9, 5s. 8d., £9, 9s. 8d. and £9, 10s. 4d.; while in the last two cases they are £9, 10s. 4d., £3, 10s. 4d., £3, 6s. 4d. and £3, 5s. 8d.
In subtracting from the left, we look ahead to see whether a 1 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.Multiplication-Table.—For multiplication and division we use amultiplication-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 0) to 10, 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 × 1, 3 × 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 largemultiple 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)) that 3·427 = 3(400 + 20 + 7) = 3·400 + 3·20 + 3·7. This, if we 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 = 3·4·100 = 12·100 = 1200; 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, 10 − 1, 10 + 1 or 10 + 2; and similarly for 13, 17, 18, 19, &c.
(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.Multiplication 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 = (200 + 30 + 1)·4273 = 200·4273 + 30·4273 + 1·4273. This gives thepartial products, the sum of which is the complete products. The process is shown fully in A below,—
and more concisely in B. To multiply 4273 by 200, we use the commutative law, which gives 200·4273 = 2 × 100 × 4273 = 2 × 4273 × 100 = 8546 × 100 = 854600; and similarly for 30·4273. In B the terminal 0’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 them 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 up to the multiplication of decimals and of approximate values of numbers, is to place the first figure of the multipler under the first figure of the multiplicand, as in E; the first figure of each partial product will then come under the corresponding figure of the multiplier.
102.Contracted Multiplication.—The partial products are sometimes omitted; the process saves time in writing, but is not easy. The principle is that,e.g.(a·102+ b·10 + c)(p·102+ q10+ r) = ap·104+ (aq + bp)·103+ (ar + bq + cp)·102+ (br + cq)·10 + cr. Hence the digits are multiplied in pairs, and grouped according to the power of 10 which each product contains. A method of performing the process is shown here for the case of 162·427. The principle is that 162·427 = 100·427 + 60·427 + 2·427 = 1·42700 + 6·4270 + 2·427; but, instead of writing down the separate products, we (in effect) write 42700, 4270, and 427 in separate rows, with the multipliers 1, 6, 2 in the margin, and then multiply each number in each column by the corresponding 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 calledaliquot parts(from Lat.aliquot, some, several). This can usually be done in a good many ways. Thus5⁄6= 1 −1⁄6, and also = ½ +1⁄3; and 15% = .15 =1⁄10+1⁄20=1⁄6−1⁄60=1⁄8+1⁄40. 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 ½ +1⁄20−1⁄60is a simpler expression for8⁄15than ½ +1⁄30.
The process may sometimes by applied two or three times in succession; thus8⁄15=4⁄5·2⁄3= (1 −1⁄5)(1 −1⁄3), and33⁄40= ¾·11⁄10= (1 − ¼)(1 +1⁄10).
104.Practice.—The above is a particular case of the method calledpractice, but the nomenclature of the method is confusing. There are two kinds of practice,simple practiceandcompound practice, but the latter is the simpler of the two. To find the cost of 2 ℔ 8 oz. of butter at 1s. 2d. a ℔, we multiply 1s. 2d. by 28⁄16= 2½. This straightforward process is called “compound” practice. “Simple” practice involves an application of the commutative law. To find the cost of n articles at £a, bs, cd. each, we express £a, bs, cd. in the form £(a + f), where f is a fraction (or the sum of several fractions); we then say that the cost, being n × £(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 469, we may express the former in any of the ways £7.675,307⁄40of £1, 153½s., 153.5s., 307 sixpences, or 1842 pence. Expression in £ 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 ¾d. A rule for approximate conversion is that 1s. = .05 of £1, and that 2½d.= .01 of £1. For accurate conversion we write .1£ for each 2s., and .001£ for each farthing beyond 2s., their number being first increased by one twenty-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 series of subtractions. Starting from the dividend, we in theory keep on subtracting the unit, and count the number of subtractions that have to be performed until nothing is left. In actual practice, of course, we subtract large multiples at a time. Thus, to divide 987063 by 427, we reverse the procedure of § 101, but with intermediate stages. We first construct the multiple-table C, and then subtract successively 200 times, 30 times and 1 times; these numbers being thepartial quotients. The theory of the process is shown fully in F. Treating x as the unknown quotient corresponding to the original dividend,we obtain successive dividends corresponding to quotients x − 200, x − 230 and x − 231. The original dividend is written as 0987063, since its initial figures are greater than those of the divisor; if the dividend had commenced with (e.g.) 3 ... it would not have been necessary to insert the initial 0. At each stage of the division the number of digits in the reduced dividend is decreased by one. The final dividend being 0000, we have x − 231 = 0, and therefore x = 231.
107.Methods of Division.—What are described as different methods of division (by a single divisor) are mainly different methods of writing the successive figures occurring in the process. Inlong divisionthe divisor is put on the left of the dividend, and the quotient on the right; and each partial product, with the remainder after its subtraction, is shown in full. Inshort divisionthe divisor and the quotient are placed respectively on the left of and below the dividend, and the partial products and remainders are not shown at all. TheAustrianmethod (sometimes called in Great Britain theItalianmethod) differs from these in two respects. The first, and most important, is that the quotient is placed above the dividend. The second, which is not essential to the method, is that the remainders are shown, but not the partial products; the remainders being obtained by working from the right, and using complementary addition. It is doubtful whether the brevity of this latter process really compensates for its greater difficulty.
The advantage of the Austrian arrangement of the quotient lies in the indication it gives of the true value of each partial quotient. A modification of the method, corresponding with D of § 101, is shown in G; the fact that the partial product 08546 is followed by two blank spaces shows that the figure 2 represents a partial quotient 200. An alternative arrangement, corresponding to E of § 101, and suited for more advanced work, is shown in H.
108.Division with Remainder.—It has so far been assumed that the division can be performed exactly,i.e.without leaving an ultimate remainder. Where this is not the case, difficulties are apt to arise, which are mainly due to failure to distinguish between the two kinds of division. If we say that the division of 41d. by 12 gives quotient 3d. with remainder 5d., we are speaking loosely; for in fact we only distribute 36d. out of the 41d., the other 5d. remaining undistributed. It can only be distributed by a subdivision of the unit;i.e.the true result of the division is 35⁄12d. On the other hand, we can quite well express the result of dividing 41d. by 1s (= 12d.) as 3 with 5d. (not “5”) over, for this is only stating that 41d. = 3s. 5d.; though the result might be more exactly expressed as 35⁄12s.
Division with a remainder has thus a certain air of unreality, which is accentuated when the division is performed by means of factors (§ 42). If we have to divide 935 by 240, taking 12 and 20 as factors, the result will depend on the fact that, in the notation of § 17,In incomplete partition the quotient is 3, and the remainders 11 and 17 are in effect disregarded; if, after finding the quotient 3, we want to know what remainder would be produced by a direct division, the simplest method is to multiply 3 by 240 and subtract the result from 935. In complete partition the successive quotients are 7711⁄12and 3[(1711⁄12)/20] = 3215⁄240. Division in the sense of measuring leads to such a result as 935d. = £3, 17s. 11d.; we may, if we please, express the 17s. 11d. as 215d., but there is no particular reason why we should do so.
109.Division by a Mixed Number.—To divide by a mixed number, when the quotient is seen to be large, it usually saves time to express the divisor as either a simple fraction or a decimal of a unit of one of the denominations. Exact division by a mixed number is not often required in real life; where approximate division is required (e.g.in determining the rate of a “dividend”), approximate expression of the divisor in terms of the largest unit is sufficient.
110.Calculation of Square Root.—The calculation of the square root of a number depends on the formula (iii) of § 60. To find the square root of N, we first find some number a whose square is less than N, and subtract a2from N. If the complete square root is a + b, the remainder after subtracting a2is (2a + b)b. We therefore guess b by dividing the remainder by 2a, and form the product (2a + b)b. If this is equal to the remainder, we have found the square root. If it exceeds the square root, we must alter the value of b, so as to get a product which does not exceed the remainder. If the product is less than the remainder, we get a new remainder, which is N − (a + b)2; we then assume the full square root to be c, so that the new remainder is equal to (2a + 2b + c)c, and try to find c in the same way as we tried to find b.
An analogous method of finding cube root, based on the formula for (a + b)3, used to be given in text-books, but it is of no practical use. To find a root other than a square root we can use logarithms, as explained in § 113.
(ii.)Approximate Calculation.
111.Multiplication.—When we have to multiply two numbers, and the product is only required, or can only be approximately correct, to a certain number of significant figures, we need only work to two or three more figures (§ 83), and then correct the final figure in the result by means of the superfluous figures.
A common method is to reverse the digits in one of the numbers; but this is only appropriate to the old-fashioned method of writing down products from the right. A better method is to ignore the positions of the decimal points, and multiply the numbers as if they were decimals between .1 and 1.0. The method E of § 101 being adopted, the multiplicand and the multiplier are written with a space after as many digits (of each) as will be required in the product (on the principle explained in § 101); and the multiplication is performed from the left, two extra figures being kept in. Thus, to multiply 27.343 by 3.1415927 to one decimal place, we require 2 + 1 + 1 = 4 figures in the product. The result is 085.9 = 85.9, the position of the decimal point being determined by counting the figures before the decimal points in the original numbers.
112.Division.—In the same way, in performing approximate division, we can at a certain stage begin to abbreviate the divisor, taking off one figure (but with correction of the final figure of the partial product) at each stage. Thus, to divide 85.9 by 3.1415927 to two places of decimals, we in effect divide .0859 by .31415927 to four places of decimals. In the work, as here shown, a 0 is inserted in front of the 859, on the principle explained in § 106. The result of the division is 27.34.
113.Logarithms.—Multiplication, division, involution and evolution, when the results cannot be exact, are usually most simply performed, at any rate to a first approximation, by means of a table of logarithms. Thus, to find the square root of 2, we have log √2 = log (21/2) = ½ log 2. We take out log 2 from the table, halve it, and then find from the table the number of which this is the logarithm. (SeeLogarithm.) Theslide-rule(seeCalculating Machines) is a simple apparatus for the mechanical application of the methods of logarithms.
When a first approximation has been obtained in this way, further approximations can be obtained in various ways. Thus, having found √2 = 1.414 approximately, we write √2 = 1.414 + θ, whence 2 = (1.414)2+ (2.818)θ + θ2. Since θ2is less than ¼ of(.001)2, we can obtain three more figures approximately by dividing 2 − (1.414)2by 2.818.
114.Binomial Theorem.—More generally, if we have obtained a as an approximate value for the pth root of N, the binomial theorem gives as an approximate formulap√N = a + θ, where N = ap+ pap − 1θ.
115.Series.—A number can often be expressed by a series of terms, such that by taking successive terms we obtain successively closer approximations. A decimal is of course a series of this kind,e.g.3.14159 ... means 3 + 1/10 + 4/102+ 1/103+ 5/104+ 9/105+ ... A series of aliquot parts is another kind,e.g.3.1416 is a little less than 3 +1⁄7−1⁄800.
Recurring Decimalsare a particular kind of series, which arise from the expression of a fraction as a decimal. If the denominator of the fraction, when it is in its lowest terms, contains any other prime factors than 2 and 5, it cannot be expressed exactly as a decimal; but after a certain point a definite series of figures will constantly recur. The interest of these series is, however, mainly theoretical.
116.Continued Products.—Instead of being expressed as the sum of a series of terms, a number may be expressed as the product of a series of factors, which become successively more and more nearly equal to 1. For example,
3.1416 = 3 ×10472⁄10000= 3 ×1309⁄1250= 3 ×22⁄21×2499⁄2500= 3(1 +1⁄21)(1 −1⁄2500).
Hence, to multiply by 3.1416, we can multiply by 31⁄7, and subtract1⁄2500(= .0004) of the result; or, to divide by 3.1416, we can divide by 3, then subtract1⁄22of the result, and then add1⁄2499of the new result.
117.Continued Fractions.—The theory ofcontinued fractions(q.v.) gives a method of expressing a number, in certain cases, as a continued product. A continued fraction, of the kind we are considering, is an expression of the formwhere b, c, d, ... are integers, and a is an integer or zero. The expression is usually written, for compactness, a + 1/b+ 1/c+ 1/d+ &c. The numbers a, b, c, d, ... are called thequotients.
Any exact fraction can be expressed as a continued fraction, and there are methods for expressing as continued fractions certain other numbers,e.g.square roots, whose values cannot be expressed exactly as fractions.
The successive values, a/1, (ab + 1)/b, ..., obtained by taking account of the successive quotients, are calledconvergents,i.e.convergents to the true value. The following are the main properties of the convergents.
(i) If we precede the series of convergents by0⁄1and1⁄0, then the numerator (or denominator) of each term of the series0⁄1,1⁄0, a/1, (ab + 1)/b ..., after the first two, is found by multiplying the numerator (or denominator) of the last preceding term by the corresponding quotient and adding the numerator (or denominator) of the term before that. If a is zero, we may regard 1/b as the first convergent, and precede the series by1⁄0and0⁄1.
(ii) Each convergent is a fraction in its lowest terms.
(iii) The convergents are alternately less and greater than the true value.
(iv) Each convergent is nearer to the true value than any other fraction whose denominator is less than that of the convergent.
(v) The difference of two successive convergents is the reciprocal of the product of their denominators;e.g.(ab + 1)/b − a/1 = 1/(1·b), and (abc + c + a)/(bc + 1) − (ab + 1)/b = −1/b(bc + 1).
It follows from these last three properties that if the successive convergents are p1/1, p2/q2, p3/q3, ... the number can be expressed in the form p1(1 + 1/p1q2) (1 − 1/p2q3) (1 + 1/p3q4) ..., and that if we go up to the factor 1 ± 1/(pnqn + 1) the product of these factors differs from the true value of the number by less than ±{1/(qnqn + 1).
In certain cases two or more factors can be combined so as to produce an expression of the form 1 ± 1/k, where k is an integer. For instance, 3.1415927 = 3(1 +1⁄3.7) (1 −1⁄22.106) (1 +1⁄333.113) ...; but the last two of these factors may be combined as (1 −1⁄22.113). Hence 3.1415927 =3⁄1·22⁄21·2485⁄2486...
XII. Applications
(i.)Systems of Measures.1
118.Metric System.—The metric system was adopted in France at the end of the 18th century. The system is decimal throughout. The principal units of length, weight and volume are themetre, gramme(orgram) andlitre. Other units are derived from these by multiplication or division by powers of 10, the names being denoted by prefixes. The prefixes for multiplication by 10, 102, 103and 104aredeca-,hecto-,kilo-andmyria-, and those for division by 10, 102and 103aredeci-,centi-andmilli-; the former being derived from Greek, and the latter from Latin. Thuskilogrammemeans 1000 grammes, andcentimetremeans1⁄100of a metre. There are also certain special units, such as thehectare, which is equal to a square hectometre, and themicron, which is1⁄1000of a millimetre.
The metre and the gramme are defined by standard measures preserved at Paris. The litre is equal to a cubic decimetre. The gramme was intended to be equal to the weight of a cubic centimetre of pure water at a certain temperature, but the equality is only approximate.
The metric system is now in use in the greater part of the civilized world, but some of the measures retain the names of old disused measures. In Germany, for instance, thePfundis ½ kilogramme, and is approximately equal to 11⁄10℔ English.
119.British Systems.—The British systems have various origins, and are still subject to variations caused by local usage or by the usage of particular businesses. The following tables are given as illustrations of the arrangement adopted elsewhere in this article; the entries in any column denote multiples or submultiples of the unit stated at the head of the column, and the entries in any row give the expression of one unit in term of the other units.
Length
Weight (Avoirdupois)
(Also 7000 grains = 1 ℔ avoirdupois.)
120.Change of System.—It is sometimes necessary, when a quantity is expressed in one system, to express it in another,The following are the ratios of some of the units; each unit is expressed approximately as a decimal of the other, and their ratio is shown as a continued product (§ 116), a few of the corresponding convergents to the continued fraction (§ 117) being added in brackets. It must be remembered that the number expressing any quantity in terms of a unit isinversely proportionalto the magnitude of the unit,i.e.the number of new units is to be found by multiplying the number of old units by the ratio of the old unit to the new unit.
(ii.)Special Applications.
121.Commercial Arithmetic.—This term covers practically all dealings with money which involve the application of the principle of proportion. A simple class of cases is that which deals with equivalence of sums of money in different currencies; these cases really come under § 120. In other cases we are concerned with a proportion stated as anumerical percentage, or as amoney percentage(i.e.a sum of money per £100), or as aratein the £ or the shilling. The following are some examples. Percentage:Brokerage, commission, discount, dividend, interest, investment, profit and loss.Rate in the £:Discount, dividend, rates, taxes.Rate in the shilling:Discount.
Text-books on arithmetic usually contain explanations of the chief commercial transactions in which arithmetical calculations arise; it will be sufficient in the present article to deal with interest and discount, and to give some notes on percentages and rates in the £.InsuranceandAnnuitiesare matters of general importance, which are dealt with elsewhere under their own headings.
122.Percentages and Rates in the £.—In dealing with percentages and rates it is important to notice whether the sum which is expressed as a percentage of a rate on another sum is a part of or an addition to that sum, or whether they are independent of one another. Income tax, for instance, is calculated on income, and is in the nature of a deduction from the income; but local rates are calculated in proportion to certain other payments, actual or potential, and could without absurdity exceed 20s. in the £.
It is also important to note that if the increase or decrease of an amount A by a certain percentage produces B, it will require a different percentage to decrease or increase B to A. Thus, if B is 20% less than A, A is 25% greater than B.
123.Interestis usually calculated yearly or half-yearly, at a certain rate per cent. on the principal. In legal documents the rate is sometimes expressed as a certain sum of money “per centum per annum”; here “centum” must be taken to mean “£100.”
Simple interestarises where unpaid interest accumulates as a debt not itself bearing interest; but, if this debt bears interest, the total,i.e.interest and interest on interest, is calledcompound interest.If 100r is the rate per cent. per annum, the simple interest on £A for n years is £nrA, and the compound interest (supposing interest payable yearly) is £[(1 + r)n− 1]A. If n is large, the compound interest is most easily calculated by means of logarithms.
124.Discountis of various kinds. Tradesmen allow discount for ready money, this being usually at so much in the shilling or £. Discount may be allowed twice in succession off quoted prices; in such cases the second discount is off the reduced price, and therefore it is not correct to add the two rates of discount together. Thus a discount of 20%, followed by a further discount of 25%, gives a total discount of 40%, not 45%, off the original amount. When an amount will fall due at some future date, thepresent valueof the debt is found by deducting discount at some rate per cent. for the intervening period, in the same way as interest to be added is calculated. This discount, of course, is not equal to the interest which the present value would produce at that rate of interest, but is rather greater, so that the present value as calculated in this way is less than the theoretical present value.
125. Applications toPhysicsare numerous, but are usually only of special interest. A case of general interest is the measurement oftemperature.The graduation of a thermometer is determined by the freezing-point and the boiling-point of water, the interval between these being divided into a certain number of degrees, representing equal increases of temperature. On the Fahrenheit scale the points are respectively 32° and 212°; on the Centigrade scale they are 0° and 100°; and on the Réaumur they are 0° and 80°. From these data a temperature as measured on one scale can be expressed on either of the other two scales.
126.Averagesoccur in statistics, economics, &c. An average is found by adding together several measurements of the same kind and dividing by the number of measurements. In calculating an average it should be observed that the addition of any numerical quantity (positive or negative) to each of the measurements produces the addition of the same quantity to the average, so that the calculation may often be simplified by taking some particular measurement as a new zero from which to measure.