Chapter 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 L3, 15s. 9d. and another of L2, 6s. 5d., we should add them by putting the coins of each denomination together and commencing the addition with the L. 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 (SS 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 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 35s., we take out 20 of the 35 shillings, and exchange them for L1.

To add from the left, we have to look ahead to see whether the next addition will require an exchange. Thus, in adding L3, 17s. 0d. to L2, 18s. 0d., we write down the sum of L3 and L2 as L6, not as L5, 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 L3, 5s. 4d. from L9, 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" L6, 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 L3, 5s. 4d. into L9, 7s. 8d. we must successively add L6, 2s. and 4d. if we work from the left, or 4d., 2s. and L6 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 L9, 5s. 4d. and L9, 7s. 4d. in the one case, and L3, 5s. 8d. and L3, 7s. 8d. in the other. If we count backwards, the intermediate values are L3, 7s. 8d. and L3, 5s. 8d. in the one case, and L9, 7s. 4d. and L9, 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 called _complementary addition_.

To subtract L3, 5s. 8d. from L9, 10s. 4d., on the grouping system, we must _change_ 1s. out of the 10s. into 12d., so that we subtract L3, 5s. 8d. from L9, 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 L3, 5s. 8d., L9, 5s. 8d., L9, 9s. 8d. and L9, 10s. 4d.; while in the last two cases they are L9, 10s. 4d., L3, 10s. 4d., L3, 6s. 4d. and L3, 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 a _multiplication-table_, which is a multiple-table, arranged as explained in S 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 X 1, 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 (S 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 (S 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 (S 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 the _partial products_, the sum of which is the complete products. The process is shown fully in A below,--

A B C
| |
| 4273 | 4273 1 - 04273
-----+-------- ------+--------- 2 - 08546
200 | 854600 | 8546 3 - 12819
30 | 128190 231 | 12819 . .
1 | 4273 | 4273 . .
-----+-------- +--------- . .
231 | 987063 | 987063 10 - 042730
============== ==========

and more concisely in B. To multiply 4273 by 200, we use the commutative law, which gives 200.4273 = 2 X 100 X 4273 = 2 X 4273 X 100 = 8546 X 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 (S 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.

D E
| | | 4273
| 4273| 231 | 231
----+-----+----- ----+---------
| 0854:6 | 08546
231 | 128:19 231 | 12819
| 04:273 | 04273
+-----:----- +---------
| 0987:063 | 0987063
============= ==========

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.10^2 + b.10 + c)(p.10^2 + q^10 + r) = ap.10^4 + (aq + bp).10^3 + (ar + bq + cp).10^2 + (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.

+---+-------
| 1 | 427
| 6 | 427
| 2 | 427
+---+-------
69174
=====

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 5/6 = 1 -1/6, and also = 1/2 + 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 1/2 + 1/20 - 1/60 is a simpler expression for 8/15 than 1/2 + 1/30.

The process may sometimes by applied two or three times in succession; thus 8/15 = 4/5 . 2/3 = (1 - 1/5)(1 - 1/3), and 33/40 = 3/4 . 11/10 = (1 -1/4)(1 + 1/10).

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 1s. 2d. a lb., we multiply 1s. 2d. by 2(8/16) = 2-1/2. This straightforward process is called "compound" practice. "Simple" practice involves an application of the commutative law. To find the cost of n articles at La, bs, cd. each, we express La, bs, cd. in the form L(a + f), where f is a fraction (or the sum of several fractions); we then say that the cost, being n X L(a + f), is equal to (a + f) X Ln, 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 L7, 13s. 6d. by 469, we may express the former in any of the ways L7.675, 307/40 of L1, 153-1/2s., 153.5s., 307 sixpences, or 1842 pence. Expression in L and decimals of L1 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 L1 unless it is a multiple of 3/4d. A rule for approximate conversion is that 1s. = .05 of L1, and that 2-1/2d.= .01 of L1. For accurate conversion we write .1L for each 2s., and .001L 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 S 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 the _partial 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.

F
+---------+----------+
| | 4273 |
+=========+==========+
| x | 0987063 |
+---------+----------+
| 200 | 0854600 |
+---------+----------+
| x - 200 | 132463 |
+---------+----------+
| 30 | 128190 |
+---------+----------+
| x - 230 | 04273 |
+---------+----------+
| 1 | 04273 |
+---------+----------+
| x - 231 | 0000 |
+---------+----------+

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. In _long division_ the 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. In _short division_ the 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. The _Austrian_ method (sometimes called in Great Britain the _Italian_ method) 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 S 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 S 101, and suited for more advanced work, is shown in H.

G H
| | | 4273
| 4273 | 2 | 2
--+------+----- --+------------
| 0987063 | 0987063
| 08546 | 08546
--+------------ --+------------
| |

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 3(5/12)d. 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 3(5/12)s.

Division with a remainder has thus a certain air of unreality, which is accentuated when the division is performed by means of factors (S 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 S 17,

(20) (12)
935 = 3 " 17 " 11.

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 77(11/12) and 3(17(11/12))/20 = 3(215/240).

Division in the sense of measuring leads to such a result as 935d. = L3, 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 S 60. To find the square root of N, we first find some number a whose square is less than N, and subtract a^2 from N. If the complete square root is a + b, the remainder after subtracting a^2 is (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 S 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 (S 83), and then correct the final figure in the result by means of the superfluous figures.

| 2734 3
| 3141 59
-----+-----------
| 0820 29
| 027 34
| 10 94
| 0 27
| 14
| 2
-----+-----------
| 0859

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 S 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 S 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.

| 3141 5927
| 2734
-----+------------
| 0859 00
| 0628 32
| -------
| 230 68
| 219 91
| -------
| 10 77
| 9 42
| ------
| 1 35
| 1 26
| ------

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 S 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 [root]2 = log (2^1/2) = 1/2 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. (See LOGARITHM.) The _slide-rule_ (see CALCULATING 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 [root]2 = 1.414 approximately, we write [root]2 = 1.414 + [theta], whence 2 = (1.414)^2 + (2.818)[theta] + [theta]^2. Since [theta]^2 is less than 1/4 of (.001)^2, we can obtain three more figures approximately by dividing 2 - (1.414)^2 by 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 formula [root p]N = a + [theta], where N = a^p + pa^(p - 1)[theta].

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 + 4/10^2 + 1/10^3 + 5/10^4 + 9/10^5 + ... A series of aliquot parts is another kind, e.g. 3.1416 is a little less than 3 + 1/7 - 1/800.

_Recurring Decimals_ are 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 X 10472/10000 = 3 X 1309/1250 = 3 X 22/21 X 2499/2500 =
3(1 + 1/21)(1 - 1/2500).

Hence, to multiply by 3.1416, we can multiply by 3(1/7), and subtract 1/2500 (= .0004) of the result; or, to divide by 3.1416, we can divide by 3, then subtract 1/22 of the result, and then add 1/2499 of the new result.

117. _Continued Fractions._--The theory of _continued 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 form

1
a + -------------
1
b + ---------
1
c + -----
d + &c.

where 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 the _quotients_.

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 called _convergents_, i.e. convergents to the true value. The following are the main properties of the convergents.

(i) If we precede the series of convergents by 0/1 and 1/0, then the numerator (or denominator) of each term of the series 0/1, 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 by 1/0 and 0/1.

(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/(p_n)(q_(n + 1)) the product of these factors differs from the true value of the number by less than [+-][1/(q_n) (q_(n + 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 ...