(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 ...