62. _Fractional Numbers._--According to the definition in S 50 the
quantity denoted by 3/6 of A is made up of a number, 3, and a unit,
which is one-sixth of A. Similarly p/n of A, q/n of A, r/n of A, ...
mean quantities which are respectively p times, q times r times, ... the
unit, n of which make up A. Thus any arithmetical processes which can be
applied to the numbers p, q, r, ... can be applied to p/n, q/n, r/n, ...,
the denominator n remaining unaltered.
If we denote the unit 1/n of A by X, then A is n times X, and p/n of n
times X is p times X; i.e. p/n of n times is p times.
Hence, so long as the denominator remains unaltered, we can deal with
performed on the numerators. The expressions p/n, q/n, r/n, ... are then
_fractional numbers_, their relation to ordinary or _integral_ numbers
being that p/n times n times is equal to p times.
This relation is of exactly the same kind as the relation of the
successive digits in numbers expressed in a scale of notation whose base
is n. Hence we can treat the fractional numbers which have any one
denominator as constituting a number-series, as shown in the adjoining
diagram. The result of taking 13 sixths of A is then seen to be the same
as the result of taking twice A and one-sixth of A, so that we may
regard 13/6 as being equal to 2(1/6). A fractional number is called a
_proper fraction_ or an _improper fraction_ according as the numerator
is or is not less than the denominator; and an expression such as 2(1/6)
is called a mixed number. An improper fraction is therefore equal either
to an integer or to a mixed number. It will be seen from S 17 that a
mixed number corresponds with what is there called a _mixed quantity_.
Thus L3, 17s. is a mixed quantity, being expressed in pounds and
shillings; to express it in terms of pounds only we must write it
L3(17/20).
Ones. Sixths.
0 0
1
2
3
4
5
1 0
1
2
3
4
5
2 0
1
:
:
63. _Fractional Numbers with different Denominators._--If we divided the
unit into halves, and these new units into thirds, we should get sixths
of the original unit, as shown in A; while, if we divided the unit into
thirds, and these new units into halves, we should again get sixths, but
as shown in B. The series of halves in the one case, and of thirds in
the other, are entirely different series of fractional numbers, but we
can compare them by putting each in its proper position in relation to
the series of sixths. Thus 3/2 is equal to 9/6, and 5/3 is equal to
10/6, and conversely; in other words, any fractional number is
equivalent to the fractional number obtained by multiplying or dividing
the numerator and denominator by any integer. We can thus find
fractional numbers equivalent to the sum or difference of any two
fractional numbers. The process is the same as that of finding the sum
or difference of 3 sixpences and 5 fourpences; we cannot subtract 3
sixpenny-bits from 5 fourpenny-bits, but we can express each as an
equivalent number of pence, and then perform the subtraction. Generally,
to find the sum or difference of two or more fractional numbers, we must
replace them by other fractional numbers having the same denominator; it
is usually most convenient to take as this denominator the L.C.M. of the
original fractional numbers (cf. S 53).
A B
Ones. Halves. Sixths. Ones. Thirds. Sixths.
0 0 0 0 0 0
1 1
2 1 0
1 0 1
1 2 0
2 1
1 0 0 1 0 0
: :
: :
64. _Complex Fractions._--A fraction (or fractional number), the
numerator or denominator of which is a fractional number, is called a
_complex_ fraction (or fractional number), to distinguish it from a
_simple_ fraction, which is a fraction having integers for numerator and
denominator. Thus 5(2/3)/11(1/3) of A means that we take a unit X such
that 11(1/3) times X is equal to A, and then take 5(2/3) times X. To
simplify this, we take a new unit Y, which is 1/3 of X. Then A is 34
times Y, and 5(2/3)/11(1/3) of A is 17 times Y, i.e. it is 1/2 of A.
65. _Multiplication of Fractional Numbers._--To multiply 8/3 by 5/7 is
to take 5/7 times 8/3. It has already been explained (S 62) that 5/7
times is an operation such that 5/7 times 7 times is equal to 5 times.
Hence we must express 8/3, which itself means 8/3 times, as being 7
times something. This is done by multiplying both numerator and
denominator by 7; i.e. 8/3 is equal to (7.8)/(7.3), which is the same
thing as 7 times 8/(7.3). Hence 5/7 times 8/3 = 5/7 times 7 times
8/(7.3) = 5 times 8/(7.3) = (5.8)/(7.3). The rule for multiplying a
fractional number by a fractional number is therefore the same as the
rule for finding a fraction of a fraction.
66. _Division of Fractional Numbers._--To divide 8/3 by 5/7 is to find a
number (i.e. a fractional number) x such that 5/7 times x is equal to
8/3. But 7/5 times 5/7 times x is, by the last section, equal to x.
Hence x is equal to 7/5 times 8/3. Thus to divide by a fractional number
we must multiply by the number obtained by interchanging the numerator
and the denominator, i.e. by the _reciprocal_ of the original number.
If we divide 1 by 5/7 we obtain, by this rule, 7/5. Thus the reciprocal
of a number may be defined as the number obtained by dividing 1 by it.
This definition applies whether the original number is integral or
fractional.
By means of the present and the preceding sections the rule given in S
63 can be extended to the statement that a fractional number is equal to
the number obtained by multiplying its numerator and its denominator by
any fractional number.
67. _Negative Fractional Numbers._--We can obtain negative fractional
numbers in the same way that we obtain negative integral numbers; thus
-(5/7) or -(5/7)A means that 5/7 or (5/7)A is taken negatively.
68. _Genesis of Fractional Numbers._--A fractional number may be
regarded as the result of a measuring division (S 39) which cannot be
performed exactly. Thus we cannot divide 3 in. by 11 in. exactly, i.e.
we cannot express 3 in. as an integral multiple of 11 in.; but, by
extending the meaning of "times" as in S 62, we can say that 3 in. is
3/11 times 11 in., and therefore call 3/11 the quotient when 3 in. is
divided by 11 in. Hence, if p and n are numbers, p/n is sometimes
regarded as denoting the result of dividing p by n, whether p and n are
integral or fractional (mixed numbers being included in fractional).
The idea and properties of a fractional number having been explained, we
may now call it, for brevity, a _fraction_. Thus "2/3 of A" no longer
means two of the units, three of which make up A; it means that A is
multiplied by the fraction 2/3, i.e. it means the same thing as "2/3
times A."
69. _Percentage._--In order to deal, by way of comparison or addition or
subtraction, with fractions which have different denominators, it is
necessary to reduce them to a common denominator. To avoid this
difficulty, in practical life, it is usual to confine our operations to
fractions which have a certain standard denominator. Thus (S 79) the
Romans reckoned in twelfths, and the Babylonians in sixtieths; the
former method supplied a basis for division by 2, 3, 4, 6 or 12, and the
latter for division by 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, or 60. The
modern method is to deal with fractions which have 100 as denominator;
such fractions are called _percentages_. They only apply accurately to
divisions by 2, 4, 5, 10, 20, 25 or 50; but they have the convenience of
fitting in with the denary scale of notation, and they can be extended
to other divisions by using a mixed number as numerator. One-fortieth,
for instance, can be expressed as (2-1/2)/100, which is called 2-1/2
_per cent._, and usually written 2-1/2%. Similarly 3(1/3)% is equal to
one-thirtieth.
If the numerator is a multiple of 5, the fraction represents twentieths.
This is convenient, e.g. for expressing _rates in the pound_; thus 15%
denotes the process of taking 3s. for every L1, i.e. a rate of 3s. in
the L.
In applications to money "per cent." sometimes means "per L100." Thus
"L3, 17s. 6d. per cent." is really the complex fraction
17(6/12)
3 -------
20
------------ .
100
70. _Decimal Notation of Percentage._--An integral percentage, i.e. a
simple fraction with 100 for denominator, can be expressed by writing
the two figures of the numerator (or, if there is only one figure, this
figure preceded by 0) with a dot or "point" before them; thus .76 means
76%, or 76/100. If there is an integral number to be taken as well as a
percentage, this number is written in front of the point; thus 23.76 X A
means 23 times A, with 76% of A. We might therefore denote 76% by 0.76.
If as our unit we take X = 1/100 of A = 1% of A, the above quantity
might equally be written 2376 X = 2376/100 of A; i.e. 23.76 X A is equal
to 2376% of A.
71. _Approximate Expression by Percentage._--When a fraction cannot be
expressed by an integral percentage, it can be so expressed
approximately, by taking the _nearest_ integer to the numerator of an
equal fraction having 100 for its denominator. Thus 1/7 = 14(2/7)/100,
so that 1/7 is approximately equal to 14%; and 2/7 = 28(4/7)/100, which
is approximately equal to 29%. The difference between this approximate
percentage and the true value is less than 1/2%, i.e. is less than
1/200.
If the numerator of the fraction consists of an integer and 1/2--e.g. in
the case of 3/8 = (37-1/2)/100--it is uncertain whether we should take
the next lowest or the next highest integer. It is best in such cases to
retain the 1/2; thus we can write 3/8 = 37-1/2 % = .37-1/2.
72. _Addition and Subtraction of Percentages._--The sum or difference of
two percentages is expressed by the sum or difference of the numbers
expressing the two percentages.
73. _Percentage of a Percentage._--Since 37% of 1 is expressed by 0.37,
37% of 1% (i.e. of 0.01) might similarly be expressed by 0.00.37. The
second point, however, is omitted, so that we write it 0.0037 or .0037,
this expression meaning 37/100 of 1/100 = 37/10000.
On the same principle, since 37% of 45% is equal to 37/100 of 45/100 =
1665/10000 = 16/100 + (65/100 of 1/100), we can express it by .1665; and
3% of 2% can be expressed by .0006. Hence, to find a percentage of a
percentage, we multiply the two numbers, put 0's in front if necessary
to make up four figures (not counting fractions), and prefix the point.
74. _Decimal Fractions._--The percentage-notation can be extended to any
fraction which has any power of 10 for its denominator. Thus 153/1000
can be written .153 and 15300/100000 can be written .15300. These two
fractions are equal to each other, and also to .1530. A fraction written
in this way is called a _decimal fraction_; or we might define a decimal
fraction as a fraction having a power of 10 for its denominator, there
being a special notation for writing such fractions.
A mixed number, the fractional part of which is a decimal fraction, is
expressed by writing the integral part in front of the point, which is
called the _decimal point_. Thus 27(1530/10000) can be written
27.1530. This number, expressed in terms of the fraction 1/10000 or
.0001, would be 271530. Hence the successive figures after the decimal
point have the same relation to each other and to the figures before the
point as if the point did not exist. The point merely indicates the
_denomination_ in which the number is expressed: the above number,
expressed in terms of 1/16, would be 271.530, but expressed in terms of
100 it would be .271530.
Fractions other than decimal fractions are usually called _vulgar
fractions_.
75. _Decimal Numbers._--Instead of regarding the .153 in 27.153 as
meaning 153/1000, we may regard the different 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 +
1/10 + 5/10^2 + 3/10^3, 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 1, 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. _Sums and Differences of Decimals._--To add or subtract decimals, we
must reduce them to the same denomination, i.e. if one has more figures
after the decimal point than the other, we must add sufficient 0'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 .013, and recombine the .013 with the sum of 3.8 and
5.4.
77. _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 (S 111).
78. _Division by Decimal._--To divide one decimal by another, we must
reduce them to the same denomination, as explained in S 76, and then
omit the decimal points. Thus 5.413 [:] 3.8 = (5413/1000) [:]
(3800/1000) = 5413 [:] 3800.
79. _Historical Development of Fractions and Decimals._--The fractions
used in ancient times were mainly of two kinds: unit-fractions, i.e.
fractions representing aliquot parts (S 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 2/3 as a single fraction. Except in the case of 2/3 and 1/2, 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 [xi]
represented 60, and [omicron] 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 _as_ (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 notation came to be generally
accepted. Under one system, for instance, the continued sum 4/5 + 1/(7 X
5) + 3/(8 X 7 X 5) would be denoted by (3 1 4)/(8 7 5); 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
A.D. 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 sexagesimal 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 1617 by H. Briggs, the computer
of logarithms.
80. _Fractions 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 _equality_ 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 oz. 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.