0 1 2 3 4 5 6 7 8 9
0 1 2 3 4 5
indicates that after getting to 4 we make a fresh start from 4 as our
zero.
To subtract, we may proceed in either of two ways. The subtraction of 4
from 9 may mean either "What has to be added to 4 in order to make up a
total of 9," or "To what has 4 to be added in order to make up a total
of 9." For the former meaning we count forwards, till we get to 4, and
then make a new count, parallel with the continuation of the old series,
and see at what number we arrive when we get to 9. This corresponds to
the concrete method, in which we have 9 objects, take away 4 of them,
and recount the remainder. The alternative method is to retrace the
steps of addition, i.e. to count backwards, treating 9 of one (the
standard) series as corresponding with 4 of the other, and finding which
number of the former corresponds with 0 of the latter. This is a more
advanced method, which leads easily to the idea of negative quantities,
if the subtraction is such that we have to go behind the 0 of the
standard series.
32. _Mixed Quantities._--The application of the above principles, and of
similar principles with regard to multiplication and division, to
numerical quantities expressed in any of the diverse British
denominations, presents no theoretical difficulty if the successive
denominations are regarded as constituting a varying scale of notation
(S 17). Thus the expression 2 ft. 3 in. implies that in counting inches
we use 0 to eleven instead of 0 to 9 as our first repeating series, so
that we put down 1 for the next denomination when we get to twelve
instead of when we get to ten. Similarly 3 yds. 2 ft. means
yds. 0 1 2 3
ft. 0 1 2 0 1 2 0 1 2 0 1 2
The practical difficulty, of course, is that the addition of two numbers
produces different results according to the scale in which we are for
the moment proceeding; thus the sum of 9 and 8 is 17, 15, 13 or 11
according as we are dealing with shillings, pence, pounds (avoirdupois)
or ounces. The difficulty may be minimized by using the notation
explained in S 17.
(iii.) _Multiples, Submultiples and Quotients._
33. _Multiplication_ and _Division_ are the names given to certain
numerical processes which have to be performed in order to find the
result of certain arithmetical operations. Each process may arise out of
either of two distinct operations; but the terminology is based on the
processes, not on the operations to which they belong, and the latter
are not always clearly understood.
34. _Repetition and Subdivision._--_Multiplication_ occurs when a
certain number or numerical quantity is treated as a _unit_ (S 11), and
is taken a certain _number_ of times. It therefore arises in one or
other of two ways, according as the unit or the number exists first in
consciousness. If pennies are arranged in groups of five, the total
amounts arranged are successively once 5d., twice 5d., three times 5d.,
...; which are written 1 X 5d., 2 X 5d., 3 X 5d., ... (S 28). This
process is _repetition_, and the quantities 1 X 5d., 2 X 5d., 3 X 5d.,
... are the successive _multiples_ of 5d. If, on the other hand, we have
a sum of 5s., and treat a shilling as being equivalent to twelve pence,
the 5s. is equivalent to 5 X 12d.; here the multiplication arises out of
a _subdivision_ of the original unit 1s. into 12d.
Although multiplication may arise in either of these two ways, the
actual process in each case is performed by commencing with the unit and
taking it the necessary number of times. In the above case of
subdivision, for instance, each of the 5 shillings is separately
converted into pence, so that we do in fact find in succession once
12d., twice 12d., ...; i.e. we find the multiples of 12d. up to 5
times.
The result of the multiplication is called the _product_ of the unit by
the number of times it is taken.
35. _Diagram of Multiplication._--The process of multiplication is
performed in order to obtain such results as the following:--
If 1 boy receives 7 apples,
then 3 boys receive 21 apples;
or
If 1s. is equivalent to 12d.,
then 5s. is equivalent to 60d.
The essential portions of these statements, from the arithmetical point
of view, may be exhibited in the form of the diagrams A and B:--
A B
+--------+-----------+ +-----+------+
| 1 boy | 7 apples | | 1s. | 12d. |
+--------+-----------+ +-----+------+
| 3 boys | 21 apples | | 5s. | 60d. |
+--------+-----------+ +-----+------+
or more briefly, as in C or C' and D or D':--
C C' D D'
+---+-----------+ | +---+------+ |
| 1 | 7 apples | | 7 apples | 1 | 12d. | | 12d.
+---+-----------+ ---+----------- +---+------+ ---+------
| 3 | 21 apples | 3 | 21 apples | 5 | 60d. | 5 | 60d.
+---+-----------+ | +---+------+ |
the general arrangement of the diagram being as shown in E or E':--
E E'
+--------+---------+ |
| 1 | Unit | | Unit
+--------+---------+ --------+---------
| Number | Product | Number | Product
+--------+---------+ |
Multiplication is therefore equivalent to completion of the diagram by
entry of the product.
36. _Multiple-Tables._--The diagram C or D of S 35 is part of a complete
table giving the successive multiples of the particular unit. If we take
several different units, and write down their successive multiples in
parallel columns, preceded by the number-series, we obtain a
_multiple-table_ such as the following:--
+---+---+----+----+----------+---------------+-------+-----
| 1 | 1 | 2 | 9 | 1s. 5d. | 3 yds. 2 ft. | 17359 | ...
+---+---+----+----+----------+---------------+-------+-----
| 2 | 2 | 4 | 18 | 2s. 10d. | 7 yds. 1 ft. | 34718 | ...
+---+---+----+----+----------+---------------+-------+-----
| 3 | 3 | 6 | 27 | 4s. 3d. | 11 yds. 0 ft. | 52077 | ...
+---+---+----+----+----------+---------------+-------+-----
| 4 | 4 | 8 | 36 | 5s. 8d. | 14 yds. 2 ft. | 69436 | ...
+---+---+----+----+----------+---------------+-------+-----
| 5 | 5 | 10 | 45 | 7s. 1d. | 16 yds. 1 ft. | 86795 | ...
+---+---+----+----+----------+---------------+-------+-----
| . | . | . | . | . | . | . | ...
| . | . | . | . | . | . | . | ...
| . | . | . | . | . | . | . | ...
| . | . | . | . | . | . | . | ...
It is to be considered that each column may extend downwards
indefinitely.
37. _Successive Multiplication._--In multiplication by repetition the
unit is itself usually a multiple of some other unit, i.e. it is a
product which is taken as a new unit. When this new unit has been
multiplied by a number, we can again take the product as a unit for the
purpose of another multiplication; and so on indefinitely. Similarly
where multiplication has arisen out of the subdivision of a unit into
smaller units, we can again subdivide these smaller units. Thus we get
successive multiplication; but it represents quite different operations
according as it is due to repetition, in the sense of S 34, or to
subdivision, and these operations will be exhibited by different
diagrams. Of the two diagrams below, A exhibits the successive
multiplication of L3 by 20, 12 and 4, and B the successive reduction of
L3 to shillings, pence and farthings. The principle on which the
diagrams are constructed is obvious from S 35. It should be noticed that
in multiplying L3 by 20 we find the value of 20.3, but that in reducing
L3 to shillings, since each L becomes 20s., we find the value of 3.20.
A B
+----+-------+ +-------+--------+
| 1 | L3 | | 1d. | 4f. |
+-----+----+-------+ +------+-------+--------+
| 1 | 20 | L60 | | 1s. | 12d. | |
+---+-----+----+-------+ +----+------+-------+--------+
| 1 | 12 | | L720 | | L1 | 20s. | | |
+---+-----+----+-------+ +----+------+-------+--------+
| 4 | | | L2880 | | L3 | 60s. | 720d. | 2880f. |
+---+-----+----+-------+ +----+------+-------+--------+
38. _Submultiples._--The relation of a unit to its successive multiples
as shown in a multiple-table is expressed by saying that it is a
submultiple of the multiples, the successive submultiples being
_one-half, one-third, one-fourth_, ... Thus, in the diagram of S 36, 1s.
5d. is one-half of 2s. 10d., one-third of 4s. 3d., one-fourth of 5s.
8d., ...; these being written "1/2 of 2s. 10d.," "1/3 of 4s. 3d.," "1/4
of 5s. 8d,"...
The relation of submultiple is the converse of that of multiple; thus if
a is 1/5 of b, then b is 5 times a. The determination of a submultiple
is therefore equivalent to completion of the diagram E or E' of S 35 by
entry of the unit, when the number of times it is taken, and the
product, are given. The operation is the converse of repetition; it is
usually called _partition_, as representing division into a number of
equal shares.
39. _Quotients._--The converse of subdivision is the formation of units
into groups, each constituting a larger unit; the number of the groups
so formed out of a definite number of the original units is called a
_quotient_. The determination of a quotient is equivalent to completion
of the diagram by entry of the number when the unit and the product are
given. There is no satisfactory name for the operation, as distinguished
from partition; it is sometimes called measuring, but this implies an
equality in the original units, which is not an essential feature of the
operation.
40. _Division._--From the commutative law for multiplication, which
shows that 3 X 4d. = 4 X 3d. = 12d., it follows that the number of pence
in one-fourth of 12d. is equal to the quotient when 12 pence are formed
into units of 4d.; each of these numbers being said to be obtained by
_dividing_ 12 by 4. The term _division_ is therefore used in text-books
to describe the two processes described in SS 38 and 39; the product
mentioned in S 34 is the _dividend_, the number or the unit, whichever
is given, is called the _divisor_, and the unit or number which is to be
found is called the _quotient_. The symbol [:] is used to denote both
kinds of division; thus A [:] n denotes the unit, n of which make up A,
and A [:] B denotes the number of times that B has to be taken to make
up A. In the present article this confusion is avoided by writing the
former as 1/n of A.
Methods of division are considered later (SS 106-108).
41. _Diagrams of Division._--Since we write from left to right or
downwards, it may be convenient for division to interchange the rows or
the columns of the multiplication-diagram. Thus the uncompleted diagram
for partition is F or G, while for measuring it is usually H; the vacant
compartment being for the unit in F or G, and for the number in H. In
some cases it may be convenient in measuring to show both the units, as
in K.
F G H K
+--------+---------+ +--------+---------+ +---------+---+ +------+-----+
| 1 | | | Number | Product | | Unit | 1 | | 12d. | 1s. |
+--------+---------+ +--------+---------+ +---------+---+ +------+-----+
| Number | Product | | 1 | | | Product | | | 60d. | |
+--------+---------+ +--------+---------+ +---------+---+ +------+-----+
42. _Successive Division_ may be performed as the converse of successive
multiplication. The diagrams A and B below are the converse (with a
slight alteration) of the corresponding diagrams in S 37; A
representing the determination of 1/20 of 1/12 of 1/4 of 2880 farthings,
and B the conversion of 2880 farthings into L.
A B
+---+--------+ +------+----+
| 4 | 2880f. | | 20s. | L1 |
+----+---+--------+ +-------+------+----+
| 12 | 1 | 720f. | | 12d. | 1s. | |
+----+----+---+--------+ +--------+-------+------+----+
| 20 | 1 | | 60f. | | 4f. | 1d. | | |
+----+----+---+--------+ +--------+-------+------+----+
| 1 | | | 3f. | | 2880f. | 720d. | 60s. | L3 |
+----+----+---+--------+ +--------+-------+------+----+
(iv.) _Properties of Numbers._
(A) Properties not depending on the Scale of Notation.
43. _Powers, Roots and Logarithms._--The standard series 1, 2, 3, ... is
obtained by successive additions of 1 to the number last found. If
instead of commencing with 1 and making successive additions of 1 we
commence with any number such as 3 and make successive multiplications
by 3, we get a series 3, 9, 27, ... as shown below the line in the
margin. The first member of the series is 3; the second is the product
of two numbers, each equal to 3; the third is the product of three
numbers, each equal to 3; and so on. These are written 3^1 (or 3), 3^2,
3^3, 3^4, ... where n^p denotes the product of p numbers, each equal to
n. If we write n^p = N, then, if any two of the three numbers n, p, N
are known, the third is determinate. If we know n and p, p is called the
_index_, and n, n^2, ... n^p are called the _first power, second power,
... pth power_ of n, the series itself being called the _power-series_.
The _second power_ and _third power_ are usually called the _square_ and
_cube_ respectively. If we know p and N, n is called the _pth root_ of
N, so that n is the _second_ (or _square_) _root_ of n^2, the _third_
(or _cube_) _root_ of n^3, the _fourth root_ of n^4, ... If we know n
and N, then p is the _logarithm_ of N to _base_ n.
0 1 = 3^0 n^0
------------------
1 3 = 3^1 n^1
2 9 = 3^2 n^2
3 27 = 3^3 n^3
4 81 = 3^4 n^4
: : : :
: : : :
The calculation of powers (i.e. of N when n and p are given) is
_involution_; the calculation of roots (i.e. of n when p and N are
given) is _evolution_; the calculation of logarithms (i.e. of p when n
and N are given) has no special name.
Involution is a direct process, consisting of successive
multiplications; the other two are inverse processes. The calculation of
a logarithm can be performed by successive divisions; evolution requires
special methods.
The above definitions of logarithms, &c., relate to cases in which n and
p are whole numbers, and are generalized later.
44. _Law of Indices._--If we multiply n^p by n^q, we multiply the
product of p n's by the product of q n's, and the result is therefore
n^(p + q). Similarly, if we divide n^p by n^q, where q is less than p,
the result is n^(p - q). Thus multiplication and division in the
power-series correspond to addition and subtraction in the index-series,
and vice versa.
If we divide n^p by n^p, the quotient is of course 1. This should be
written n^0. Thus we may make the power-series commence with 1, if we
make the index-series commence with 0. The added terms are shown above
the line in the diagram in S 43.
45. _Factors, Primes and Prime Factors._--If we take the successive
multiples of 2, 3, ... as in S 36, and place each multiple opposite the
same number in the original series, we get an arrangement as in the
adjoining diagram. If any number N occurs in the vertical series
commencing with a number n (other than 1) then n is said to be a
_factor_ of N. Thus 2, 3 and 6 are factors of 6; and 2, 3, 4, 6 and 12
are factors of 12.
1 .. .. .. .. .. .. ..
2 2 .. .. .. .. .. ..
3 .. 3 .. .. .. .. ..
4 4 .. 4 .. .. .. ..
5 .. .. .. 5 .. .. ..
6 6 6 .. .. 6 .. ..
7 .. .. .. .. .. 7 ..
8 8 .. 8 .. .. .. 8
9 .. 9 .. .. .. .. ..
10 10 .. .. 10 .. .. ..
11 .. .. .. .. .. .. ..
12 12 12 12 .. 12 .. ..
: : : : : : : :
: : : : : : : :
A number (other than 1) which has no factor except itself is called a
_prime number_, or, more briefly, a _prime_. Thus 2, 3, 5, 7 and 11 are
primes, for each of these occurs twice only in the table. A number
(other than 1) which is not a prime number is called a _composite_
number.
If a number is a factor of another number, it is a factor of any
multiple of that number. Hence, if a number has factors, one at least of
these must be a prime. Thus 12 has 6 for a factor; but 6 is not a prime,
one of its factors being 2; and therefore 2 must also be a factor of 12.
Dividing 12 by 2, we get a submultiple 6, which again has a prime 2 as a
factor. Thus any number which is not itself a prime is the product of
several factors, each of which is a prime, e.g. 12 is the product of 2,
2 and 3. These are called _prime factors_.
The following are the most important properties of numbers in reference
to factors:--
(i) If a number is a factor of another number, it is a factor of any
multiple of that number.
(ii) If a number is a factor of two numbers, it is a factor of their sum
or (if they are unequal) of their difference. (The words in brackets are
inserted to avoid the difficulty, at this stage, of saying that every
number is a factor of 0, though it is of course true that 0.n = 0,
whatever n may be.)
(iii) A number can be resolved into prime factors in one way only, no
account being taken of their relative order. Thus 12 = 2 X 2 X 3 = 2 X 3
X 2 = 3 X 2 X 2, but this is regarded as one way only. If any prime
occurs more than once, it is usual to write the number of times of
occurrence as an index; thus 144 = 2 X 2 X 2 X 2 X 3 X 3 = 2^4 . 3^2.
The number 1 is usually included amongst the primes; but, if this is
done, the last paragraph requires modification, since 144 could be
expressed as 1 . 2^4 . 3^2, or as 1^2 . 2^4 . 3^2, or as 1^p . 2^4 .
3^2, where p might be anything.
If two numbers have no factor in common (except 1) each is said to be
_prime to_ the other.
The multiples of 2 (including 1.2) are called _even_ numbers; other
numbers are _odd_ numbers.
46. _Greatest Common Divisor._--If we resolve two numbers into their
prime factors, we can find their _Greatest Common Divisor_ or _Highest
Common Factor_ (written G.C.D. or G.C.F. or H.C.F.), i.e. the greatest
number which is a factor of both. Thus 144 = 2^4 . 3^2, and 756 = 2^2 .
3^3 . 7, and therefore the G.C.D. of 144 and 756 is 2^2 . 3^2 = 36. If
we require the G.C.D. of two numbers, and cannot resolve them into their
prime factors, we use a process described in the text-books. The process
depends on (ii) of S 45, in the extended form that, if x is a factor of
a and b, it is a factor of pa - qb, where p and q are any integers.
The G.C.D. of three or more numbers is found in the same way.
47. _Least Common Multiple._--The _Least Common Multiple_, or L.C.M., of
two numbers, is the least number of which they are both factors. Thus,
since 144 = 2^4 . 3^2, and 756 = 2^2 . 3^3 . 7, the L.C.M. of 144 and
756 is 2^4 . 3^3 . 7. It is clear, from comparison with the last
paragraph, that the product of the G.C.D. and the L.C.M. of two numbers
is equal to the product of the numbers themselves. This gives a rule for
finding the L.C.M. of two numbers. But we cannot apply it to finding the
L.C.M. of three or more numbers; if we cannot resolve the numbers into
their prime factors, we must find the L.C.M. of the first two, then the
L.C.M. of this and the next number, and so on.
(B) Properties depending on the Scale of Notation.
48. _Tests of Divisibility._--The following are the principal rules for
testing whether particular numbers are factors of a given number. The
number is divisible--
(i) by 10 if it ends in 0;
(ii) by 5 if it ends in 0 or 5;
(iii) by 2 if the last digit is even;
(iv) by 4 if the number made up of the last two digits is divisible by 4;
(v) by 8 if the number made up of the last three digits is divisible by 8;
(vi) by 9 if the sum of the digits is divisible by 9;
(vii) by 3 if the sum of the digits is divisible by 3;
(viii) by 11 if the difference between the sum of the 1st, 3rd, 5th, ...
digits and the sum of the 2nd, 4th, 6th, ... is zero or divisible by
11.
(ix) To find whether a number is divisible by 7, 11 or 13, arrange the
number in groups of three figures, beginning from the end, treat each
group as a separate number, and then find the difference between the sum
of the 1st, 3rd, ... of these numbers and the sum of the 2nd, 4th, ...
Then, if this difference is zero or is divisible by 7, 11 or 13, the
original number is also so divisible; and conversely. For example, 31521
gives 521 - 31 = 490, and therefore is divisible by 7, but not by 11 or
13.
49. _Casting out Nines_ is a process based on (vi) of the last
paragraph. The remainder when a number is divided by 9 is equal to the
remainder when the sum of its digits is divided by 9. Also, if the
remainders when two numbers are divided by 9 are respectively a and b,
the remainder when their product is divided by 9 is the same as the
remainder when a.b is divided by 9. This gives a rule for testing
multiplication, which is found in most text-books. It is doubtful,
however, whether such a rule, giving a test which is necessarily
incomplete, is of much educational value.
(v.) _Relative Magnitude._
50. _Fractions._--A _fraction_ of a quantity is a submultiple, or a
multiple of a submultiple, of that quantity. Thus, since 3 X 1s. 5d. =
4s. 3d., 1s. 5d. may be denoted by 1/3 of 4s. 3d.; and any multiple of
1s. 5d., denoted by n X 1s. 5d., may also be denoted by n/3 of 4s. 3d.
We therefore use "n/a of A" to mean that we find a quantity X such that
a X X = A, and then multiply X by n.
It must be noted (i) that this is a definition of "n/a of," not a
definition of "n/a," and (ii) that it is not necessary that n should be
less than a.
51. _Subdivision of Submultiple._--By 5/7 of A we mean 5 times the unit,
7 times which is A. If we regard this unit as being 4 times a lesser
unit, then A is 7.4 times this lesser unit, and 5/7 of A is 5.4 times
the lesser unit. Hence 5/7 of A is equal to (5.4)/(7.4) of A; and,
conversely, (5.4)/(7.4) of A is equal to 5/7 of A. Similarly each of
these is equal to (5.3)/(7.3) of A. Hence the value of a fraction is not
altered by substituting for the numerator and denominator the
corresponding numbers in any other column of a multiple-table (S 36). If
we write (5.4)/(7.4) in the form (4.5)/(4.7) we may say that the value
of a fraction is not altered by multiplying or dividing the numerator
and denominator by any number.
52. _Fraction of a Fraction._--To find 11/4 of 5/7 of A we must convert
5/7 of A into 4 times some unit. This is done by the preceding
paragraph. For 5/7 of A = (5.4)/(7.4) of A = (4.5)/(7.4) of A; i.e. it
is 4 times a unit which is itself 5 times another unit, 7.4 times, which
is A. Hence, taking the former unit 11 times instead of 4 times,