III

fig. 2. fig. 3. 26. _representation of geometrical magnitude by number._--the application of arithmetical methods to geometrical measurement presents some difficulty. in reality there is a transition from a cardinal to an ordinal system, but to an ordinal system which does not agree with the original ordinal system from which the cardinal system was derived. to see this, we may represent ordinal numbers by the ordinary numerals 1, 2, 3, ... and cardinal numbers by the roman i, ii, iii, ... then in the earliest stage each object counted is indivisible; either we are counting it as a whole, or we are not counting it at all. the symbols 1, 2, 3, ... then refer to the individual objects, as in fig. 1; this is the primary ordinal stage. figs. 2 and 3 represent the cardinal stage; fig. 2 showing how the i, ii, iii, ... denote the successively larger groups of objects, while fig. 3 shows how the name ii of the whole is determined by the name 2 of the last one counted. when now we pass to geometrical measurement, each "one" is a thing which is itself divisible, and it cannot be said that at any moment we are counting it; it is only when one is completed that we can count it. the names 1, 2, 3, ... for the individual objects cease to have an intelligible meaning, and measurement is effected by the cardinal numbers i, ii, iii, ..., as in fig. 4. these cardinal numbers have now, however, come to denote individual points in the line of measurement, i.e. the points of separation of the individual units of length. the point iii in fig. 4 does not include the point ii in the same way that the number iii includes the number ii in fig. 2, and the points must therefore be denoted by the ordinal numbers 1, 2, 3, ... as in fig. 5, the zero 0 falling into its natural place immediately before the commencement of the first unit. 1 2 3 _____ _____ _____ _____ _____ _____ ... 0 1 2 3 ... i ii iii \____/ / / fig. 5 \____________/ / \_____________________/ fig. 4 thus, while arithmetical numbering refers to units, geometrical numbering does not refer to units but to the intervals between units. iii. arithmetic of integral numbers (i.) _preliminary_ 27. _equality and identity._--there is a certain difference between the use of words referring to equality and identity in arithmetic and in algebra respectively; what is an _equality_ in the former becoming an _identity_ in the latter. thus the statement that 4 times 3 is equal to 3 times 4, or, in abbreviated form, 4 x 3 = 3 x 4 (s 28), is a statement not of identity but of equality; i.e. 4 x 3 and 3 x 4 mean different things, but the operations which they denote produce the same result. but in algebra a x b = b x a is called an identity, in the sense that it is true whatever a and b may be; while n x x = a is called an equation, as being true, when n and a are given, for one value only of x. similarly the numbers represented by 6/12 and 1/2 are not identical, but are equal. 28. _symbols of operation._--the failure to observe the distinction between an identity and an equality often leads to loose reasoning; and in order to prevent this it is important that definite meanings should be attached to all symbols of operation, and especially to those which represent elementary operations. the symbols - and [:] mean respectively that the first quantity mentioned is to be reduced or divided by the second; but there is some vagueness about + and x. in the present article a + b will mean that a is taken first, and b added to it; but a x b will mean that b is taken first, and is then multiplied by a. in the case of numbers the x may be replaced by a dot; thus 4.3 means 4 times 3. when it is necessary to write the multiplicand before the multiplier, the symbol [x] will be used, so that b [x] a will mean the same as a x b. 29. _axioms._--there are certain statements that are sometimes regarded as axiomatic; e.g. that if equals are added to equals the results are equal, or that if a is greater than b then a + x is greater than b + x. such statements, however, are capable of logical proof, and are generalizations of results obtained empirically at an elementary stage; they therefore belong more properly to the laws of arithmetic (s 58). (ii.) _sums and differences._ 30. _addition and subtraction._--_addition_ is the process of expressing (in numeration or notation) a whole, the parts of which have already been expressed; while, if a whole has been expressed and also a part or parts, _subtraction_ is the process of expressing the remainder. except with very small numbers, addition and subtraction, on the grouping system, involve analysis and rearrangement. thus the sum of 8 and 7 cannot be expressed as ones; we can either form the whole, and regroup it as 10 and 5, or we can split up the 7 into 2 and 5, and add the 2 to the 8 to form 10, thus getting 8 + 7 = 8 + (2 + 5) = (8 + 2) + 5 = 10 + 5 = 15. for larger numbers the rearrangement is more extensive; thus 24 + 31 = (20 + 4) + (30 + 1) = (20 + 30) + (4 + 1) = 50 + 5 = 55, the process being still more complicated when the ones together make more than ten. similarly we cannot subtract 8 from 15, if 15 means 1 ten + 5 ones; we must either write 15 - 8 = (10 + 5) - 8 = (10 - 8) + 5 = 2 + 5 = 7, or else resolve the 15 into an inexpressible number of ones, and then subtract 8 of them, leaving 7. numerical quantities, to be added or subtracted, must be in the same denomination; we cannot, for instance, add 55 shillings and 100 pence, any more than we can add 3 yards and 2 metres. 31. _relative position in the series._--the above method of dealing with addition and subtraction is synthetic, and is appropriate to the grouping method of dealing with number. we commence with processes, and see what they lead to; and thus get an idea of sums and differences. if we adopted the counting method, we should proceed in a different way, our method being analytic. one number is less or greater than another, according as the symbol (or ordinal) of the former comes earlier or later than that of the latter in the number-series. thus (writing ordinals in light type, and cardinals in heavy type) 9 comes after 4, and therefore 9 is greater than 4. to find how much greater, we compare two series, in one of which we go up to 9, while in the other we stop at 4 and then recommence our counting. the series are shown below, the numbers being placed horizontally for convenience of printing, instead of vertically (s 14):-- 1 2 3 4 5 6 7 8 9 1 2 3 4 1 2 3 4 5 this exhibits 9 as the sum of 4 and 5; it being understood that the sum of 4 and 5 means that we add 5 to 4. that this gives the same result as adding 4 to 5 may be seen by reckoning the series backwards. it is convenient to introduce the zero; thus 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, 11/4 of 5/7 of a = (11.5)/(7.4) of a a fraction of a fraction is sometimes called a _compound fraction._ 53. _comparison, addition and subtraction of fractions._--the quantities 3/4 of a and 5/7 of a are expressed in terms of different units. to compare them, or to add or subtract them, we must express them in terms of the same unit. thus, taking 1/28 of a as the unit, we have (s 51) 3/4 of a = 21/28 of a; 5/7 of a = 20/28 of a. hence the former is greater than the latter; their sum is 41/28 of a; and their difference is 1/28 of a. thus the fractions must be reduced to a _common denominator_. this denominator must, if the fractions are in their lowest terms (s 54), be a multiple of each of the denominators; it is usually most convenient that it should be their l.c.m. (s 47). 54. _fraction in its lowest terms._--a fraction is said to be _in its lowest terms_ when its numerator and denominator have no common factor; or to be reduced to its lowest terms when it is replaced by such a fraction. thus 8/22 of a is said to be reduced to its lowest terms when it is replaced by 4/11 of a. it is important always to bear in mind that 4/11 of a is not the _same_ as 8/22 of a, though it is _equal_ to it. +------+-----------+ | 1 | 7d. | +------+-----------+ | 10 | 5s.10d. | +------+-----------+ | 24 | 14s. | +------+-----------+ 55. _diagram of fractional relation._--to find 10/24 of 14s. we have to take 10 of the units, 24 of which make up 14s. hence the required amount will, in the multiple-table of s 36, be opposite 10 in the column in which the amount opposite 24 is 14s.; the quantity at the head of this column, representing the unit, will be found to be 7d. the elements of the multiple-table with which we are concerned are shown in the diagram in the margin. this diagram serves equally for the two statements that (i) 10/24 of 14s. is 5s. 10d., (ii) 24/10 of 5s. 10d. is 14s. the two statements are in fact merely different aspects of a single relation, considered in the next section. a +------+-----------+ | 10 | 5s. 10d. | +------+-----------+ | 24 | 14s. | +------+-----------+ b +------+-----------+ | 5 | 5s. 10d. | +------+-----------+ | 12 | 14s. | +------+-----------+ 56. _ratio._--if we omit the two upper compartments of the diagram in the last section, we obtain the diagram a. this diagram exhibits a relation between the two amounts 5s. 10d. and 14s. on the one hand, and the numbers 10 and 24 of the standard series on the other, which is expressed by saying that 5s. 10d. is to 14s. in the _ratio_ of 10 to 24, or that 14s. is to 5s. 10d. in the ratio of 24 to 10. if we had taken 1s. 2d. instead of 7d. as the unit for the second column, we should have obtained the diagram b. thus we must regard the ratio of a to b as being the same as the ratio of c to d, if the fractions a/b and c/d are equal. for this reason the ratio of a to b is sometimes written a/b, but the more correct method is to write it a:b. if two quantities or numbers p and q are to each other in the ratio of p to q, it is clear from the diagram that p times q = q times p, so that q = q/p of p. 57. _proportion._--if from any two columns in the table of s 36 we remove the numbers or quantities in any two rows, we get a diagram such as that here shown. the pair of compartments on either side may, as here, contain numerical quantities, or may contain numbers. but the two pairs of compartments will correspond to a single pair of numbers, e.g. 2 and 6, in the standard series, so that, denoting them by m, n and p, q respectively, m will be to n in the same ratio that p is to q. +----------+---------------+ | 2s. 10d. | 7 yds. 1 ft. | +----------+---------------+ | 8s. 6d. | 22 yds. | +----------+---------------+ this is expressed by saying that m is to n as p to q, the relation being written m : n :: p : q; the four quantities are then said to be _in proportion_ or to be _proportionals_. +---+---+ | m | p | +---+---+ | n | q | +---+---+ this is the most general expression of the relative magnitude of two quantities; i.e. the relation expressed by proportion includes the relations expressed by multiple, submultiple, fraction and ratio. if m and n are respectively m and n times a unit, and p and q are respectively p and q times a unit, then the quantities are in proportion if mq = np; and conversely. iv. laws of arithmetic 58. _laws of arithmetic._--the arithmetical processes which we have considered in reference to positive integral numbers are subject to the following laws:-- (i) _equalities and inequalities._--the following are sometimes called _axioms_ (s 29), but their truth should be proved, even if at an early stage it is assumed. the symbols ">" and "<" mean respectively "is greater than" and "is less than." the numbers represented by a, b, c, x and m are all supposed to be positive. (a) if a = b, and b = c. then a = c; (b) if a = b, then a + x = b + x, and a - x = b - x; (c) if a > b, then a + x > b + x, and a - x > b - x; (d) if a < b, then a + b < b + x, and a - x < b - x; (e) if a = b, then ma = mb, and a [:] m = b [:] m; (f) if a > b, then ma > mb, and a [:] m > b [:] m; (g) if a < b, then ma < mb, and a [:] m < b [:] m. (ii) _associative law for additions and subtractions._--this law includes the _rule of signs_, that a - (b - c) = a - b + c; and it states that, subject to this, successive operations of addition or subtraction may be grouped in sets in any way; e.g. a - b + c + d + e - f = a - (b - c) + (d + e - f). (iii) _commutative law for additions and subtractions_, that additions and subtractions may be performed in any order; e.g. a - b + c + d = a + c - b + d = a - b + c - b. (iv) _associative law for multiplications and divisions._--this law includes a rule, similar to the rule of signs, to the effect that a[:](b[:]c) = a [:] b [x] c; and it states that, subject to this, successive operations of multiplication or division may be grouped in sets in any way; e.g. a b [x] c [x] d [x] e [:] f = a [:] (b [:] c) [x] (d [x] e [:] f). (v) _commutative law for multiplications and divisions_, that multiplications and divisions may be performed in any order: e.g. a [:] b [x] c [x] d = a [x] c [:] b [x] d = a [x] d [x] c [:] b. (vi) _distributive law_, that multiplications and divisions may be distributed over additions and subtractions, e.g. that m(a + b - c) = m.a + m.b - m.c, or that (a + b - c) [:] n = (a [:] n) + (b [:] n) + (c [:] n). in the case of (ii), (iii) and (vi), the letters a, b, c, ... may denote either numbers or numerical quantities, while m and n denote numbers; in the case of (iv) and (v) the letters denote numbers only. 59. _results of inverse operations._--addition, multiplication and involution are direct processes; and, if we start with positive integers, we continue with positive integers throughout. but, in attempting the inverse processes of subtraction, division, and either evolution or determination of index, the data may be such that a process cannot be performed. we can, however, denote the result of the process by a symbol, and deal with this symbol according to the laws of arithmetic. in this way we arrive at (i) negative numbers, (ii) fractional numbers, (iii) surds, (iv) logarithms (in the ordinary sense of the word). 60. _simple formulae._--the following are some simple formulae which follow from the laws stated in s 58. (i) (a + b + c + ...)(p + q + r + ...) = (ap + aq + ar + ...) + (bp + bq + br + ...) + (cp + cq + cr + ...)+ ...; i.e. the product of two or more numbers, each of which consists of two or more parts, is the sum of the products of each part of the one with each part of the other. (ii) (a + b)(a - b) = a^2 - b^2; i.e. the product of the sum and the difference of two numbers is equal to the difference of their squares. (iii) (a + b)^2 = a^2 + 2ab + b^2 = a^2 + (2a + b)b. v. negative numbers 61. _negative numbers_ may be regarded as resulting from the commutative law for addition and subtraction. according to this law, 10 + 3 + 6 - 7 = 10 + 3 - 7 + 6 = 3 + 6 - 7 + 10 = &c. but, if we write the expression as 3 - 7 + 6 + 10, this means that we must first subtract 7 from 3. this cannot be done; but the result of the subtraction, if it could be done, is something which, when 6 is added to it, becomes 3 - 7 + 6 = 3 + 6 - 7 = 2. the result of 3 - 7 is the same as that of 0 - 4; and we may write it "-4," and call it a _negative number_, if by this we mean something possessing the property that -4 + 4 = 0. this, of course, is unintelligible on the grouping system of treating number; on the counting system it merely means that we count backwards from 0, just as we might count inches backwards from a point marked 0 on a scale. it should be remembered that the counting is performed with something as unit. if this unit is a, then what we are really considering is -4a; and this means, not that a is multiplied by -4, but that a is multiplied by 4, and the product is taken negatively. it would therefore be better, in some ways, to retain the unit throughout, and to describe -4a as a _negative quantity_, in order to avoid confusion with the "negative numbers" with which operations are performed in formal algebra. the positive quantity or number obtained from a negative quantity or number by omitting the "-" is called its _numerical value_. vi. fractional and decimal numbers 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. vii. approximation 81. _approximate character of numbers._--the numbers (integral or decimal) by which we represent the results of arithmetical operations are often only approximately correct. all numbers, for instance, which represent physical measurements, are limited in their accuracy not only by our powers of measurement but also by the accuracy of the measure we use as our unit. also most fractions cannot be expressed exactly as decimals; and this is also the case for surds and logarithms, as well as for the numbers expressing certain ratios which arise out of geometrical relations. even where numbers are supposed to be exact, calculations based on them can often only be approximate. we might, for instance, calculate the exact cost of 3 lb. 5 oz. of meat at 9-1/2 d. a lb., but there are no coins in which we could pay this exact amount. when the result of any arithmetical operation or operations is represented approximately but not exactly by a number, the excess (positive or negative) of this number over the number which would express the result exactly is called the _error_. 82. _degree of accuracy._--there are three principal ways of expressing the degree of accuracy of any number, i.e. the extent to which it is equal to the number it is intended to represent. (i) a number can be _correct to_ so many _places of decimals_. this means (cf. s 71) that the number differs from the true value by less than one-half of the unit represented by 1 in the last place of decimals. for instance, .143 represents 1/7 correct to 3 places of decimals, since it differs from it by less than .0005. the final figure, in a case like this, is said to be _corrected_. this method is not good for comparative purposes. thus .143 and 14.286 represent respectively 1/7 and 100/7 to the same number of places of decimals, but the latter is obviously more exact than the former. (ii) a number can be correct to so many _significant figures_. the significant figures of a number are those which commence with the first figure other than zero in the number; thus the significant figures of 13.027 and of .00013027 are the same. this is the usual method; but the relative accuracy of two numbers expressed to the same number of significant figures depends to a certain extent on the magnitude of the first figure. thus .14286 and .85714 represent 1/7 and 6/7 correct to 5 significant figures; but the latter is relatively more accurate than the former. for the former shows only that 1/7 lies between .142855 and .142865, or, as it is better expressed, between .14285-1/2 and .14286-1/2; but the latter shows that 6/7 lies between .85713-1/2 and .85714-1/2, and therefore that 1/7 lies between .14285-7/12 and .14285-9/12. in either of the above cases, and generally in any case where a number is known to be within a certain limit on each side of the stated value, the _limit of error_ is expressed by the sign [+-]. thus the former of the above two statements would give 1/7 = .14286 [+-] .000005. it should be observed that the numerical value of the error is to be subtracted from or added to the stated value according as the error is positive or negative. (iii) the limit of error can be expressed as a fraction of the number as stated. thus 1/7 = .143 [+-] .0005 can be written 1/7 = 143(1 [+-] 1/286). 83. _accuracy after arithmetical operations._--if the numbers which are the subject of operations are not all exact, the accuracy of the result requires special investigation in each case. additions and subtractions are simple. if, for instance, the values of a and b, correct to two places of decimals, are 3.58 and 1.34, then 2.24, as the value of a-b, is not necessarily correct to two places. the limit of error of each being [+-].005, the limit of error of their sum or difference is [+-].01. for multiplication we make use of the formula (s 60 (i)) (a' [+-] [alpha])(b' [+-] [beta]) = a'b' + [alpha][beta] [+-] (a'[beta] + b'[alpha]). if a' and b' are the stated values, and [+-][alpha] and [+-][beta] the respective limits of error, we ought strictly to take a'b' + [alpha][beta] as the product, with a limit of error [+-](a'[beta] + b'[alpha]). in practice, however, both [alpha][beta] and a certain portion of a'b' are small in comparison with a'[beta] and b'[alpha], and we therefore replace a'b' + [alpha][beta] by an approximate value, and increase the limit of error so as to cover the further error thus introduced. in the case of the two numbers given in the last paragraph, the product lies between 3.575 x 1.335 = 4.772625 and 3.585 x 1.345 = 4.821825. we might take the product as (3.58 x 1.34) + (.005)^2 = 4.797225, the limits of error being [+-].005(3.58 + 1.34) = [+-].0246; but it is more convenient to write it in such a form as 4.797 [+-] .025 or 4.80 [+-] .03. if the number of decimal places to which a result is to be accurate is determined beforehand, it is usually not necessary in the actual working to go to more than two or three places beyond this. at the close of the work the extra figures are dropped, the last figure which remains being corrected (s 82 (i)) if necessary. viii. surds and logarithms 84. _roots and surds._--the pth root of a number (s 43) may, if the number is an integer, be found by expressing it in terms of its prime factors; or, if it is not an integer, by expressing it as a fraction in its lowest terms, and finding the pth roots of the numerator and of the denominator separately. thus to find the cube root of 1728, we write it in the form 2^6 . 3^3, and find that its cube root is 2^2 . 3 = 12; or, to find the cube root of 1.728, we write it as 1728/1000 = 216/125 = (2^3 . 3^3)/5^3, and find that the cube root is (2.3)/5 = 1.2. similarly the cube root of 2197 is 13. but we cannot find any number whose cube is 2000. it is, however, possible to find a number whose cube shall approximate as closely as we please to 2000. thus the cubes of 12.5 and of 12.6 are respectively 1953.125 and 2000.376, so that the number whose cube differs as little as possible from 2000 is somewhere between 12.5 and 12.6. again the cube of 12.59 is 1995.616979, so that the number lies between 12.59 and 12.60. we may therefore consider that there is some number x whose cube is 2000, and we can find this number to any degree of accuracy that we please. a number of this kind is called a _surd_; the surd which is the pth root of n is written [root p]n, but if the index is 2 it is usually omitted, so that the square root of n is written [root]n. 85. _surd as a power._--we have seen (ss 43, 44) that, if we take the successive powers of a number n, commencing with 1, they may be written n^0, n^1, n^2, n^3, ..., the series of indices being the standard series; and we have also seen (s 44) that multiplication of any two of these numbers corresponds to addition of their indices. hence we may insert in the power-series numbers with fractional indices, provided that the multiplication of these numbers follows the same law. the number denoted by n^(1/3) will therefore be such that n^(1/3) x n^(1/3) x n^(1/3) = n^(1/3 + 1/3 + 1/3) = n; i.e. it will be the cube root of n. by analogy with the notation of fractional numbers, n^(2/3) will be n^(1/3 + 1/3) = n^(1/3) x n^(1/3); and, generally, n^(p/q) will mean the product of p numbers, the product of q of which is equal to n. thus n^(2/6) will not mean the _same_ as n^(1/3), but will mean the square of n^(1/6); but this will be _equal_ to n^(1/3), i.e. ([root 6]n)^2 = [root 3]n. 86. _multiplication and division of surds._--to add or subtract fractional numbers, we must reduce them to a common denominator; and similarly, to multiply or divide surds, we must express them as power-numbers with the same index. thus ([root 3]2) x ([root]5) = 2^(1/3) x 5^(1/2) = 2^(2/6) x 5^(3/6) = 4^(1/6) x 125^(1/6) = 500^(1/6) = [root 6]500. 87. _antilogarithms._--if we take a fixed number, e.g. 2, as base, and take as indices the successive decimal numbers to any particular number of places of decimals, we get a series of _antilogarithms_ of the indices to this base. thus, if we go to two places of decimals, we have as the integral series the numbers 1, 2, 4, 8, ... which are the values of 2^0, 2^1, 2^2, ... and we insert within this series the successive powers of x, where x is such that x^100 = 2. we thus get the numbers 2^.01, 2^.02, 2^.03, ..., which are the antilogarithms of .01, .02, .03, ... to base 2; the first antilogarithm being 2^.00 = 1, which is thus the antilogarithm of 0 to this (or any other) base. the series is formed by successive multiplication, and any antilogarithm to a larger number of decimal places is formed from it in the same way by multiplication. if, for instance, we have found 2^(.31), then the value of 2^(.316) is found from it by multiplying by the 6th power of the 1000th root of 2. for practical purposes the number taken as base is 10; the convenience of this being that the increase of the index by an integer means multiplication by the corresponding power of 10, i.e. it means a shifting of the decimal point. in the same way, by dividing by powers of 10 we may get negative indices. 88. _logarithms._--if n is the antilogarithm of p to the base a, i.e. if n = a^p, then p is called the logarithm of n to the base a, and is written log_a n. as the table of antilogarithms is formed by successive multiplications, so the logarithm of any given number is in theory found by successive divisions. thus, to find the logarithm of a number to base 2, the number being greater than 1, we first divide repeatedly by 2 until we get a number between 1 and 2; then divide repeatedly by [root 10]2 until we get a number between 1 and [root 10]2; then divide repeatedly by [root 100]2; and so on. if, for instance, we find that the number is approximately equal to 2^3 x ([root 10]2)^5 x ([root 100]2)^7 x ([root 1000]2)^4, it may be written 2^(3.574), and its logarithm to base 2 is 3.574. for a further explanation of logarithms, and for an explanation of the treatment of cases in which an antilogarithm is less than 1, see logarithm. for practical purposes logarithms are usually calculated to base 10, so that log10 10 = 1, log10 100 = 2, &c. ix. units 89. _change of denomination_ of a numerical quantity is usually called _reduction_, so that this term covers, e.g., the expression of l153, 7s. 4d. as shillings and pence and also the expression of 3067s. 4d. as l, s. and d. the usual statement is that to express l153, 7s. as shillings we multiply 153 by 20 and add 7. this, as already explained (s 37), is incorrect. l153 denotes 153 units, each of which is l1 or 20s.; and therefore we must multiply 20s. by 153 and add 7s., i.e. multiply 20 by 153 (the unit being now 1s.) and add 7. this is the expression of the process on the grouping method. on the counting method we have a scale with every 20th shilling marked as a l; there are 153 of these 20's, and 7 over. a +-------------+---------+ | 1s. | 12d. | +-------------+-------------+---------+ |l1 | 20s. | | +-------------+-------------+---------+ |l153, 7s. 4d.| 3067s. 4d. | 36808d. | +-------------+-------------+---------+ b +-----------+---------------+ | 20s. | l1 | +---------+-----------+---------------+ | 12d. | 1s. | | +---------+-----------+---------------+ | 36808d. | 3067s. 4d.| l153, 7s. 4d. | +---------+-----------+---------------+ the simplest case, in which the quantity can be expressed as an integral number of the largest units involved, has already been considered (ss 37, 42). the same method can be applied in other cases by regarding a quantity expressed in several denominations as a fractional number of units of the largest denomination mentioned; thus 7s. 4d. is to be taken as meaning 7(4/12)s., but l0, 7s. 4d. as l0 (7(4/12))/20 (s 17). the reduction of l153, 7s. 4d. to pence, and of 36808d. to l, s. d., on this principle, is shown in diagrams a and b above. for reduction of pounds to shillings, or shillings to pounds, we must consider that we have a multiple-table (s 36) in which the multiples of l1 and of 20s. are arranged in parallel columns; and similarly for shillings and pence. 90. _change of unit._--the statement "l153 = 3060s." is not a statement of _equality_ of the same kind as the statement "153 x 20 = 3060," but only a statement of _equivalence_ for certain purposes; in other words, it does not convey an absolute truth. it is therefore of interest to see whether we cannot replace it by an absolute truth. to do this, consider what the ordinary processes of multiplication and division mean in reference to concrete objects. if we want to give, to 5 boys, 4 apples each, we are said to multiply 4 apples by 5. we cannot multiply 4 apples by 5 boys, for then we should get 20 "boy-apples," an expression which has no meaning. or, again, to distribute 20 apples amongst 5 boys, we are not regarded as dividing 20 apples by 5 boys, but as dividing 20 apples by the number 5. the multiplication or division here involves the omission of the unit "boy," and the operation is incomplete. the complete operation, in each case, is as follows. (i) in the case of multiplication we commence with the conception of the number "5" and the unit "boy"; and we then convert this unit into 4 apples, and thus obtain the result, 20 apples. the conversion of the unit may be represented as multiplication by a factor (4 apples)/(1 boy), so that the operation is [(4 apples)/(1 boy)] x (5 boys) = 5 x [(4 apples)/(1 boy)] x (1 boy) = 5 x 4 apples = 20 apples. similarly, to convert l153 into shillings we must multiply it by a factor 20s./l1, so that we get (20s./l1) x l153 = 153 x (20s./l1) x l1 = 153 x 20s. = 3060s. hence we can only regard l153 as being equal to 3060s. if we regard this converting factor as unity. (ii) in the case of partition we can express the complete operation if we extend the meaning of division so as to enable us to divide 20 apples by 5 boys. we thus get (20 apples)/(5 boys) = (4 apples)/(1 boy), which means that the distribution can be effected by distributing at the rate of 4 apples per boy. the converting factor mentioned under (i) therefore represents a _rate_; and partition, applied to concrete cases, leads to a rate. in reference to the use of the sign x with the converting factor, it should be observed that "(7 lb.)/(4 lb.) x" symbolizes the replacing of so many times 4 lb. by the same number of times 7 lb., while "(7/4) x" symbolizes the replacing of 4 times something by 7 times that something. x. arithmetical reasoning 91. _correspondence of series of numbers._--in ss 33-42 we have dealt with the parallelism of the original number-series with a series consisting of the corresponding multiples of some unit, whether a number or a numerical quantity; and the relations arising out of multiplication, division, &c., have been exhibited by diagrams comprising pairs of corresponding terms of the two series. this, however, is only a particular case of the correspondence of two series. in considering addition, for instance, we have introduced two parallel series, each being the original number-series, but the two being placed in different positions. if we add 1, 2, 3, ... to 6, we obtain a series 7, 8, 9, ... , the terms of which correspond with those of the original series 1, 2, 3,... again, in ss 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. 5^3 involves 5 and 3, and log2 8 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, 5^2, 5^3, ..., or the numbers [root 3]1, [root 3]2, [root 3]3, ... or log10 1.001, log10 1.002, log10 1.003 ... we obtain a series in which each term corresponds with a term of the original number-series. a b c +---+-----+ +---+----+ +---+---------+ | n |6 + n| | n | 4n | | n | [root]n | +---+-----+ +---+----+ +---+---------+ | 0 | 6 | | 0 | 0 | | 0 | .000 | | 1 | 7 | | 1 | 4 | | 1 | 1.000 | | 2 | 8 | | 2 | 8 | | 2 | 1.414 | | 3 | 9 | | 3 | 12 | | 3 | 1.732 | | . | . | | . | . | | . | . | | . | . | | . | . | | . | . | | . | . | | . | . | | . | . | +---+-----+ +---+----+ +---+---------+ this correspondence is usually shown by _tabulation_, 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 s 89, we have considered cases of multiple-tables of numerical quantities, where each quantity in one series is _equivalent_ to the corresponding quantity in the other series. we might extend this principle to cases in which the terms of two series, whether of numbers or of numerical quantities, merely _correspond_ with 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. d +-----------+-------------+ | length of | volume | | edge in | of | | inches. | cube. | +-----------+-------------+ | 0 | nil. | | 1 | 1 cub. in. | | 2 | 8 cub. in. | | 3 | 27 cub. in. | | . | . | | . | . | | . | . | +-----------+-------------+ 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 1-1/2 in. has a definite volume, viz. 3(3/8) cub. in. the determination of any such intermediate value is performed by _interpolation_ (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 as _ordinal_, not cardinal. the transition is similar to that which arises in the case of geometrical measurement (s 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 a _unitary_ method. 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 s 57, this relation being then called the _rule 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 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.,