exist potentially. Probably the latter is almost universally the case. without any corresponding ideas of number; the series not being necessarily correct. When numbering begins, the names of the successive numbers are attached to the individual objects; thus the numbers are These forms are developed spontaneously, without suggestion from outside. The possibility of replacing them by a standard form, which could be utilized for performing arithmetical opera-originally ordinal, not cardinal. tions, is worthy of consideration; some of the difficulties in the way of standardization have already been indicated (§ 14). The general tendency to prefer an upward direction is important; and our current phraseology suggests that this is the direction which increase is naturally regarded as taking. Thus we speak of counting up to a certain number; and similarly mathematicians speak of high and ascending powers, while engineers speak of high pressure, high speed, high power, &c. This tendency is probably aided by the use of bricks or cubes in elementary number-teaching. 24. Primitive Ideas of Number.-The names of numbers give an idea of the way in which the idea of number has developed. Where civilization is at all advanced, there are usually certain names, the origin of which cannot be traced; but, as we go farther back, these become fewer, and the names are found to be composed on certain systems. The systems are varied, and it is impossible to lay down any absolute laws, but the following seem to be the main conclusions. The conception of number as cardinal, i.e. as something belonging to a group of objects as a whole, is a comparatively late one, and does not arise until the idea of a whole consisting of its parts has been formed. This is the quantitative aspect of number. The development from the name-series to the quantitative conception is aided by the numbering of material objects and the performance of elementary processes of comparison, addition, &c., with them. It may also be aided, to a certain extent, by the tendency to find rhythms in sequences of sounds. This tendency is common in adults as well as in children; the strokes of a clock may, for instance, be grouped into fours, and thus eleven is represented as two fours and three. Finger-counting is of course natural to children, and leads to grouping into fives, and ultimately to an understanding of the denary system of notation. 26. Representation of Geometrical Magnitude by NumberThe application of arithmetical methods to geometrical measurement presents some difficulty. In reality there is a transition (i) Amongst some of the lowest tribes, as (with a few excep- from a cardinal to an ordinal system, but to an ordinal system tions) amongst animals, the only differentiation is between one which does not agree with the original ordinal system from which and many, or between one, two and many, or between one, two, the cardinal system was derived. To see this, we may represent three and many. As it becomes necessary to use higher but still ordinal numbers by the ordinary numerals 1, 2, 3, ... and small numbers, they are formed by combinations of one and two, cardinal numbers by the Roman I, II, III, Then in the or perhaps of three with one or two. Thus many of the Austral-earliest stage each object counted is indivisible; either we are asian and South American tribes use only one and two; seven, for instance, would be two two two one. (ii) Beyond ten, and in many cases beyond five, the names have reference to the use of the fingers, and sometimes of the toes, for counting; and the scale may be quinary, denary or vigesimal, according as one hand, the pair of hands, or the hands and feet, are taken as the new unit. Five may be signified by the word for hand; and either ten or twenty by the word for man. Or the words signifying these numbers may have reference to the completion of some act of counting. Between five and ten, or beyond ten, the names may be due to combinations, e.g. 16 may be 10+5+1; or they may be the actual names of the fingers last counted. (iii) There are a few, but only a few, cases in which the number 6 or 8 is named as twice 3 or twice 4; and there are also a few cases in which 7, 8 and 9 are named as 6+1, 6+2 and 6+3. In the large majority of cases the numbers 6, 7, 8 and 9 are 5+1, 5+2, 5+3 and 5+4, being named either directly from their composition in this way or as the fingers on the second hand. (iv) There is a certain tendency to name 4, 9, 14 and 19 as being one short of 5, 10, 15 and 20 respectively; the principle being thus the same as that of the Roman IV, IX, &c. It is possible that at an early stage the number of the fingers on one hand or on the two hands together was only thought of vaguely as a large number in comparison with 2 or 3, and that the number did not attain definiteness until it was linked up with the smaller by insertion of the intermediate ones; and the linking up might take place in both directions. (v) In a few cases the names of certain small numbers are the names of objects which present these numbers in some conspicuous way. Thus the word used by the Abipones to denote 5 was the name of a certain hide of five colours. It has been suggested that names of this kind may have been the origin of the numeral words of different races; but it is improbable that direct visual perception would lead to a name for a number unless a name based on a process of counting had previously been given to it. 25. Growth of the Number-Concept.-The general principle that the development of the individual follows the development of the race holds good to a certain extent in the case of the numberconcept, but it is modified by the existence of language dealing with concepts which are beyond the reach of the child, and also, of course, by the direct attempts at instruction. One result is the formation of a number-series as a mere succession of names I 2 3 FIG. 1. ... 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 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 deterr ined 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 inaividual 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 o falling into its natural place immediately before the commencement of the first unit. 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 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, 4X3=3X4(§ 28), is a statement not of identity but of equality; i.e. 4X3 and 3X4 mean different things, but the operations which they denote produce the same result. But in algebra axb bxa is called an identity, in the sense that it is true whatever a and b may be; while nXX-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 and 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 aXb 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 will be used, so that ba will mean the same as axb. 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 (§ 58). = 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 (§ 14): 5 6 7 8 9 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 o 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 o of the standard seriés. 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 (§17). Thus the expression 2 ft. 3 in. implies that in counting inches we use o to eleven instead of o 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 0 I 2 I 0 I 2 2 O I 2 3 O I 2 yds. o 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 § 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 (§ 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 1X5d., 2X5d., 3×5d., . (§ 28). This process is repetition, and the quantities IX 5d., ... are the successive multiples of 5d. If, on 2X5d., 3X5d.,. 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 is. 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 | reducing £3 to shillings, since each £ becomes 20s., we find the is performed in order to obtain such results as the following:value of 3.20. If boy receives 7 apples, then 3 boys receive 21 apples; A or If Is. 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 I 20 £60 IS. or more briefly, as in C or C' and D or D': 12d. C C' D D 60d. £2880 £3 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 § 36, 1s. 5d. is one-half of 25. 10d., one-third of 4s. 3d., one-fourth of 5s. 8d., ... ; these being written" of 2s. 1od.," "of 4s. 3d.," " of 5s. 8d.," The relation of submultiple is the converse of that of multiple; thus if a is of b, then bis 5 times a. The determination of a sub the general arrangement of the diagram being as shown in E multiple is therefore equivalent to 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 34, or to subdivision, and these operations will be exhibited by different diagrams. Of the two diagrams below, A exhibits the successive multiplication of £3 by 20, 12 and 4, and B the successive reduction of £3 to shillings, pence and farthings. The principle on which the diagrams are constructed is obvious from § 35. It should be noticed that in multiplying £3 by 20 we find the value of 20.3, but that in completion of the diagram E or E of § 35 by entry of the unit, when the number of times it is taken, and the product, are given. The operation is the converse or 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 3X4d. =4X3d. = 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 §§ 38 and 39; the product mentioned in § 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 An 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 of A. Methods of division are considered later (§§ 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 G H K of | 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. (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 I 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 mem0 1=3° n° ber of the series is 3; the second is the product of 13-3 n two numbers, each equal to 3; the third is the pro2 9-3 duct of three numbers, each equal to 3; and so on. 327=33 n3 These are written 3' (or 3), 32, 33, 3, ... where 481=3' 22' no denotes the product of p numbers, each equal to n. If we write PN, 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, n2, . . . n" 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 n2, the third (or cube) root of n3, the fourth root of n',. . . If we know n and N, then p is the logarithm of N to base n. : : 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 o, though it is of course true that o. n=o, 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=2X2X3=2X3X2=3X2X2, 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 144-2X2X2X2X3X3=2* 32. 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. 32, or as 12. 24. 32, or as 1o. 2. 32, 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.), The calculation of powers (i.e. of N when # 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 whenni.e. the greatest number which is a factor of both. Thus 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 are whole numbers, and are generalized later. 44. Law of Indices.-If we multiply no by no, we multiply the product of pn's by the product of q n's, and the result is therefore +. Similarly, if we divide n by n, where q is less than p, the result is "P-. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa. 144-21 32, and 756=22 33 7, and therefore the G.C.D. of 144 and 756 is 22 32-36. 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 $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 1442 32, and 756-22 3 7, the L.C.M. of 144 and 756 is 2 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 suc-L.C,M. of this and the next number, and so on. If we divide n by no, the quotient is of course 1. This should be written no. Thus we may make the power-series commence with 1, if we make the index-series commence with o. The added terms are shown above the line in the diagram in § 43. 45. Factors, Primes and Prime Factors. If we take the cessive multiples of 2, 3, . . 2 4 as in 36, and place each 2 multiple opposite the same 3 number in the original series, 4 we get an arrangement as in the adjoining diagram. If any number N occurs in the 8 8 vertical series commencing 2 with a number n (other than 10 10 II. 8 9 10 1) then n is said to be a factor 12 12 12 12 of N. Thus 2, 3 and 6 are factors of 6; and 2, 3, 4, and 12 are factors of 12. A number (other than 1) which has no factor except itself is (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 o; (ii) by 5 if it ends in o 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 ri 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 3X1s. 5d. =4s. 3d., 1s. 5d. may be denoted by of 4s. 3d.; and any multiple of is. 5d., denoted by nX1s. 5d., may also be denoted by of 4s. 3d. We therefore use " of A" to mean that we find a quantity X such that aXX-A, and then multiply X by n. It must be noted (i) that this is a definition of "" of, " not a definition of "," and (ii) that it is not necessary that ʼn should be less than a. 51. Subdivision of Submultiple.-By 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 of A is 5.4 times the lesser unit. Hence of A is equal to 54 of A; and, conversely, 54 of A is equal to of A. Similarly each of 7.4 7.3 7.4 these is equal to 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 5.4 4.5 (§ 36). If we write in the form 7.4 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 of of A we must convert of A into 4 times some unit. This is done by the pre4.5 ceding paragraph. For of A-54 of A= of A; i.e. it is 7.4 7.4 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, of of A=11.5 of A. 7.4 A fraction of a fraction is sometimes called a compound fraction. 53. Comparison, Addition and Subtraction of Fractions.—The quantities of A and 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 of A as the unit, we have (§ 51) of A of A; ‡ of A =}} of A. 10 24 14s. 5s. 10d., (ii) 7d. 55. Diagram of Fractional Relation.-To find of 145. we have to take 10 of the units, 24 of which make up 14s. Hence the required amount will, in the multiple-table of § 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 5s. 10d. 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) of 145. is of 5s. 10d. is 14s. The two statements are in fact merely different aspects of a single relation, considered in the next section. 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 A amounts 5s. rod. 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 and are equal. For this reason the ratio of a to b is sometimes written, but ΤΟ 5s. Iod. 57. Proportion. If from any two columns in the table of § 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. 22 yds. 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. This is expressed by saying that M is to N as P to Q, the relation being written M:NPQ; the four quantities are then said to be in proportion or to be proportionals. 58. Laws of Arithmetic.-The arithmetical processes which we Hence the former is greater than the latter; their sum is of A; have considered in reference to positive integral numbers are and their difference is of A. Thus the fractions must be reduced to a common denominator. This denominator must, if the fractions are in their lowest terms (§ 54), be a multiple of each of the denominators; it is usually most convenient that it should be their L.C.M. (§ 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 subject to the following laws: (i) Equalities and Inequalities.-The following are sometimes called Axioms (§ 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. |