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 beingone-half, one-third, one-fourth, ... Thus, in the diagram of § 36, 1s. 5d. is one-half of 2s. 10d., one-third of 4s. 3d., one-fourth of 5s. 8d., ...; these being written “½ of 2s. 10d.,” “1⁄3of 4s. 3d.,” “¼ of 5s. 8d,”...
The relation of submultiple is the converse of that of multiple; thus if a is1⁄5of b, then b is 5 times a. The determination of a submultiple is therefore equivalent to 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 of repetition; it is usually calledpartition, 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 aquotient. 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 × 4d. = 4 × 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 bydividing12 by 4. The termdivisionis therefore used in text-books to describe the two processes described in §§ 38 and 39; the product mentioned in § 34 is thedividend, the number or the unit, whichever is given, is called thedivisor, and the unit or number which is to be found is called thequotient. 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 as1⁄nof 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 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.
42.Successive Divisionmay be performed as the converse of successive multiplication. The diagrams A and B below are the converse (with a slight alteration) of the corresponding diagramsin § 37; A representing the determination of1⁄20of1⁄12of ¼ of 2880 farthings, and B the conversion of 2880 farthings into £.
(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 31(or 3), 32, 33, 34, ... where npdenotes the product of p numbers, each equal to n. If we write np= 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 theindex, and n, n2, ... npare called thefirst power, second power, ... pth powerof n, the series itself being called thepower-series. Thesecond powerandthird powerare usually called thesquareandcuberespectively. If we know p and N, n is called thepth rootof N, so that n is thesecond(orsquare)rootof n2, thethird(orcube)rootof n3, thefourth rootof n4, ... If we know n and N, then p is thelogarithmof N tobasen.
The calculation of powers (i.e.of N when n and p are given) isinvolution; the calculation of roots (i.e.of n when p and N are given) isevolution; 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 npby nq, we multiply the product of p n’s by the product of q n’s, and the result is therefore np + q. Similarly, if we divide npby nq, where q is less than p, the result is np − q. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.
If we divide npby np, the quotient is of course 1. This should be written n0. 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 § 43.
45.Factors, Primes and Prime Factors.—If we take the successive multiples of 2, 3, ... as in § 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 afactorof N. Thus 2, 3 and 6 are factors of 6; and 2, 3, 4, 6 and 12 are factors of 12.
A number (other than 1) which has no factor except itself is called aprime number, or, more briefly, aprime. 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 acompositenumber.
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 calledprime 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 × 2 × 3 = 2 × 3 × 2 = 3 × 2 × 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 × 2 × 2 × 2 × 3 × 3 = 24·32.
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 1·24·32, or as 12·24·32, or as 1p·24·32, where p might be anything.
If two numbers have no factor in common (except 1) each is said to beprime tothe other.
The multiples of 2 (including 1·2) are calledevennumbers; other numbers areoddnumbers.
46.Greatest Common Divisor.—If we resolve two numbers into their prime factors, we can find theirGreatest Common DivisororHighest 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 = 24·32, 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.—TheLeast Common Multiple, or L.C.M., of two numbers, is the least number of which they are both factors. Thus, since 144 = 24·32, and 756 = 22·33·7, the L.C.M. of 144 and 756 is 24·33·7. 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 Ninesis 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.—Afractionof a quantity is a submultiple, or a multiple of a submultiple, of that quantity. Thus, since 3 × 1s. 5d. = 4s. 3d., 1s. 5d. may be denoted by1⁄3of 4s. 3d.; and any multiple of 1s. 5d., denoted by n × 1s. 5d., may also be denoted by n/3 of 4s. 3d. We therefore use “n⁄aof A” to mean that we find a quantity X such that a × 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.—By5⁄7of 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, and5⁄7of A is 5·4 times the lesser unit. Hence5⁄7of A is equal to5·4⁄7·4of A; and, conversely,5·4⁄7·4of A is equal to5⁄7of A. Similarly each of these is equal to5·3⁄7·3of 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 (§ 36). If we write5·4⁄7·4in the form4·5⁄4·7we 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 find11⁄4of5⁄7of A we must convert5⁄7of A into 4 times some unit. This is done by the preceding paragraph. For5⁄7of A =5·4⁄7·4of A =4·5⁄7·4of 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,
A fraction of a fraction is sometimes called acompound fraction.
53.Comparison, Addition and Subtraction of Fractions.—The quantities ¾ of A and5⁄7of 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, taking1⁄28of A as the unit, we have (§ 51)
¾ of A =21⁄28of A;5⁄7of A =20⁄28of A.
Hence the former is greater than the latter; their sum is41⁄28of A; and their difference is1⁄28of A.
Thus the fractions must be reduced to acommon 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 bein its lowest termswhen its numerator and denominator have no common factor; or to be reduced to its lowest terms when it is replaced by such a fraction. Thus8⁄22of A is said to be reduced to its lowest terms when it is replaced by4⁄11of A. It is important always to bear in mind that4⁄11of A is not thesameas8⁄22of A, though it isequalto it.
55.Diagram of Fractional Relation.—To find10⁄24of 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 § 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)10⁄24of 14s. is 5s. 10d., (ii)24⁄10of 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 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 theratioof 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 § 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. 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 bein proportionor to beproportionals.
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 calledAxioms(§ 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.
(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 therule 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×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×c×d×e ÷ f = a ÷ (b ÷ c)×(d×e ÷ f).
(v)Commutative Law for Multiplications and Divisions, that multiplications and divisions may be performed in any order:e.g.a ÷ b×c×d = a×c ÷ b×d = a×d×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 § 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) = a2− b2;i.e.the product of the sum and the difference of two numbers is equal to the difference of their squares.
(in) (a + b)2= a2+ 2ab + b2= a2+ (2a + b)b.
V. Negative Numbers
61.Negative Numbersmay 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 anegative 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 anegative 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 itsnumerical value.
VI. Fractional and Decimal Numbers
62.Fractional Numbers.—According to the definition in § 50 the quantity denoted by3⁄6of 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 p/n, q/n, r/n, ... exactly as if they were numbers, any operations being performed on the numerators. The expressions p/n, q/n, r/n, ... are thenfractional numbers, their relation to ordinary orintegralnumbers 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 regard13⁄6as being equal to 21⁄6. A fractional number is called aproper fractionor animproper fractionaccording as the numerator is or is not less than the denominator; and an expression such as 21⁄6is called a mixed number. An improper fraction is therefore equal either to an integer or to a mixed number. It will be seen from § 17 that a mixed number corresponds with what is there called amixed quantity. Thus £3, 17s. is a mixed quantity, being expressed in pounds and shillings; to express it in terms of pounds only we must write it £317⁄20.
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. Thus3⁄2is equal to9⁄6, and5⁄3is equal to10⁄6, 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 ofpence, 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. § 53).
64.Complex Fractions.—A fraction (or fractional number), the numerator or denominator of which is a fractional number, is called acomplexfraction (or fractional number), to distinguish it from asimplefraction, which is a fraction having integers for numerator and denominator. Thus 52⁄3/ 111⁄3of A means that we take a unit X such that 111⁄3times X is equal to A, and then take 52⁄3times X. To simplify this, we take a new unit Y, which is1⁄3of X. Then A is 34 times Y, and 52⁄3/ 111⁄3of A is 17 times Y,i.e.it is ½ of A.
65.Multiplication of Fractional Numbers.—To multiply8⁄3by5⁄7is to take5⁄7times8⁄3. It has already been explained (§ 62) that5⁄7times is an operation such that5⁄7times 7 times is equal to 5 times. Hence we must express8⁄3, which itself means8⁄3times, as being 7 times something. This is done by multiplying both numerator and denominator by 7;i.e.8⁄3is equal to7·8⁄7·3, which is the same thing as 7 times8⁄7·3. Hence5⁄7times8⁄3=5⁄7times 7 times8⁄7·3= 5 times8⁄7·3=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 divide8⁄3by5⁄7is to find a number (i.e.a fractional number) x such that5⁄7times x is equal to8⁄3. But7⁄5times5⁄7times x is, by the last section, equal to x. Hence x is equal to7⁄5times8⁄3. Thus to divide by a fractional number we must multiply by the number obtained by interchanging the numerator and the denominator,i.e.by thereciprocalof the original number.
If we divide 1 by5⁄7we 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 § 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 −5⁄7or −5⁄7A means that5⁄7or5⁄7A is taken negatively.
68.Genesis of Fractional Numbers.—A fractional number may be regarded as the result of a measuring division (§ 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 § 62, we can say that 3 in. is3⁄11times 11 in., and therefore call3⁄11the 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, afraction. Thus “2⁄3of A” no longer means two of the units, three of which make up A; it means that A is multiplied by the fraction2⁄3,i.e.it means the same thing as “2⁄3times 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 (§ 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 calledpercentages. 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 2½/100, which is called 2½per cent., and usually written 2½%. Similarly 31⁄3% is equal to one-thirtieth.
If the numerator is a multiple of 5, the fraction represents twentieths. This is convenient,e.g.for expressingrates in the pound; thus 15% denotes the process of taking 3s. for every £1,i.e.a rate of 3s. in the £.
In applications to money “per cent.” sometimes means “per £100.” Thus “£3, 17s. 6d. per cent.” is really the complex fraction
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%, or76⁄100. 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 × A means 23 times A, with 76% of A. We might therefore denote 76% by 0.76.
If as our unit we take X =1⁄100of A = 1% of A, the above quantity might equally be written 2376 X =2376⁄100of A;i.e.23.76 × 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 thenearestinteger to the numerator of an equal fraction having 100 for its denominator. Thus1⁄7= 142⁄7/ 100, so that1⁄7is approximately equal to 14%; and2⁄7= (284⁄7)/100, which is approximately equal to 29%. The difference between this approximate percentage and the true value is less than ½%,i.e.is less than1⁄200.
If the numerator of the fraction consists of an integer and ½—e.g.in the case of3⁄8= (37½)/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 ½; thus we can write3⁄8= 37½% = .37½.
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 meaning37⁄100of1⁄100=37⁄10000.
On the same principle, since 37% of 45% is equal to37⁄100of45⁄100=1665⁄10000=16⁄100+ (65⁄100of1⁄100), 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. Thus153⁄1000can be written .153 and15300⁄100000can be written .15300. These two fractions are equal to each other, and also to .1530. A fraction written in this way is called adecimal 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 thedecimal point. Thus 271530⁄10000} can be written 27.1530. This number, expressed in terms of the fraction1⁄10000or .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 thedenominationin which the number is expressed: the above number, expressed in termsof1⁄10, would be 271.530, but expressed in terms of 100 it would be .271530.
Fractions other than decimal fractions are usually calledvulgar fractions.
75.Decimal Numbers.—Instead of regarding the .153 in 27.153 as meaning153⁄1000, 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 + 5/102+ 3/103, 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 adecimal number, or, more briefly, adecimal. 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 (§ 111).
78.Division by Decimal.—To divide one decimal by another, we must reduce them to the same denomination, as explained in § 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 (§ 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 of2⁄3as a single fraction. Except in the case of2⁄3and ½, 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 ξ represented 60, and ο 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 asunciae(ounces), being twelfths of theas(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 × 5) + 3/(8 × 7 × 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 centuryA.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 untilA.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 ofequalityas a necessary characteristic of the units or subunits that are used. The third involves also the idea ofcontinuityand therefore of unlimited subdivision. In weighing an object with ounce-weights the fact that it weighs more than 1 ℔ 3 oz. but less than 1 ℔ 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 ℔ 5 oz. of meat at 9½ d. a ℔, 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 theerror.
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 becorrect toso manyplaces of decimals. This means (cf. § 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 represents1⁄7correct 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 becorrected.
This method is not good for comparative purposes. Thus .143 and 14.286 represent respectively1⁄7and100⁄7to 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 manysignificant 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 represent1⁄7and6⁄7correct to 5 significant figures; but the latter is relatively more accurate than the former. For the former shows only that1⁄7lies between .142855 and .142865, or, as it is better expressed, between .14285½ and .14286½; but the latter shows that6⁄7lies between .85713½ and .85714½, and therefore that1⁄7lies between .142857⁄12and .142859⁄12.
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, thelimit of erroris expressed by the sign ±. Thus the former of the above two statements would give1⁄7= .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. Thus1⁄7= .143 ± .0005 can be written1⁄7= 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 (§ 60 (i)) (a′ ± α)(b′ ± β) = a′b′ + aβ ± (a′β + b′α). If a′ and b′ are the stated values, and ±α and ±β the respective limits of error, we ought strictly to take a′b′ + αβ as the product, with a limit of error ±(a′β + b′α). In practice, however, both αβ and a certain portion of a′b′ are small in comparison with a′β and b′α, and we therefore replace a′b′ + αβ 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 × 1.335 = 4.772625 and 3.585 × 1.345 = 4.821825. We might take the product as (3.58 × 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 (§ 82 (i)) if necessary.
VIII. Surds and Logarithms
84.Roots and Surds.—The pth root of a number (§ 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 26}·33, and find that its cube root is 22·3 = 12; or, to find the cube root of 1.728, we write it as1728⁄1000=216⁄125= 23·33/53, 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 asurd; the surd which is the pth root of N is writtenp√N, but if the index is 2 it is usually omitted, so that the square root of N is written √N.
85.Surd as a Power.—We have seen (§§ 43, 44) that, if we take the successive powers of a number N, commencing with 1, they may be written N0, N1, N2, N3, ..., the series of indices being the standard series; and we have also seen (§ 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 N1/3will therefore be such that N1/3× N1/3× N1/3= N1/3 + 1/3 + 1/3= N;i.e.it will be the cube root of N. By analogy with the notation of fractional numbers, N2/3will be N1/3 + 1/3= N1/3× N1/3; and, generally, Np/qwill mean the product of p numbers, the product of q of which is equal to N. Thus N2/6will not mean thesameas N1/3, but will mean the square of N1/6; but this will beequalto N1/3,i.e.(6√N)2=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. Thus3√2 × √5 = 21/3× 51/2= 22/6× 53/6= 41/6× 1251/6= 5001/6=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 ofantilogarithmsof 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 20, 21, 22, ... and we insert within this series the successive powers of x, where x is such that x100= 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 2.316is 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 = ap, then p is called the logarithm of N to the base a, and is written logaN. As the table of antilogarithms is formed by successive multiplications, so the logarithm of any givennumber 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 by10√2 until we get a number between 1 and10√2; then divide repeatedly by100√2; and so on. If, for instance, we find that the number is approximately equal to 23× (10√2)5× (100√2)7× (1000√2)4, it may be written 23.574, 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, seeLogarithm.
For practical purposes logarithms are usually calculated to base 10, so that log1010 = 1, log10100 = 2, &c.
IX. Units
89.Change of Denominationof a numerical quantity is usually calledreduction, so that this term covers,e.g., the expression of £153, 7s. 4d. as shillings and pence and also the expression of 3067s. 4d. as £, s. and d.
The usual statement is that to express £153, 7s. as shillings we multiply 153 by 20 and add 7. This, as already explained (§ 37), is incorrect. £153 denotes 153 units, each of which is £1 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 £; there are 153 of these 20’s, and 7 over.
The simplest case, in which the quantity can be expressed as an integral number of the largest units involved, has already been considered (§§ 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 74⁄12s., but £0, 7s. 4d. as £0[(74⁄12) / 20] (§ 17). The reduction of £153, 7s. 4d. to pence, and of 36808d. to £, 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 (§ 36) in which the multiples of £1 and of 20s. are arranged in parallel columns; and similarly for shillings and pence.
90.Change of Unit.—The statement “£153 = 3060s.” is not a statement ofequalityof the same kind as the statement “153 × 20 = 3060,” but only a statement ofequivalencefor 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) × (5 boys) = 5 × (4 apples)/(1 boy) × (1 boy) = 5 × 4 apples = 20 apples. Similarly, to convert £153 into shillings we must multiply it by a factor 20s./£1, so that we get
Hence we can only regard £153 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 arate; and partition, applied to concrete cases, leads to a rate.
In reference to the use of the sign × with the converting factor, it should be observed that “(7 ℔)/(4 ℔) ×” symbolizes the replacing of so many times 4 ℔ by the same number of times 7 ℔, while “7⁄4×” symbolizes the replacing of 4 times something by 7 times that something.
X. Arithmetical Reasoning
91.Correspondence of Series of Numbers.—In §§ 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,...