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It is to be considered that each column may extend downwards indefinitely.

37. Successive M ultiplication.—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 .liagrams 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 bv 20 we find the value of 20.3, but that in

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38. Submultiples.—T he 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 2s. lOd., one-third of 4s. 3d., one-fourth of 5s. 8d., . . . ; these being written “ } of :5. 10¢," “ l of 4s. 3d.,” “ k of 55. 8d.," . . .

The relation of submultiple is the converse of that of multiple; thus if a is t of b, then b is 5 times a. The determination of a sub multiple 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 or repetition; it is usually called partition, as representing division into a number of equal shares.

39. Quotient:.—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 1 2 pence are formed into units of 4d.; each of these numbers being said to be obtained by dividing I 2 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 A —I- n denotes the unit, it of which make up A, and A+ 8 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

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(iv.) Proberties 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 3l (or 3), 3’, 3', 3‘, . . . where n” denotes 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 the index, and n, n’, . . . 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 1: and N, n is called the pth root of N, so that n is the second (or square) root of n’, the third (or cube) root of n', the fourth root of n‘, . . . If we know u and N, then 9 is the logarithm of N to base n.

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" by n", we multiply the product of p n’s by the product of q n’s, and the result is therefore n'+'. Similarly, if we divide n” by n", where q is less than p, the result is n"-°. Thus multiplication and division in the power-series correspond to addition and subtraction in the index-series, and vice versa.

If we divide n, by n”, the quotient is of course I. This should be written 21". ' 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 § 4 3.

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 __ H H __ __ 1) then n is said to be afactor n 12 n 12 __ n of 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 i) which has no factor except itself is

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called a prime number, or, more briefly, a Prime. Thus 2, 3, 5, 7 and 1 1 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 hasG 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=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 x44=2X2X2X2X3X3=2f 3'.

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 r. 2‘. 3’, or as 1’. 2‘. 3’, or as 1'. 2‘. 3', where 9 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 Divis0r.—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 r44=2‘_ 3’, and 756=2'_ 3‘, 7, and therefore the G.C.D. of 144 and 756 is 2’, 3’=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 M ample—The Least Common Multiple, or L.C.M., of two numbers, is the least number of which they are both factors. Thus, since 144= 21 3', and 756= 2’. 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 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 divisiblr—

(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 it if the difference between the sum of the let, 3rd, 5th, . . . digits and the sum of the and, 4th, 6th, . . . is zero or divisible by rr.

(ix) To find whether a number is divisible by 7, 11 or :3, 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 and, 4th, . . . Then, if this difiercnce is zero or is divisible by 7, u or 13, the original number is also so divisible; and conversely. For examplengtsar gives 521-31 =4oo, 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 diVide 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.

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A fraction of a fraction is sometimes called a comPound fraction.

53. Comparison, Addition and Subtraction of Froctiom.-—The quantities } of A and Q 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 11 of A as the unit, we have (§ 51)

1 of A=fl ofA; ! ofA=fl of A. Hence the former is greater than the latter; their sum is of A; and their difference is 1); 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 it: 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 will; of A is said to be reduced to its lowest terms when it is replaced by T4; of A. It is important always to bear in mind that {I of A is not the some as 15; of A, though it is equal to it.

55. Diagram of Fractional Relation.—T0 find 1% of :45. we have to take 10 of the units, 24 of which make up :45. Hence the required amount will, in the multiple-table of § 36, be opposite 10 in the column in which the amount opposite 24 is 145.; 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) H of 145. is 55. rod., (ii) H of 5s. rod. is 145. 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

A diagram exhibits a relation between the two amounts 59. rod. and r4s. on the one hand,

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,0 5?“ 10¢ and the numbers 10 and 24 of the standard series on the other, which is expressed by saying that 55. rod. is to 145. in the ratio of IO

24 l48- to 24, or that 145. is to 55. rod. in the ratio of 24 to 10. If we had taken IS. 2d. instead of

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ratio of a to b is sometimes written %, but the more correct method is to write it azb.

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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 expremed 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 probortion or to be proportionals.

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.

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(a) If a=b, and b=c, then a—c;
(b) If a=b, then a+x=b+x, and a—x=b-:;
(e) If a>b, then a+x>b+x, and a—x>b—-x;
(d) If a<b, then a+x<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.o—b+c+d+e—f =a—(b—()+(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+d+c—b.

(iv) Associative Law for M ultiplicalions and Divisions—This law includes a rule, similar to the rule of signs, to the effect that 0+ (b+c) =a+ b><e; 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, Lg. that m(a+b—c)=m.a+m.b—m.c, or that (a+b—c)+n=(a+n)+ (11+ n) — (c-t— n).

In the case of (ii), (iii) and (vi), the letters a, b, c, . . . may denote either numbers or numerical quantities, while In 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) (4+b+c+ - - - ) (P+q+r+ . . . )=(HP+aq+ar+ - - ~ )+ (bp+bq+br+ . . . )+(cp+eq+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) =az—b’; i.e. the product of the sum and the difierence of two numbers is equal to the difference of their squares.

(iii) (a+b)’=a’+zab+b’=a’+(za+b)b.

V. Neoarrve 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 = ro+3— 7+6=3+6— 7+ro=&c. But, if we write the expression as 3—7+6+io, 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

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n, n, 7;, . . . are then fractional numbers, their relation to

ordinary or integral numbers being that 5 times n times is equal to p times. a This relation is of exactly the same kind as the relation of the successive digits in numbers expremed 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 15‘ as being equal to 2;}. 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 2b 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 5 17 that a mixed number corresponds with what is there called a mixed quantity. Thus £3, 175. is a mixed quantity, being expressed in pounds and shillings; to express it in terms of pounds only we must write it {Sib- _ 63. Fractional Numbers 'un'lh diflerenl Denommators.——If we divided the unit into halves, and these new units into thirds, we should get sixths of the original unit, as A shown in A; while, if we divided the Ones. Halves. Siltha- unit into thirds, and these new units

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o o 0 into halves, we should again get sixths, _ l butas shown in B. The series of halves

I g in the one case, and of thirds in the

1 other, are entirely different series of

a fractional numbers, but we can com

! ° 0 pare them by putting each in its proper

position in relation to the series of sixths. Thus 1} is equal to RP, andfi is equal to 11,", and conversely; in other Words, any fractional number is equivalent to the

B fractional number obtained by multiOnes. Thirds. Sixths. plying or dividing the numerator and We can

0 o o- denominator by any integer. ' 1 thus find fractional numbers equivalent ‘ "I, to the sum or difference of any two 2 o fractional numbers. The process is the '1 same as that of finding the sum or differ! ° ° ence 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 0!

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a unit X such that 11} times X is equal 'to A, and then take 5} times X. To simplify this, we take a new unit Y, which is § of X. _Then A is 34 times Y, and IsllrofA is 17 times Y, i.e. it is} ofA. 65. M ultiplicatian of Fractional Numbers.—To multiply Q by Q is to take ? times It has already been explained (§ 62) that times is an operation such that if times 7 times is equal to 5 times. Hence we must express 5, which itself means i times, as being 7 times something. This is done by multiplying both numerator

and denominator by 7; i.e. i is equal to 9—2, which is the same . ' . . 8 thing as7 times Hence times times 7 times n= 5

times 7.37la. 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 fi by is to find a number (i.e. a fractional number) x such that 2 times 2: is equal to But § timesil times x is, by the last section, equal to 1. Hence x is equal to § times §. 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 i we obtain, by this rule, Thus the reciprocal of a number may be defined as the number obtained by dividing r 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 Number:.——We can obtain negative fractional numbers in the same way that we obtain negative integral numbers ; thus — Q or — ’QA means that Qor QA is taken negatively.

68. Genesis 0] Fractional Number:.—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 r r in. exactly, i.e. we cannot express 3 in. as an integral multiple of n in. ;but, by extending the meaning of“ times " as in §6z, we can say that 3 in. is 1", times it in.. and therefore call x“; the quotient when 3 in. is divided by u in. Hence, if p and n are

numbers, n is sometimes regarded as denoting the result of

dividing p by n, whether 9 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 “ Q of A ” no longer means two of the units, three of which make up A; it means that A is multiplied by the fraction i, i.e. it means the same thing as “ i times A."

60. Percentage—In order to deal, by way of comparison or addition or subtraction, with fractions which have difierent 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 (§ 70) 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, l0, :2, i5, 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, 2o, 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 . . . 100, which 15 called 2% per cent., and usually written 2% %.

Similarly 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 35. for every £r, i.e. a rate of 35. in the l. .

In applications to money “ per cent.” sometimes means “ per £100." Thus “£3, 175. 6d. per cent." is really the complex 31113

20 ,

too _

7o. 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 ineans 76 ‘70, or 170%. 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= 151; of A = 1 % of A, the above quantity might equally be written 2376 X = ’15,)? of A; i.e. 23-76XA is equal to 2376 % of A.

71. A pProximate 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.


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take the next lowest or the next highest integer. It is best in such cases to retain the l; thus we can write %=37§ %= -37§.

72. Addition and Subtraction of Pcrccntages.——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 00!) 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 13070 of 1 in; = 307,, 3.

0n the same principle, since 37 %of 45 % is equal to 307,-, of 14350 = ,‘oo°°56= 11066+(16056 of Th), 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 trout if necessary to make up four figures (not counting fractions). and prefix the point.

74. Decimal Fraction:,—The percentage-notation can be extended to any fraction which has any power of 10 for its denominator. Thus {050% can be written -153 and ,‘dloaooolo can be written 45300. 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 271'05050“; can be written 27-1530. This number, expressed in terms of the fraction 11,30 0 or -ooor, would be 271.530. 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

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