For

for the numerator, and the divisor for the denominator. example, 4, we should have for the quotient 3, and 7 for the remainder; hence, we conclude that is the same as 3.

67. Thus we see how fractions, whose numerators are greater than the denominators, are resolved into two parts; one of which is an integer, and the other a fractional number. Such fractions are called improper fractions, to distinguish them from common fractions.

68. The nature of fractions is frequently considered in another way, which may throw additional light on the subject. For example, the fraction, it is evident is three times as great as . Now this fraction means that, if we divide 1 into 4 equal parts this will be the value of one of those parts; it is obvious then that by taking 3 of those parts, we shall have the value of the fraction 4.

69. As it is easy to understand what is, when we know the signification of, we may consider the fractions, whose numerator is unity, as the foundation of all others. And it is observable that these fractions go on continually diminishing; for the more you divide an integer, or the greater number of parts into which you distribute it, the less does each of those parts become. Thus To is less than 6.

70. As we have seen, that the more we increase the denominator of such fractions, their value become less ; it may be askedwhether it is possible to make the denominator so great, that the fraction shall be reduced to nothing? I answer no; for into whatever number of parts unity (the length of a foot for instance) is divided; let those parts be ever so small, they will still preserve a certain magnitude, and therefore can never be absolutely reduced to nothing. This circumstance has introduced the expression, that the denominator must be infinite, or infinitely great, in order that the fraction may be reduced to 0, or to nothing; and the word infinite in reality signifies here, that we should never arrive at the end of the series of the above mentioned fractions.

71. It may be necessary also in this place to correct the mistake of those who assert, that a number infinitely great is not susceptible of increase. This opinion is inconsistent with just principles, for signifies a number infinitely great, and being incontestibly the double of, it is evident that a number, though infinitely great may still become two or more times greater.

72. To reduce a given fraction to its least terms, it is required to find a number by which both the numerator and denominator may be divided. Such a number is called a common divisor, and so long as we can find a common divisor to the numerator and the denominator, it is certain that the fraction may be reduced to a lower form; but, on the contrary, when we see that except unity no other common divisor can be found, this shows that the fraction is already in the simplest form that it admits of.

73. This property of fractions preserving an invariable value, whether we divide or multiply the numerator and denominator by the same number, is of the greatest importance, and is the principal foundation of the doctrine of fractions. For example, we can scarcely add together two fractions, or subtract them from each

other, before we have, by means of this property, reduced them to other forms, that is to say, to expressions whose denominations are equal.

Addition and Subtraction of Fractions.

ART. 74. When fractions have equal denominators, there is no difficulty in adding and subtracting them; for,+ is equal to and — is equal to 4. In this case, either for addition or subtraction, we alter only the numerators, and place the common denominator under the line, thus + is equal to, or 1, that is to say, an integer; and -- is equal to 4, that is to say, nothing, or 0.

75. But when fractions have not equal denominators, we can always change them into other fractions that have the same de nominators. For example, add together and, we must consider that is the same as 3, and that is equivalent to , we have therefore, the sum of which is §.

76. We may have a greater number of fractions to be reduced to a common denominator, for example, †, 4, 4, §; in this case the whole depends on finding a number which may be divisible by all the denominators of these fractions. In this instance, 60 is the number which has that property, and which consequently becomes the common denominator. We shall therefore have 38 instead of ; instead of ; 45 instead of 3; 48 instead of ; and 8 instead of. Now if it be required to add together all these fractions 18, 48, 45, 48, and 58, we have only to add all the numerators, and under the sum place the common denominator 60; that is to say, we shall have 213, or three integers, and 33, or 317.

77. When it is required to subtract a fraction from an integer, it is sufficient to change one of the units of that integer into a fraction having the same denominator as the fraction to be subtracted; the rest of the operation is performed without any difficulty. If it be required, for example, to subtract from 1, we write instead of 1, and say, that taken from leave. So subtracted from 1, leave.

If it were required to subtract from 2, we should write 1 and instead of 2, and we should immediately see that after the subtrac tion there must remain 14.

78. It sometimes happens, that having added two or more frae tions together, we obtain more than an integer; that is to say, a numerator greater than the denominator; for example +, or

, makes 3, or 15 We have only to perform the actual divi sion of the numerator by the denominator, to see how many integers there are for the quotient, and to set down the remainder. Nearly the same must be done to add together numbers compounded of integers and fractions; (called mixed numbers ;) we first add the fractions, and if their sum produces one or more integers, these are added to the other integers. For example, add together 34 and 24; we first take the sum of and, or of and 4. It is 7 or 17; then the sum total is 6.

Multiplication and Division of Fractions.

ART. 79. The rule for the multiplication of a fraction by an integer is to multiply the numerator by the given number, and not to change the denominator, thus:

2 times, or twice makes, or one integer.

But instead of this rule, we may use that of dividing the denominator by the given integer; and this is preferable when it can be used, because it shortens the operation. If it be required for example, to multiply by 3; we divide the denominator by the integer, and find immediately or 23 for the given product.

80. We have shown how a fraction is to be multiplied by an integer; let us now consider how a fraction is to be divided by an integer. It is evident if we have to divide the fraction by 2, that the result must be ; and that the quotient of divided by 3 is 4. The rule therefore is, to divide the numerator by the integer with out changing the denominator. Thus:

divided by 2 gives .

This rule may be easily practiced, provided the numerator be divisible by the number proposed; but very often it is not: for example, to divide by 2, we should change the fraction into §, and then dividing the numerator by 2, we should immediately have for the quotient sought.

81. When a fraction is to be divided by an integer, we have only to multiply the denominator by that number, and leave the nume rator as it is. Thus & divided by 4 gives 56.

This operation becomes easier when the numerator itself is divi sible by the integer. For example, divided by 3 would give, according to the last rule, ; but by the first rule, which is appli cable here, we obtain, an expression equivalent to, but more simple.

Hence the following rule for multiplying fractions :-multiply separately the numerators, and the denominators together. Thus: by gives the product, or 3.

82. It remains to show how one fraction may be divided by another. We remark first, that if the two fractions have the same number for a denominator, the division takes place only with respect to the numerators; it is evident, that is contained as many times in as 3 in 9, that is to say, thrice; and in the same manner, in order to divide by 2, we have only to divide 8 by 9, which gives .

83. But when the fractions have not equal denominators, we must have recourse to the method already mentioned for reducing them to a common denominator.

Hence the following rule: Multiply the numerator of the dividend by the denominator of the divisor, and the denominator of the dividend by the numerator of the divisor; the first product will be the numerator of the quotient, and the second will be its denomi

nator.

81. Applying this rule to the division of by, we shall have the quotient 1, the division of a by will give, or or 1 and . 85. This rule is often represented in a manner more easily remem

bered as follows: Invert the fraction which is the divisor, so that the denominator may be in the place of the numerator, and the latter be written under the line: then multiply the fraction, which is the dividend by this inverted fraction, and the product will be the quotient sought. Thus divided by is the same as multiplied.

86. Every number when divided by itself produces unity, and it is evident that a fraction divided by itself must also give 1 for the quotient. The same follows from our rule, for, in order to divide by, we must multiply by, and we obtain + or 1; and if it be required to divide by, we multiply by ; now the product is equal to 1.

87. We have still to explain an expression which is frequently used. It may be asked, for example, what is the half of; this means that we must multiply 3 by. So likewise, if the value of

of were required, we should multiply & oy, which produces ; and of is the same as multiplied by, which produces 4.

A DICTIONARY OF

ARITHMETICAL TERMS.

Addition, the first of the four fundamental rules in Arithmetic, Annuity, a periodical payment; whereby several small sums are annuities generally are yearly, halfadded or collected into one that is yearly, or quarterly. Annuities are larger. The only method of prov-of two kinds, certain and contingent. ing addition, which can properly be called a proof is by subtraction.

Arithmetic is properly the science which treats of numbers. It is called a science, because it inAliquot parts, such part of a num-vestigates the properties upon as will divide or measure a whole which its rules depend; and an art,