CHAP. IX.

OF FRACTIONS IN GENERAL.

86. WHEN a number, as for example 7, is said not to be divisible by another number, such as 3, it is meant that the quotient cannot be expressed by an entire number or integer, but it must not be supposed that no idea can be formed of such a quotient.

We have only to imagine a line 7 feet long, and no one will doubt that it is possible to divide it into three equal parts, and to form an idea of the length of any one of them.

87. Since, therefore, we can form a precise idea of the quotient that is obtained in these cases, although the quotient is not an entire number, we are led to consider a particular kind of numbers called fractions or broken numbers.

88. If then we have to divide 7 by 3, the quotient which should result is easily represented, and is expressed by writing the divisor under the dividend, and separating the two numbers by a line as 3. And in general when a is to be divided by b, the quotient is expressed by a form of expression which is called a fraction.

a

a

89. We cannot therefore give a more precise idea of a fraction, than by saying that we represent in this manner the quotient resulting from the division of the upper number by the lower. It is necessary also to remember, that in all fractions the upper number is called the numerator, and the lower the denominator of the fraction. Thus in the above fraction, which is expressed seven thirds, 7 is the numerator and 3 the denominator.

90. That

90. That we may arrive at a more perfect knowledge of the nature of fractions, we shall begin by considering the case, where the numerator is equal to the denominator, as in the fraction

Now since by this is expressed the quotient which we obtain by the division of a by a, it is clear that this quotient is unity, and consequently that this fraction is of a value equal to 1, or one integer; and it follows therefore that the fractions

a

a

are all equal to each other; that is to say, that they are all equal to 1, or one integer.

91. We have now seen that a fraction whose numerator is equal to its denominator, is itself equal to unity. It follows therefore that all fractions, whose numerators are less than their denominators, have a value less than unity. For if we have a number which is to be divided by another number greater than itself, the quotient must evidently be less than 1. For example, if a line of two feet long, be divided into three parts, any one of those parts must necessarily be less than a foot. It is evident therefore, that 2 divided by 3, or must be less than 1, and for this reason, viz. that the numerator or dividend is less than the denominator or divisor.

92. If, on the contrary, the numerator be greater than the denominator, the value of the fraction is greater than unity. For instance, 3 divided by 2 or is greater than 1. For is the same as with added to it. Now has been shown to 9244

be equal to 1, therefore & must be greater than 1 by 1; i. e. it is one and a half. In like manner is greater than 1 by , is greater than 1 by, and is greater than 2 by : and in general it is only necessary in these cases to divide the upper number by the lower, and to the quotient to add the remainder in the form of a fraction, having the remainder itself for a numerator, and the divisor for a denominator.

For instance, if the given fraction were ; by actual division we should have 3 for the quotient with 7 for the remainder, from which it follows that 3 is the same thing as 8 and 7.

93. From hence it appears that fractions whose numerators are greater than their denominators may be resolved into two numbers, one of which is an integer or whole number, and the other a fraction, whose numerator is less than its denominator. These fractions are called improper fractions, to distinguish them from real fractions, or fractions properly so called, which having a numerator less than their denominator are less than unity or whole numbers,

94. We may likewise form an idea of the nature of fractions in another way, by which additional light may be thrown upon the subject. If we consider for example the fraction, it is evident that this is 3 times as great as 4. Now the fraction shows the value of one of the parts of a line divided into four equal parts; it is therefore clear that by taking 3 of those parts we shall have the value of the fraction 4.

95. In the same manner we may consider all other fractions, as for example. For if we divide unity or any whole number into 12 equal parts, 7 of those parts will be the value of the fraction proposed.

96. To this method of considering fractions, the terms numerator and denominator owe their origin. For in the preceding fraction, the number under the line denotes that unity is to be divided into twelve equal parts. This number therefore names or denotes those parts, and is therefore called the denominator.

97. Again, as the number above the line, as 7, shows that to obtain the value of the fraction, we must take or collect together 7 of those parts, and consequently that this number

may

may be said to compute or number them, it has been thought proper to call it the numerator.

98. Since then it is easy to comprehend the meaning of the fraction when we understand what is meant by, we may consider those fractions whose numerator is unity or 1, as the foundation of all others. Such are the fractions

1, t, t, t, t, t, b, d, to, t, 7, &c.

And it should be observed, that these fractions go on continually diminishing: for the more you divide an integer, or the greater the number of equal parts into which it is divided, the smaller does each of those parts become. So that is less than, Too is less than, and is less than Too, &c.

99. Having thus seen that the more we increase the denominator of a fraction the less will be the value of that fraction, it may be asked whether it would not be possible to make the denominator so large that the fraction would be reduced to 0. We answer no: for into whatever number of parts unity be divided (as the length of a foot for instance), these parts, however small, will still possess a certain magnitude, and consequently can never be absolutely reduced to nothing. How ever great, therefore, the denominator of a fraction may be, it is certain that we can never actually reduce a fraction so as to arrive completely at 0, or nothing, and consequently that as those fractions will always preserve a certain magnitude, we may continue them in the series contained in the 98th article, without end.

100. From this property of fractions it has been said that the denominator must be infinite, or infinitely great, to reduce a fraction to 0, or nothing; and this word infinity means, in fact, that we never can arrive at such a fraction in the series of fractions above mentioned.

CHAP. X.

OF THE PROPERTIES OF FRACTIONS.

101. WE have already seen that each of the fractions

4, 4, 4, 5, 6, 7, , &c.

make an integer, and consequently that they are all equal to each other. The same equality is preserved in the following fractions,

each of which make an integer also; for the numerator of each being divided by the denominator gives a quotient of 2.

are equal to each other, as their common value is 3.

102. We may also represent the value of any fraction in an infinite variety of ways. For if we multiply both the numerator and denominator of a fraction by the same number, which may be taken at pleasure, the value of the fraction will not be altered; and for this reason all the fractions

are all equal to each other, for the value of each is 1. In like manner,

a

are all equal, since the value of each is ; and hence we may conclude in general that the fraction may be represented in any of the following ways, of which each is equal to