74. If the numerator, on the contrary, be greater than the de- ' nominator, the value of the fraction is greater than unity. Thus is equal to together with. Now is ex

is greater than 1, for actly 1, consequently

3

is equal to 1+, that is, to an integer and a half. In the same manner is equal to 14, to 1, and to 24. And in general, it is sufficient in such cases to divide the upper number by the lower, and to add to the quotient a fraction having the remainder for the numerator, and the divisor for the denominator. If the given fraction were, for example, 4, we should have for the quotient 3, and 7 for the remainder; whence we conclude that is the same as 37.

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75. 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, having the numerator less than the denominator. Such fractions as contain one or more integers, are called improper fractions, to distinguish them from fractions properly so called, which, having the numerator less than the denominator, are less than unity, or than an integer.

76. The nature of fractions is frequently considered in another way, which may throw additional light on the subject. If we consider, for example, the fraction, it is evident that it is three times greater than 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 2.

In the same manner we may consider every other fraction; for example, if we divide unity into 12 equal parts, 7 of those parts will be equal to this fraction.

77. From this manner of considering fractions, the expressions numerator and denominator are derived. For, as in the preceding fraction, the number under the line shows, that 12 is the number of parts into which unity is to be divided; and as it may be said to denote, or name the parts, it has not improperly been called the denominator.

Further, as the upper number, namely 7, shows that, in order to have the value of the fraction, we must take, or collect 7 of those parts, and therefore may be said to reckon, or number them, it has been thought proper to call the number above the line the numerator.

78. 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. Such are the fractions,

1 1

1, 1, 1, 1, 1, 4, b, b, Yo, I'ù, 12, &c.,

and it is observable that these fractions go on continually diminishing; for the more you divide an integer, or the greater the number of parts into which you distribute it, the less does each of those parts become. Thus is less than '; ' is less than ; and is less than 1000

10

000

100

79. As we have seen, that the more we increase the denominator of such fractions, the less their values become; it may be asked, whether it is not 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.

80. It is true, if we divide the length of a foot into 1000 parts; those parts will not easily fall under the cognizance of our senses; but view them through a good microscope, and each of them will appear large enough to be subdivided into 100 parts and more.

At present, however, we have nothing to do with what depends on ourselves, or with what we are capable of performing, and what our eyes can perceive; the question is rather, what is possible in itself. And, in this sense of the word, it is certain, that however great we suppose the denominator, the fraction will never entirely vanish, or become equal to 0.

81. We never therefore arrive completely at nothing, however great the denominator may be; and these fractions always preserving a certain value, we may continue the series of fractions in the 78th article without interruption. 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.

82. To express this idea, which is extremely well founded, we make use of the sign co, which consequently indicates a number infinitely great; and we may therefore say that this fraction is really nothing, for the very reason that a fraction cannot be reduced to nothing, until the denominator has been increased to infinity.

83. It is the more necessary to pay attention to this idea of infinity, as it is derived from the first foundations of our knowledge, and as it will be of the greatest importance in the following part of this treatise.

We may here deduce from it a few consequences, that are extremely curious and worthy of attention. The fraction represents the quotient resulting from the division of the dividend 1 by the divisor co. Now we know that if we divide the dividend 1 by the quotient, which is equal to 0, we obtain again the divisor; hence we acquire a new idea of infinity; we learn that it arises from the division of 1 by 0; and we are therefore entitled to say, that 1 divided by O expresses a number infinitely great, or ∞.

84. 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 the just principles which we have laid down; for signifying a number infinitely great, and being incontestably the double of, it is evident that a number, though infinitely great, may still become two or more times greater.

CHAPTER VIII.

Of the Properties of Fractions.

85. WE have already seen, that each of the fractions,

makes an integer, and that consequently they are all equal to one another. The same equality exists in the following fractions,

each of them making two integers; for the numerator of each, divided by its denominator, gives 2. So all the fractions

are equal to one another, since 3 is their common value.

86. We may likewise represent the value of any fraction, in an infinite variety of ways. For if For if we multiply both the numerator and the denominator of a fraction by the same number, which may be assumed at pleasure, this fraction will still preserve the same value.

are equal fractions, the value of each of which is . The fractions,

have likewise all the same value; and lastly, we may conclude in

a

general, that the fraction may be represented by the following ex

pressions, each of which is equal to

a

b

; namely,

a

2 a За

&c.

b' 26' 36' 46' 56' 66' 76'

87. To be convinced of this we have only to write for the value

a

of the fraction a certain letter c representing by this letter c the

quotient of the division of a by b; and to recollect that the multiplication of the quotient c by the divisor b must give the dividend. For since c multiplied by b gives a, it is evident that c multiplied by 26 will give 2 a, that c multiplied by 3b will give 3a, and that in general e multiplied by mb must give m a. Now changing this into an example of division, and dividing the product ma, by mb one of the factors, the quotient must be equal to the other factor c; but

ma divided by mb gives also the fraction

ma

m b'

which is consequently equal to c; and this is what was to be proved: for c having been

α

assumed as the value of the fraction, it is evident that this fraction

is equal to the fraction

ma

m b'

whatever be the value of m.

88. We have seen that every fraction may be represented in an infinite number of forms, each of which contains the same value; and it is evident that of all these forms, that, which shall be composed of the least numbers, will be most easily understood. For example, we might substitute instead of the following fractions,

but of all these expressions is that of which it is easiest to form an idea. Here, therefore, a problem arises, how a fraction, such as

12, which is not expressed by the least possible numbers, may be reduced to its simplest form, or to its least terms, that is to say, in our present example, to .

89. It will be easy to resolve this problem, if we consider that a fraction still preserves its value, when we multiply both its terms, or its numerator and denominator, by the same number. For from this it follows also, that if we divide the numerator and denominator of a fraction by the same number, the fraction still preserves the same value. This is made more evident by means of the general expresfor if we divide both the numerator m a and the denomi

sion

ma
m

α

nator m b by the number m, we obtain the fraction, which, as was

90. In order, therefore, 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.

91. To make this more clear, let us consider the fraction 4 We see immediately that both the terms are divisible by 2, and that there results the fraction. Then that it may again be divided by 2, and reduced to ; and this also, having 2 for a common divisor, it is evident, may be reduced to. But now we easily perceive, that the numerator and denominator are still divisible by 3; performing this division, therefore, we obtain the fraction, which is equal to the fraction proposed, and gives the simplest expression to which it can be reduced; for 2 and 5 have no common divisor but 1, which cannot diminish these numbers any further.

92. 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 Eul. Alg.

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