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preceding fraction, the number under the line shews 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, shews that, in order 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. fractions,

1

Such are the

1, 1, 1, 1, 1, 4, 1, 1, 70, 71, 72, &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; To is less

1

than and Too is less than Too

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 oo, 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. Now we know that if we divide the dividend 1 by the quotient, which is equal to 0, we obtain again the divisor o 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 0 expresses a number infinitely great, or o.

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.

9

CHAPTER VIII.

Of the properties of Fractions.

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

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makes an integer, and that consequently they are all equal to The same equality exists in the following frac

one another.

tions,

each of them

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

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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 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. For this reason all the fractions

5

6

9

1, 4, 3, 4, fo, 12, 77, 78, 77, 10, &c.,

are equal, the value of each being.

2 19

4

5

6
8

Also

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3, 3, 3, 72, 73, TT, ZT, 74, F. 10, &c.,

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

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

a

in general, that the fraction may be represented by the fol

b

lowing expressions, each of which is equal to ; namely,

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

α

b

value 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 e multiplied by 2b will give 2 a, that c multiplied by 3 b will give

s a, and that in general c 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 m a divided by mb gives also the fraction which is consequently equal to c; and this is what was to

ma

m b'

:

be proved for c having been assumed as the value of the frac

tion, it is evident that this fraction is equal to the fraction

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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,

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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, 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

ma

m b

of the general expression ; for if we divide both the numerator ma and the denominator mb by the number m, we obtain the fraction, which, as was before proved, is equal to m b

ma

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 lower form; but, on the con

Eul. Alg.

4

trary, when we see that except unity no other common divisor can be found, this shews that the fraction is already in the simplest form that it admits of.

48

120

91. To make this more clear, let us consider the fraction We see immediately that both the terms are divisible by 2, and that there results the fraction 24. 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 other forms, that is to say, to expressions whose denominators are equal. Of this we shall treat in the following chapter.

98. We conclude the present by remarking, that all integers may also be represented by fractions. For example, 6 is the same as, because 6 divided by 1 makes 6; and we may, in the same manner, express the number 6 by the fractions 12, 18, 3o, and an infinite number of others, which have the same value.

36

24

CHAPTER IX.

Of the Addition and Subtraction of Fractions.

94. WHEN fractions have equal denominators, there is no difficulty in adding and subtracting them; for + is equal to , and is equal to 2. In this case, either for addition or

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