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so doing be led into any absurdity or contradiction. We are then at liberty to call the operation of taking a part of a quantity "multiplication by a fraction,” and to express it by the same sign, and we may interpret the meaning of a product of a quantity by a fraction to be the fractional part of the quantity, which is expressed by the multiplier. 2 4
2 4 x 2 Hence £ Х
3 4 x 2 3 3 4 X 2 Simly. £ Х Х
2 3 X 4 2
5 In a similar manner it may be shewn that the result of the multiplication of a quantity by several fractions in succession is the same as the result of multiplication by one fraction, which is compounded of them all.
Hitherto it has been supposed that the multiplicand is concrete, but if it be abstract, and therefore represent one or two operations, according as it is a whole number or a fraction, what meaning are we to assign to its "multiplication by a fraction ?" and what is to be the interpretation of the product ?
It is plain that we cannot conceive any idea of taking a fractional part of an operation ; therefore the meaning of the product cannot be the same as in the former case. To determine the meaning, which is to be assigned, let us revert to the idea which we have conceived of the product of two abstract numbers. This is, that it represents an operation which is equivalent to the repetition of that indicated by the multiplicand, as often as there are units in the multiplier. But this repetition is also equivalent to the successive operations indicated by the multiplicand and the multiplier. For multiplication by two numbers in succession is equivalent to the repetition of the multiplication by one of them, as often as there are units in another.
Therefore the operation indicated by the product of two abstract numbers is equivalent to the successive operations indicated by the two. This view of the term product evidently is applicable to the case of fractions, as well as of whole numbers, since every abstract number represents an operation, and may represent two, as every fraction does; therefore we may with equal justice talk of the product of the one kind of number, as of the other. Adopting then this view of the case, we shall see that the product of an abstract number, fractional or not, by a fraction, is the indication of the operation denoted by the fraction of the number. For the successive multiplication by two fractions has been shewn in the previous part of this Prop. to be equivalent to multiplication by the fraction compounded of them; therefore the operation indicated by the product of two fractions is the same as is indicated by the fraction compounded of them. In other words the product of two fractions is equal to the fraction of a fraction.
Hence, generally it appears that the Rule for multiplication by fractions is the same as that for finding the fraction of the multiplicand, or, if this be fractional, the same as for finding the fraction of a fraction.
Prop. 35.-To explain the meaning of Division by a frac
tion, and to deduce a Rule for forming the quotient. Division is the operation by which (according to our first notion of it) is found some number or quantity, which repeated a given number of times, equal to the divisor, becomes equal to a given number or quantity, the dividend. But since the number of units in the quantity sought is the same as the number of times the abstract units in the dividend contain those in the divisor, therefore the operation of division is defined to be that, by which is ascertained how often one number is contained in another. In order to explain the meaning of Division by a fraction, it will be necessary to consider both these notions of the operation.
Now every integer is expressible in the form of a fraction, and every multiplication by an integer might be performed by the repetition of equal parts of the multiplicand, therefore division by an integer might be expressed, as division by a fraction, and the quantity sought in the division would be defined to be such, that the parts of it denoted by the divisor would be equal to the dividend. It appears then that if division by a fraction be defined to be the operation by which is found the number or quantity, of which the part or parts denoted by the divisor are equal to the dividend, no new idea of division is introduced, but one, which is included in the idea of division by an integer. Such then is defined to be the operation of division by a fraction, according to the first notion of division.
If it be considered that the object of division is to find how often the abstract units in the dividend contain those in the divisor, we need no extension of ideas to explain the case of division by a fraction. For, by the number of times which one abstract nnmber contains another is meant the number of times which the operation represented by the one must be repeated in order to produce the effect of that represented by the other; and this number of times is the symbol of the operations of multiplication and division which must be performed upon the result of the one to produce that of the other, so that the object of division might be defined to be the discovering what operations of multiplication and division must be performed upon the result of the operation denoted by the divisor
in order to produce the result of that denoted by the dividend. There is nothing in this definition to exclude a fraction as a divisor.
We have now to deduce a Rule for the formation of the quotient.
1st. Let the dividend be concrete, and represented by the straight line A B; let the divisor be s or i. А D
c Then we seek a line, of which or shall be equal to A B. Divide A B into three equal parts, then each of these parts is in the one case of the line sought, and in the other is of the same. Therefore the line sought is equal to 8 or 2 of these parts; but each of these parts is equal to Ž A B, therefore quotient is equal to f A B or A B. Hence to divide a concrete quantity by a fraction, we must invert the divisor and multiply by it the dividend.
2nd. Let the dividend be abstract, then we seek the symbol of some operation, such that by performing upon its result the operation denoted by the divisor, a result will be obtained equal to that from performing the operation indicated by the dividend. Now since # of g of an unit = & of an unit, therefore it is the symbol of the operations, upon whose result, if the operations denoted by be performed, the same result is obtained as by performing the operations denoted by & ; hence it is the quotient of Ở = $; but = ở x }; therefore the rule for obtaining the quotient is the same as before, "invert the divisor and multiply by it the dividend."
3rd. Let it be required to find how often a fraction is contained in a given abstract number; i.e. to find what operation must be performed upon the result of the operations denoted by the divisor to produce the same result as the operations denoted by the dividend. Here, since i of of an unit = - of an unit, therefore we must perform the operations denoted by & upon the result of those denoted by 4 in order to produce the result of those denoted by & ; hence it denotes the “number of times" that a contains *; and is obtained from these two fractions as before.
Prop. 36.-- To explain the meaning of a complex fraction,
and to deduce a Rule for its reduction to a simple fraction. A complex fraction is one in which the numerator and denominator are, one or both, fractions or fractional expressions. If they are fractional expressions, these may be reduced to simple fractions, therefore it will be necessary only to consider the case of their being simple fractions.
Whatever meaning may be assigned to this class of fractions must be applicable to all cases of them: now every ordinary fraction may be expressed as a complex fraction, as for instance Ś may be put into the form iz;
hence whatever may be the interpretation assigned to such a fraction, as this, must be also assigned to every other fraction of the class, of which this is only a particular case. Now s inch denotes a quantity composed of parts, of which 3 compose an inch, 2 of these parts being contained in the quantity, i.e. in other words, it denotes a quantity composed of parts, of which compose an inch, of these parts being contained the quantity.
13 Hence we must interpret the meaning of such an expression as 4 inch to be, that it denotes a quantity composed of parts of which 54 compose an inch, 13 of these parts being contained in the quantity.
Again ġ inch denotes a quantity such that, if the operation denoted by the denominator, viz. multiplication by 3 be performed upon it, it will become equal to the numerator; in other words, it denotes a quantity such that, if the operations denoted by ** (which are equivalent to multiplication by 3) be performed upon it, it will become equal to the numerator. Hence we may interpret the meaning of the above-mentioned complex fraction to be, that it denotes a quantity such that, if 5% of it be taken, 13 inch will be formed. Now the operation by which such a quantity is found is that which has been defined to be division by the fractional denominator; 90 that such a fraction may be used to represent the quotient of any quantity by a fraction.
Again: an abstract fraction as ģ is to be regarded as the symbol of division by 3 and of multiplication by 2 ; so also an abstract complex fraction must be regarded as the symbol of division of the unit by the denominator, and of multiplication by the numerator. Hence also it will denote the part, which the concrete units in the nnmerator are of those in the denominator; for evidently in order to form the quantity 1š inch from the quantity 54 inch, we must divide the latter by 54 and multiply it by 13, i.e. we
must perform the operations denoted by the symbol 3
Lastly: since 13 inch = of 54 inch, therefore it follows that is and
represents the opera52 tions which have to be performed upon the result of the operations denoted by 54, in order to produce the effect of performing those denoted by 13, i.e. it represents the quotient of the abstract fraction 13 by 54.
Hence it appears that in all respects a complex fraction admits of the same interpretation as an ordinary fraction. We have now to deduce a Rule for the reduction of a complex to a simple fraction. Let the fraction already used serve as an example. According to the first explanation of the meaning of the fraction, we have to find a quantity of which 54 compose an unit. This is effected by the previous proposition by inverting the
divisor, and multiplying by it so inverted; doing this we find the quantity to be sig of an unit; therefore the value of the fraction is 13 x 39 of an unit or 13 = 54 of an unit. This is the same result as we should obtain by using any of the other explanations given. Therefore in general to reduce a complex fraction, divide the numerator by the denominator,
If the numerator and denominator be composed by the sum or difference of several fractions, it manifestly will not affect the result to bring them all to a C.D. And if this be done, the denominator will vanish in the divi. sion, and may therefore be omitted altogether. Henee the Rule 2 page 24 in the 56 Practice."
Prop. 37.-To explain the system of notation of decimal
fractions, In the ordinary system of notation, the number represented by a figure is always one-tenth of that which it would represent if it occupied the place of the figure on its left. Thus in the number 11111, the 1 on the left denotes 10000; that next 1000, the next 100, the next 10, the last 1. Now if we plaee a point after this last 1 to shew that it is in the units' place, and then write a series of l's after it, the first of them should on the same principle represent one-tenth, the second one-hundredth, the third onethousandth, and so on, so that the first, second, third, &c. places to the right of the unit's place may be called the tenths', the hundredths', the thousandths', &c. places. Now every fraction with 10 or some power of 10 for its denominator may be resolved into the sum of a series of fractions having inferior powers of 10 for their denominators, every numerator being less than 10, i.e. may be resolved into the sum of a number of tenths, hundredths, &c. less than 10, and may therefore be denoted by writing the numerators in succession, supplying ciphers where any denomination of fraction is wanting, in order to keep the other figures in their proper places. 374 300 70 4
7 4 Thus + +
+ =24 + + = 24.706 1000 1000 1000 1000
10 1000 24 20
2 4 +
-.0024 10000 10000 10000
1000 10000 Hence it appears that in order to denote a fraction with any power of 10 as its denominator decimally, we must mark off by a point as many figures from the right of the numerator as there are ciphers in the denominator; and if there are not a sufficient number of figures we must supply ciphers on the left.