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the unit is divided ; that above the line shews how many of these parts there are in the given quantity.
3. The expression & inch means 2-fifths of 1 inch, or 1-fifth of 2 inches. 4. It also denotes the quotient of 2 inches divided by 5.
5. The expression s is the symbol of the multiplication by 2, and division by 5.
6. It also denotes the part (or fraction) which 2 is of 5. 7. Or the "number of times” which 5 units are contained in 2 units. 8. Or the quotient of 2 divided by 5.
Cor. Hence is apparent the method of converting an improper fraction into a mixed number. For since such an expression as a represents the quotient of 23 divided by 8, let this division be performed as far as possible, from which we obtain the integral quotient 2, with a remainder 7, which has still to be divided by 8; but this division not being capable of being actually performed, must be denoted by the fraction š. Hence the whole quotient is 2+ , written 2%: or 22 = 27.
Prop. 27.—The numerator and denominator of a fraction
may be both multiplied or divided by any the same num
ber, without altering its value. Take a line A B, divide it into 3 equal parts, and each third part into
B 4 equal parts; then the whole A B will be divided into 12 equal parts. And it appears that A C, which is equal to šof A B, contains 8 of the twelfth parts of A B; or 'of A B = 18, of A B: i.e. the multiplication of numerator and denominator by 4 has not altered the value of the fraction. The same may be shewn of any fraction, and any multiplier. Also since of A B=} of A B, the division of numerator and denominator by 4 has not altered the fraction. The same may be shewn in every case. The same Prop. otherwise :8
of an unit = 8 units 12 = (8 units - 4) = 3 (Prop. 15)
3 Or exhibited algebraically :
a X o
units = (a X c) units (6 X c) Хc
= a units b=-units,
Cor. 1. Hence the correctness of the Rule for reducing fractions to their lowest terms is evident. For numerator and denominator may be divided by any the same number without altering the value of the fraction, and therefore by their G.C.M., or by their common factors in succession.
Cor. 2. Hence also the Rule for conversion of an integer into an improper fraction. For every integer may be expressed in the fractional notation with denominator ]; thus 3 = , because this expresses a quantity composed by 3 of such parts, of which 1 forms the unit, i.e. 3 units.
3 X 5 15 Hence 3=-=
&c. &c. 1 1 x 5 5
Prop. 28.- To prove the Rule for reducing fractions to
Since the terms of a fraction may be both multiplied by the same number without altering its value, therefore if multipliers can be found, which shall convert all the denominators of several fractions into the same number, and we multiply the terms of each fraction by the proper multiplier, we shall have converted all the fractions into others of equivalent value, but having the same denominator. Now it is evident that the multipliers in question cannot be found, unless the common denominator be a number capable of exact division by each of the denominators, i.e. unless the C.D. be a multiple of all the denominators. Hence it follows that the least possible common denominator to which several fractions can be reduced is the L.C.M. of all the denominators. And the multipliers, which shall convert each denominator into the L.C.D. are the quotients of the L.C.D. by each.
Prop. 29 —To explain the Rule for Addition of frac
It is an axiom that it is impossible to add together concrete numbers of different denominations, i.e. to express their sum by a concrete number of either denomination equal to the sum of the abstract numbers. Thus it would be absurd to say that £2 and 28. added together are equal to £4 or 4s.
Now the denominator of a fraction denotes the kind of sub-unit, (or part of unit,) of which the numerator denotes the number : therefore fractions with different denominators are of different denominations; therefore it is impossible to add them, as they are. They must therefore be expressed as fractions with the same denominator, and then their sum may be obtained by adding the numerators, and retaining the same common denominator which only shews the denomination : for just as we say £4 + £2 = £6; so 12 ti or 4-twelfths + 2-twelfths = 6-twelfths = n.
If any of the fractions be mixed numbers, we may separate them into their parts of integers and fractions, add each separately, and then add the sums.
Cor. Hence the correctness of the Rule for conversion of a mixed number into an improper fraction is evident. For the mixed number is the sum of an integer, and a fraction, indicated but not performed; but if the integer be expressed in the same denomination as the fraction, the addition may be actually performed. Therefore the integer is expressed as a fraction with the denominator of the given fraction by multiplying it by the denominator, and the addition is performed in the ordinary way.
7 3X 8 7 3 X8+7 31
8 Prop. 30.-To explain the Rule for Subtraction of fractions.
As it is impossible to add numbers of different denominations, so is it impossible to subtract them; therefore fractions must be expressed with the same denominator, before they can be subtracted. This being done, the difference may be obtained by subtracting the numerator of the subtrahend from that of diminuend. This is evident, if we bear in mind that the denominator of a fraction shews the denomination of the numerator. Thus -}, or 5-sevenths 3-sevenths = 2-sevenths=4.
If the diminuend be a mixed number, or both diminuend and subtrahend, then we may subtract in any parts we please (Prop. 5). Let then the fractions be reduced to their L.C.D., and let the fraction of the subtrahend be subtracted first, which may be done by subtracting it from the fraction of the diminuend; but if the fractional subtrahend be less than the fractional diminuend, this is not possible; therefore increase the numerator of the diminuend by the C.D. and the integer of the subtrahend by 1, (thus increasing both numbers by 1, which does not alter the difference,) and subtract. Having subtracted the fraction, the integer may now be subtracted from that of the diminuend.
Prop. 31.-To prove the Rule for Multiplication of a
fraction by an integer. Multiplication being only a short method of addition, may be performed on the same principle. Therefore as in Addition of Fractions with a C.D. the numerators are added for the numerator of the sum, and the depominator left the same, so in forming the product of a fraction by an integer, or in forming the sum of a fraction repeated as often as there are units in an integer, we must find the sum of the numerator repeated the same number of times, i.e. find the product of the numerator and integer, for the numerator of the product, and leave the denominator untouched.
2 2 2 2 4 X 2 8 Thus —£. X4=-£. +-£.+-£. +-£. = -£. =-£. 3 3 3 3 3
3 Hence also the repetition of the operations denoted by four times is equivalent to the performance of those indicated by f; or X 4 = f.
If the terms of the numerator and denominator have any common measure, they may be divided by it (Prop. 27.)
Prop. 32.–To prove the Rule for division of a fraction
by an integer. As in division of a concrete quantity, the number of the units is divided by the divisor, so in division of a fraction, considered as a concrete quantity, the number of the units being denoted by the numerator, and their kind by the denominator, we must divide the numerator by the divisor. Thus }} = 4 is equal to it; for the fourth part of 12 seventeenths is evidently 3 seventeenths, just on the same principle as the fourth part of 12 shillings is 3 shillings.
But if the numerator be not exactly divisible by the divisor, the quotient may be obtained by multiplying the denominator by the divisor.
units = 5 units (7 X 2) = units ; 2. 7 X 2
7 Or exhibited algebraically:
units = a units (6 X c) = (a units ; 6) -C= - units ЪХc
If the fraction be considered as abstract, then we seek the symbol of some operations, which, repeated a number of times equal to the divisor, shall produce the same result as those indicated by the fractional dividend. Now since the operations indicated by 1: are équivalent to the repetition of those indicated by i, four times, therefore 13 -4.
5 Simly. since
X 2 I
7X2 Prop. 33.—To prove the Rule. for finding the value of a
compound fraction. Let the fraction be of $. In order to ascertain what is the meaning, and equivalence of this expression, we consider that is the symbol of two operations, to be performed upon the quantity with which it is joined (Prop. 26.) Thus if it be required to find the value of f£ or of £l, the result would be obtained by dividing £1 by 3 and multiplying the quotient by 2, or vice versa. So if it be required to find the value of $ of £, the result will be obtained by dividing 9£ by 3 and multiplying the quotient by
10 2, and will therefore be
2 X 5
£. Hence it appears that the re
21 sult of the successive operations upon the unit £1 (or anything else), of
taking first f of it and then f of this s, is the same as the result of taking H4 of it; that is g of ļ and if represent equivalent operations and are therefore equal
2 5 2 X 5 10 Or,
3 7 3 x 7 21 Similarly it may also be shewn that the value of such expressions as of of of &c. however many fractions there may be, is obtained by multiplying the numerators for a new numerator, and the denominators for a new denominator. For the last two fractions may be reduced to one, as above, then this and the next, and so on. Prop. 34.—To explain the meaning of the Multiplication
by a fraction, and to deduce a Rule for finding the
product. The operation of Multiplication is that by which is found the result of the repetition of a number or a quantity, as often as there are units in the multiplier. Hence it is plain that the multiplier must be an abstract number, and the multiplicand either abstract or concrete. But the meaning of the result is somewhat different in these two cases; for in the former the result is an abstract number, and as an abstract nụmber is the symbol of an operation, viz. that of repetition, the meaning of the result is that the operation represented by it is equivalent to the repetition of the operation denoted by the multiplicand as often as there are units in the multiplier, In the case of the multiplicand being concrete the meaning of the result is, that the quantity which it represents is equal to as many quantities, equal to the multiplicand, as there are units in the multiplier. We have now to see whether these ideas of multiplication will hold good, when the multiplier is a fraction. Now it is plain, that in the strict sense of the expressions " as often," "as many times," as many,” we cannot speak of repeating an operation "as often," or taking “as many” quantities, as there are units in a fraction. Bụt if we adopt the extended notion of “ą time,” viz, that it may mean such a repetition of a subdivision of a quantity, as takes place in forming a part of it, then the forming a part of a quantity may be called multiplication by that which expresses the part, viz, the fraction. This operation (of forming a part) is strictly one of multiplication and division combined, but every multiplication might be performed by these combined operations (as for instance to multiply by 5 we might multiply by 10 and divide by 2), that is by such an operation as that which has been called "multiplication by a fraction;" so that the strict sense of multiplication includes the sense of multiplication by a fraction, viz. the repetition of a part of a quantity so as to form a "multiple-part," Therefore in assigning this meaning to the term multiplication," we introduce no idea contrary to the general one, and shall not by