to 1, or 2; if 300, or 100 added to 100 added to 100, be divided by 3 or 9, the remainder will be 1 added to 1 added to 1, or 3. In general, if we resolve in the same manner a number expressed by one significant figure, followed, on the right, by a number of ciphers, in order to divide it by 3 or 9; the remainder of this division will be equal to as many times 1, as there are units in the significant figure, that is, it will be equal to the significant figure itself. Now any number being resolved into units, tens, hundreds, &c. is formed by the union of several numbers expressed by a single significant figure; and, if each of these last be divided by 3 or 9, the remainder will be equal to one of the significant figures of the given number; for instance, the division of hundreds will give, for a remainder, the figure occupying the place of hundreds; that of tens, the figure occupying the place of tens; and so of the others. If then, the sum of all these remainders be divisible by 3 or 9, the division of the given number by 3 or 9 can be performed exactly; whence it follows, that if the sum of the figures, constituting any number, be divisible by 3 or 9, the number itself is divisible by 3 or 9. Thus the numbers, 423, 4251, 15342, are divisible by 3, because the sum of the significant figures is 9 in the first, 12 in the second, and 15 in the third. Also, 621, 8280, 934218, are divisible by 9, because the sum of the significant figures is 9 in the first, 18 in the second, and 27 in the third. It must be observed, that every number divisible by 9 is also divisible by 3, although every number divisible by 3 is not also divisible by 9. Observations might be made on several other numbers analogous to those just given on 2, 3, 5, and 9; but this would lead me too far from the subject. The numbers 1, 3, 5, 7, 11, 13, 17, &c. which can be divided only by themselves, and by unity, are called prime numbers ; two numbers, as 12 and 35, having, each of them, divisors, but neither of them any one, that is common to it with the other, are called prime to each other. Consequently, the numerator and denominator of an irreducible fraction are prime to each other. Examples for practice under Article 61. Ans. 12. Ans. 5. What is the greatest common divisor of 24 and 36? Examples for practice under articles 57, 58, and 60. 65. After this digression we will resume the examination of the table in article 55, By multiplying By dividing the numerator, the fraction is By multiplying the denominator, the fraction is that we may deduce from it some new inferences. We see at once, by an inspection of this table, that a fraction can be multiplied in two ways, namely, by multiplying its numerator, or dividing its denominator, and that it can also be divided in two ways, namely, by dividing its numerator, or multiplying its denominator; hence it follows, that multiplication alone, according as it is performed on the numerator or denominator, is sufficient for the multiplication and division of fractions by whole numbers. Thus, multiplied by 7 units, makes }; , divided by 3, makes →→7. 66. The doctrine of fractions enables us to generalize the definition of multiplication given in article 21. When the multiplier is a whole number, it shows how many times the multiplicand is to be repeated; but the term multiplication, extended to fractional expressions, does not always imply augmentation, as in the case of whole numbers. To comprehend in one statement every possible case, it may be said, that to multiply one number by another is, to form a number by means of the first, in the same manner as the second is formed, by means of unity. In reality, when it is required to multiply by 2, by 3, &c. the product consists of twice, three times, &c. the multiplicand, in the same way as the multiplier consists of two, three, &c. units; and to multiply any number by a fraction, for example, is to take the fifth part of it, because the multiplier being the fifth part of unity, shows that the product ought to be the fifth part of the multiplicand.* Also, to multiply any number by is to take out of this number or the multiplicand, a part, which shall be four fifths of it, or equal to four times one fifth. Hence it follows, that the object in multiplying by a fraction, whatever may be the multiplicand, is, to take out of the multiplicand a part, denoted by the multiplying fraction; and that this operation is composed of two others, namely, a division and a multiplication, in which the divisor and multiplier are whole numbers. Thus, for instance, to take of any number, it is first necessary to find the fifth part, by dividing the number by 5, and to repeat this fifth part four times, by multiplying it by 4. We see, in general, that the multiplicand must be divided by the denominator of the multiplying fraction, and the quotient be multiplied by its numerator. The multiplier being less than unity, the product will be smaller than the multiplicand, to which it would be only equal, if the multiplier were 1. * We are led to this statement, by a question which often presents itself; namely, where the price of any quantity of a thing is required, the price of the unity of the thing being known. The question evidently remains the same, whether the given quantity be greater or less than this unity. 67. If the multiplicand be a whole number divisible by 5, for instance, 35, the fifth part will be 7; this result, multiplied by 4, will give 28 for the of 35, or for the product of 35 by . If the multiplicand, always a whole number, be not exactly divisible by 5, as, for instance, if it were 32, the division by 5 will give for a quotient 63; this quotient repeated 4 times will give 249. This result presents a fraction in which the numerator exceeds the denominator, but this may be easily explained. The expression, in reality denoting 8 parts, of which 5, taken together, make unity, it follows, that is equivalent to unity added to three fifths of unity, or 13; adding this part to the 24 units, we have 253 for the value of of 32. 68. It is evident, from the preceding example, that the fraction contains unity, or a whole one, and, and the reasoning, which led to this conclusion, shows also, that every fractional expression, of which the numerator exceeds the denominator, contains one or more units, or whole ones, and that these whole ones may be extracted by dividing the numerator by the denominator; the quotient is the number of units contained in the fraction, and the remainder, written as a fraction, is that which must accompany the whole ones. The expression 307, for instance, denoting 307 parts, of which 53 make unity, there are, in the quantity represented by this expression, as many whole ones, as the number of times 53 is contained in 307; if the division be performed, we shall obtain 5 for the quotient, and 42 for the remainder; these 42 are fiftythird parts of unity; thus, instead of 307, may be written 53. Examples for practice. Reduce the fraction to its equivalent whole number. Ans. 2. Reduce to its equivalent whole or mixed number. Ans. 31⁄2. Reduce 15 to its equivalent whole or mixed number. Ans. 33. 4 8 20 Reduce to its equivalent whole or mixed number. Ans. 242 Reduce to its equivalent whole or mixed number. Ans. 121. Reduce to its equivalent whole or mixed number. Ans. 10%. 69. The expression 543, in which the whole number is given, being composed of two different parts, we have often occasion to convert it into the original expression 307, which is called, reducing a whole number to a fraction. To do this, the whole number is to be multiplied by the denominator of the accompanying fraction, the numerator to be added to the product, and the denominator of the same fraction to be given to the sum. In this case, the 5 whole ones must be converted into fiftythirds, which is done by multiplying 53 by 5, because each unit must contain 53 parts; the result will be 25; joining this part with the second, 4, the answer will be 37. 53 Examples for practice. 53 70. We now proceed to the multiplication of one fraction by another. If, for instance, were to be multiplied by ; according to article 66, the operation would consist in dividing by 5, and multiplying the result by 4; according to the table in article 65, the first operation is performed by multiplying 3, the denominator of the multiplicand, by 5; and the second, by multiplying 2, the numerator of the multiplicand, by 4; and the required product is thus found to be It will be the same with every other example, and it must consequently be concluded from what precedes, that to obtain the product of two fractions, the two numerators must be multiplied, one by the other, and under the product must be placed the product of the denominators. |