have found 9, and should consequently, have been obliged to make three useless trials. We easily perceive that this can never give too great a quotient, for if it could, then might 400 be contained in the dividend a greater number of times than 397, which is impossible, Because, 19397319, (60 and 156,) the above example is proved thus: 6 X 397 +3192701. This expression is the same as to say, multiply the divisor and the integral part of the quotient together, and to the product add the remainder or numerator of the fractional part, which will give the dividend. The student may, in like manner, prove all similar divisions: also (161) by casting out the nines. 164. It is sometimes so nice a point to determine the true quotient figure, that most persons, however skilful, are liable to err; but the multiplication of the divisor by the quotient figure immediately shows when this figure is too great, because, in this case, the product will be greater than the partial dividend from which it is to be subtracted. Also, when the subtraction of the product from the partial dividend leaves a remainder equal to or greater than the divisor, as the divisor still contained in this remainder, the quotient figure is too small. We here first say 1 in 6, or rather 11 in 68, six times, and place 6 in the quotient; but, having multiplied the divisor by this figure and found that the product 681492 is greater than the partial dividend 681483, and consequently the figure 6 too great, we efface it, and, having substituted the figure 5, proceed as usual. Or thus, taking 6 for the first quotient figure, say 6 times 11 is 66, which taken from 68, leaves 2. This 2, with the next figure, makes 21. Then 6 times 3 is 18, which taken from 21, leaves 3. This, with 4, makes 34. Then 6 times 5 is 30, and 30 from 34 leaves 4. This, with 8, makes 48. Then 6 times 8 is 48, and 48 from 48, leaves 0. Lastly, as 6 times 2 is not contained in 3, we see that 6 is too great, and put 5 in its stead as before. Proof by multiplication. 113582 59999 thousandths 1135706418 567910 6814806418 thousandths 27582 thousandths 6814834,000 Here it is plain that the product, formed by repeating 59999 thousandths 113582 times, is a number of thousandths, as well as the remainder 27582. Hence, the sum of these, which is 6814834000, is also a number of thousandths; wherefore, to obtain the units in this number, (144,) we divide by 1000, which is done (56) by cutting off the three ciphers on the right, the number on the left being the dividend, as required. 1. 728976-907855, and 728976-803855 165. The greater the dividend, or number to be divided into equal parts, the greater the quotient, or value of each part. Therefore, to multiply the dividend (150) is to multiply the the quotient; and, inversely, to divide the dividend is to divide the quotient. Again, the greater the divisor, or number of equal parts, the less the quotient, or value of each. Therefore, to multiply the divisor is to divide the quotient; and inversely, to divide the divisor is to multiply the quotient; that is to say, the operation performed upon the "dividend produces upon the quotient the same effect; but that performed upon the divisor, the contrary. But (154) multiplication and division destroy the effect of each other; wherefore, to multiply or divide both dividend and divisor by the same number has no effect upon the quotient. 166. The decimal value of the generic unit in each of the fractions,, and 13, is found by dividing a unit by the denominator, and that of the fraction by multiplying the value of the generic unit by the numerator, thus: 0,125 and 1 0,0625 and 16 1 32 0,125 × 3 = 0,375 0,03125 and 10,03125 × 12=0,37500 The ciphers on the right of the numbers 0,3750 and 0,37500 may be omitted, as having no effect upon the value of the number 0,375, because (165) 3750 3750 37500 &c. 1000 10000 1000007 167. When any two numbers are each multiplied by the same number, the two products are called equimultiples of the two numbers. Thus, if we multiply the two terms of the fraction each by 2, the resulting terms 6 and 16 are equimultiples of 3 and 8. But we have seen (165) that the quotient is the same as the quotient or 12, &c. Wherefore, if any equimultiples whatever be taken of the terms of a fraction, the resulting fraction is still of the same value. Hence, it is evident that the same fraction may be represented by an infinity of different numbers, seeing that the two terms may be multiplied by each of the numbers in the natural scale 2, 3, 4, 5, 6, 7, &c., ad. inf. Thus, === 13=18 181, &c., ad. inf. The least numbers by which a fraction can be expressed are called its lowest terms. Thus, the terms 3 and 8 are the lowest terms of the above fraction. Each of the other expressions, 1, 2, 3, &c., is called a value of 3. We may further elucidate this by the smaller Spanish coins or parts of a dollar, namely, halves, quarters, eighths, and sixteenths, which, though they do not accord with the decimal Federal Money of the United States, are still in general use. It will also be useful to the student to render himself betimes familiar with the following decimal values of these coins, thus: 168. When a fraction is expressed by equimultiples of its lowest terms, it is often necessary, in order to expedite calculation, to find those terms; for, if we would multiply a number by 122936 and divide the product by 138303, which is the same as to multiply the number by the fraction 122936, we should obtain the result much more readily if we could conHeniently discover that 122936 8, in which case we should simply multiply by 8 and divide by 9 or, subtract from the number one-ninth of itself. This we shall illustrate thus: 428695731 X 122936 138303 138303 First method 428695731 122936 3858261579 nine hundred times 15433046316 thirty-six times 1383037 857391462 5144348772 two thousand times 120 thousand times To facilitate the research above alluded to, as well as for other important purposes, the properties of numbers, with which we commence the succeeding book, will be found of great utility. |