Contractions in Division. RULE I. When the divisor consists of a single figure. (16.) Set down the number as before directed, and draw a line beneath the dividend. See how often the divisor is contained in as many of the left hand figures of the dividend as will contain it once, and place the quotient below the dividend; if there be a remainder, consider it as written before the next figure of the dividend, and divide as before; and continue the operation in the same manner through the rest of the dividend. This rule is but an abbreviated method of dividing by the preceding rule; the multiplication of the divisor by the quotient, and the subtraction of this product from the dividend, being performed mentally, instead of being written out at length. When there are 0's at the right of the divisor. (17.) Cut off the O's from the divisor, and an equal number of figures from the right of the dividend, and divide with the remaining figures according to the preceding rules; if there be a remainder, write the figures cut off from the dividend on the right of this remainder; this will give the true remainder. EXAMPLE. Divide 3976478 by 63000. 378 196 189 7478 The reasons for this rule will best be understood by a reference to the proof of division by multiplication. For, since the dividend, when it can be exactly divided, is equal to the product of the quotient by the divisor, it will follow that there can be nó figures of a lower order in such a dividend than there is in the divisor; for if the last significant figure in the divisor be tens, the product of the divisor by the quotient can contain no significant figure less than tens (12). If the divisor have none less than hundreds or thousands, then the product can have none less than hundreds or thousands. And therefore, if there be significant figures in the dividend of a less order of units than in the divisor, they must belong to the remainder; inasmuch as no quotient whatever, multiplied into the divisor, could produce them. In the preceding example, the product of 63,000 by any quotient whatever, can contain no figure of a lower order than thousands; and hence the number 478 cut off from the right hand of the divisor, will remain after the division is com pleted. Moreover, in dividing the remaining figures, we are dividing thousands by thousands, and the remainder 7 is, therefore, 7 thousands, and to this 478 is to be added to make up the whole remainder, and this is done by writing 478 at the right of the remainder. NOTE. To divide by 10, 100, 1000, 10,000, or any number consisting of unity and a number of O's, it is only necessary to cut off from the dividend a number of figures equal to the number of O's in the divisor. EXAMPLES. Divide 137272 by 7800. 20000 Ans. 174673. 10000 CHAPTER II. FRACTIONS. (18.) A fraction is an expression to denote a part, or parts of an unit. It is expressed by two numbers, placed one below the other, with a horizontal line between them. The upper number is called the numerator, and the lower the denominator. They are also called the terms of the fraction. 5 ,,, read, two-thirds, five-eighths, one-fifth, are fractions. The denominator of the fraction denotes the number of parts into which the unit is divided, and the numerator the number of parts expressed by the fraction. Thus, if we consider one foot the unit of measure, the expression of a foot signifies, that one foot is conceived to be divided into five equal parts, and that the line expressed by the fraction, contains four of these parts. Fractions may also be considered unexecuted divisions, or ) as resulting from the remainders after division. Suppose it were required to divide a line nineteen feet in length into five equal parts. By dividing 19 by 5, we should have the value of each of these parts. But this division cannot be executed in whole numbers; for, since 5 is contained more than three times, and less than four times, in 19, the length of each part is more than three feet, and less than four feet. The excess of each of these parts above three feet, must, therefore, be expressed by a fraction. Haying divided 19 by 5, we have 3 for a quotient, and a remainder of 4. We may now conceive each foot in this remainder to be divided into five equal parts; and since there are four feet in the remainder, we should have four of these parts, or of a fcot for the excess of the fifth part of nineteen feet above three feet. The expression for this part would, therefore, be 3 feet. The fifth part of nineteen feet might also be written, which would signify, that each foot is divided into five parts, and nineteen of these parts are expressed by the fraction. This expression is equivalent to the former, for nineteen fifth parts is the same as 3 and 4. (19.) A proper fraction, is one which has its numerator less than its denominator, and is, consequently, less than unity. An improper fraction, has its numerator greater than its denominator, and is therefore greater than unity. If the numerator is equal to the denominator, the fraction is equal to unity; for, in this case, the fraction expresses all the parts into which unity is divided. ,, 11, and 11, are proper fractions. 8 ,, and 4, are improper fractions. A mixed number, is composed of a whole number and a fraction together, as, 34, 21, 75, &c. A compound fraction, is the fraction of a fraction, & of §, of, and of, are compound fractions. 23 A complex fraction, is one which has a fraction for its numerator, or denominator, or both; fractions. are complex An integer, is a unit or whole number, and is so called as distinguished from a fraction, which is a part, or parts, of an unit. A prime number, is one which is not divisible by any number greater than unity; 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, &c., are prime numbers. Two numbers are said to be prime to each other, when they have no common divisor greater than unity: thus, 25 and 32 are prime to each other. The prime factors of a number, are such prime numbers as when multiplied together will produce that number, thus: 3,5 and 2 are the prime factors of 30; 2, 3 and 2, the prime factors of 12; and 2, 2, 2, and 2, the prime factors of 16. MULTIPLICATION AND DIVISION OF FRACTIONS. To multiply a fraction by a whole number. RULE. (20.) Multiply the numerator of the fraction, or divide the denominator, by the given number. Demonstration of the Rule.-Since the numerator denotes the number of parts expressed by the fraction; as many times as the numerator is repeated, so many times the value of the fraction is increased; and therefore, to multiply the numerator of a fraction by any number, multiplies the fraction by that number; thus, three times is evidently, for the value of the parts remaining the same, three times as many parts are expressed by the latter fraction. Dividing the denominator of a fraction by a whole number also multiplies the fraction by that number: for since the denominator of a fraction expresses the number of parts into which the unit is divided, if the denominator be divided by any number, it will be as many times less, as that number contains units; and hence the value of the parts expressed by the denominator, just so many times greater; and consequently the value of the fraction will be repeated as many times as there are units in the given number. Thus, × 2, for is double, inasmuch as the number of parts expressed is the same, and the value of each part in the first, is double that of each part in the second. NOTE. When a fraction is multiplied by a number equal to its denominator, the numerator becomes a whole number, for in that case the denominator becomes unity: 17 Thus, 53, 11×12 17. To divide a fraction by a whole number. RULE. (21.) Divide the numerator, or multiply the denominator of the fraction by the given number. Demonstration of the Rule.-Since the numerator expresses the value of the fraction, to divide the numerator, while the denominator remains the same, is to divide the fraction; and since the denominator shows the number of parts into which the unit is divided, it is evident, that as many times as the denominator is repeated, just so many times less each part |