13. Required the solid content of a wall 53f. 6' long, 10f. 3' high, and 2f. thick. Ans. 1310f. 9'. 1. VULGAR FRACTIONS, FRACTIONS are expressions for parts of an inte ger or whole. Vulgar Fractions are represented by two numbers, placed one above the other, with a line between them. 2. The number above the line is called the numerator; and that below the line, the denominator. The denominator shows how many parts the integer is divided into; and the numerator shows how many of those parts are contained in the fraction. 3. A proper fraction is one, whose numerator is less than the denominator; as,,, &6. 4. An improper fraction is one, whose numerator exceeds the denominator; as, 110, &c. 5. A single fraction is a simple expression for any number of parts of the integer. 6. A compound fraction is the fraction of a fraction; as 14 of, of, &c. 7. A mixed number is composed of a whole number and a fraction; as 8, 176, &c. NOTE. Any whole number may be expressed like a fraction by writing 1 under it; as 3. 8. The common measure of two or more numbers is that number, which will divide each of them without a remainder. Thus 3 is the common measure of 12 and 15; and the greatest number, that will do this, is called the greatest common measure. 9. A number, which can be measured by two or more numbers, is called their common multiple; and if it be the least number, which can be so measured, it is called their least common multiple; thus 30, 45, 60, and 75, are multiples of 3 and 5; but their least common multiple is 15.* PROBLEM I. To find the greatest common measure of two or more numbers. RULE.t 1. If there be two numbers only, divide the greater by the less, and this divisor by the remainder, and so on till nothing A prime number is that, which can only be measured by an unit. That number, which is produced by multiplying several numbers together, is called a composite number. A perfect number is equal to the sum of all its aliquot parts, The following perfect numbers are taken from the Petersburgh acts, and are all, that are known at present, 6 28 496 8128 33550336 8589869056 137438691328 2305843008139952128 2417851639228158837784576 9903520314282971830448816128 There are several other numbers, which have received different denominations, but they are principally of use in Algebra, and the higher parts of mathematics. + The truth of the rule may be shown from the first example. For since 54 measures 108, it also measures 108+54, of 162. Again, since 54 measures 108, and 162, it also measures 5X162+108, or 918. In the same manner it will be found to measure 2x918+162, or 1998, and so on. Therefore 54 measures both 918 and 1998. remains, always dividing the last divisor by the last remainder; then will the last divisor be the greatest common measure required. 2. When there are more than two numbers, find the greatest common measure of two of them as before; and of that common measure and one of the other numbers; and so on through all the numbers to the last; then will the greatest common measure, last found, be the answer. 3. If 1 happen to be the cammon measure, the given numbers are prime to each other, and found to be incommeasurable. EXAMPLES. 1. Required the greatest common measure of 918, 1998, and 522. Therefore 18 is the answer required. 2. What is the greatest common measure of 612 and 540? Ans. 36. 3. What is the greatest common measure of 720, 336, and 1736? Ans. 8. It is also the greatest common measure; for suppose there be a greater, then since the greater measures 918 and 1998, it also measures the remainder 162; and since it measures 162 and 918, it also measures the ramainder 108; in the same manner it will be found to measure the remainder 54; that is, the greater measures the less, which is absurd. Therefore 54 is the greatest common measure. In the very same manner, the demonstration may be applied to 3 or more numbers. PROBLEM TI. To find the least common multiple of two or more numbers. RULE.* 1. If there be only two numbers, divide their product by their greatest common measure; and the quotient will be their least common multiple. 2. When there are more than two numbers, find the least common multiple of two of them as before; and of that common multiple and one of the other numbers; and so on through all the numbers to the last; then will the least common multiple, last found, be the answer. 3. If the numbers be prime to each other, their product is their least common multiple. EXAMPLES. 1. What is the least common multiple of 3, 5, 8, and 10? *3 15 the least common multiple of 3 and 5. 8 120 the least common multiple of 3, 5, and 8. 10 10)1200(120 the answer. 10)1200(120, hence 10 is the 2. What is the least common multiple of 4 and 6?, Ans. 12. 3. What is the least number, that 3, 4, 8, and 12 will measure? Ans. 24. 4. What is the least number, that can be divided by the nine digits without à remainder? Ans. 2520. REDUCTION OF VULGAR FRACTIONS. Reduction of Vulgar Fractions is the bringing them out of one form into another, in order to prepare them for the operations of addition, subtraction, &c. The truth of this rule may in some measure be seen by an examination of the first example. It may be easily ascertained CASE I. To abbreviate or reduce fractions to their lowest terms. RULE*. Divide the terms of the given fraction by any number, that will divide them without a remainder, and these quotients that 15 is the least number, that can be divided by 3 and 5 without a remainder; and that 120 is the least number, that can be divided by 3, 5, and 8 without a remainder; but this can also be divided by 10 without a remainder; therefore 120 appears to be the least common multiple of 3, 5, 8, and 10. * That dividing both the terms of the fraction equally by any number whatever will give another fraction, equal to the former, is evident. And if those divisions be performed as often as can be done, or the common divisor be the greatest possible, the terms of the resulting fraction must be the least possible. NOTE 1. Any number, ending with an even number or a cypher, is divisible by 2. 2. Any number, ending with 5 or 0, is divisible by 5. 3. If the first place of any number on the right be 0, the whole is divisible by 10. 4. If the first two figures on the right of any number be divisible by 4, the whole is divisible by 4. 5. If the first three figures on the right of any number be divisible by 8, the whole is divisible by 8. 6. If the sum of the digits, constituting any number, be divisible by 3, or 9, the whole is divisible by 3, or 9. 7. All prime numbers, except 2 and 5, have 1, 3, 7, or 9, in the place of units; and all other numbers are composite. 8. When numbers, with the sign of addition or subtraction between them, are to be divided by any number, each of the 4+8+10 numbers must be divided. 2 Thus 2+4+5=11. 9. But if the numbers have the sign of multiplication between them, only one of them must be divided. Thus 3X8X10 2X6 |