division is performed by means of three factors, and there are three remainders. Here let the first dividend be considered as 7951 units, the second as 1987 fours, the third as 496 sixteens; then there are 4 sixteens, 3 fours, and 3 units over, making together 79. Hence the rule when the number of factors is more than two: Multiply each remainder by all the previous divisors and take the sum of these products, adding in the first remainder. As multiplication and division are contrary operations, either of them furnishes a proof of the other. Thus, to prove an example in multiplication: Divide the product by the multiplier, and the quotient should be the multiplicand. To prove an example in division: Multiply the quotient by the divisor, add the remainder, to the product, and the result should be the dividend. 6. 5,843721÷290; 632,087631+43100. 7. 31,973256÷4720; 8,124970-5673. 8. Divide 827579 by 108, and also successively by 9 and by 12, and show that the remainder and quotient are the same in each case. 9. Similarly divide 932854 by 63 and by 7 and 9. 10. Similarly divide 8,437985 by 132 and by 11 and 12. 11. Similarly divide 2,147952 by 168 and by 6, 4 and 7. 12. Similarly divide 85,437894 by 480 and by 10, 12 and 4. 13. Multiply 53729 by 229 and prove by division. The processes of addition, subtraction, multiplication, and division have now been sufficiently explained, and the pupil may be supposed able to work out any examples of them. But there are some considerations connected with multiplication and division which naturally come in at this part of the subject, and which it is especially important to notice, because they point out the connection between these first elements of arithmetic and the higher parts of the subject to be considered presently. Let the instance already mentioned of 59 divided by 8 be farther considered. The question was then put in the form 'How many times is 8 contained in 59?' and the answer was, 'It is contained 7 times, and there is besides a remainder 3 left over.' It is certain, then, that here, out of the given number 59, 56 has been divided by 8, and 3 has been left undivided. There is evidently in this something incomplete. Next suppose that the question were put in another form: 'What number would, when multiplied by 8, produce 59?' The answer to this cannot be more exact at present than 'A number greater than 7 and less than 8; and such a number we have no means of expressing.' The same incompleteness is found here, and it must be considered whether this arises from some actual impossibility in the question itself, or some imperfection in our mode of expressing results. This doubt is at once resolved by a reference to facts. It is evidently possible that 59 pounds of bread, or 59 gallons of ale, should be divided equally among eight people, and one share multiplied by eight, must be equal to the 59 pounds or 59 gallons. The difficulty, therefore, merely resolves itself into this, that we have, at present, no means of expressing the value of such a share; and consequently we must, if possible, extend our powers of expression. Now all the considerations that have been stated point to one conclusion, namely, that 59 divided by 8 is greater than 7 by a quantity that expresses the unknown result of dividing 3 by 8. We may therefore say that 59 divided by 8, is equal to 7 together with 3 divided by 8, or expressing the result with signs instead of words, 59÷8= 7+(3÷8); or, more shortly, 59÷8=7+3; or, the sign + being left out, 59+8=73. Here it must be distinctly understood, that the expression is put for the result of a process that we have no simpler means of expressing, but which result has a certain definite value. Such an expres sion is called a fraction, the upper number being called the numerator, and the lower the denominator. 53 53 The fractions,,, are read 'one-half,' 'one-fourth,' or one-quarter,' and 'three-fourths,' or 'three-quarters.' All other fractions may be read in two different ways, either by using the word 'by' to signify division, as when 3, 5, are called '39 by 81,' '53 by 97,' or by using the ordinal number expressed by the denominator, as when 3, 54, are called 'thirty-nine eighty-firsts,' fifty-three ninety-sevenths.' The latter form is almost exclusively used in the case of fractions with small denominators, as 'three-fifths,' 'foursevenths,' and the reasons which led to its adoption will be explained in a future chapter. 97 Some knowledge of the nature and properties of fractions is absolutely necessary before the subject of Concrete Numbers can be considered, and what is sufficient for that purpose will be briefly explained here, postponing the full consideration of fractions to an after part of the book. 72 35 means the result of a division, which result cannot be expressed in our system of numbers. In this division 35 is the dividend, 72 is the divisor, and the value of the fraction 35 is the quotient. And the laws that apply to dividends, divisors, and quotients, generally, must be the same whether the result is capable of expression as a whole number or not. Now, in the case of 84, which we know is equal to 14, we find the following facts, from which, and similar instances, certain laws may be deduced. 8414; 16828, therefore multiplying the dividend multiplies the quotient. 8414; 8=7, therefore multiplying the divisor divides the quotient. 84=14; 42=7, therefore dividing the dividend divides the quotient. 8=14; 8=28, therefore dividing the divisor multiplies the quotient. Consequently 84x2= either 168 or 84; 84+2 either or 42. Hence, in the case of 35, the same laws must apply, and 3x6 either 2 or 35, 210 +5=either or 35 360 Expressing these rules generally, we have the following rules: To multiply a fraction: Either multiply the numerator, or divide the denominator. To divide a fraction: Either multiply the denominator, or divide the numerator. Consequently, if both the numerator and denominator of a fraction are multiplied or divided by the same number, the value of the fraction will be unaltered, Thus 3=18-50-25-275-55. It is evidently convenient, however, to express fractions in a form involving as low numbers as possible. Since the order of factors in multiplication is immaterial, that is to say, since 8 × 11=11x8, it follows that 12×5= 5×1, and this latter must therefore be equal to §. Therefore 5, when multiplied by 1, gives as product, that is to say, the quantity which would result from multiplying 5 by 12, and dividing the product by 37. Hence to multiply by a fraction, the rule is, Multiply by the numerator, and divide by the denominator. Thus, As Multiplication and Division are contrary operations, it follows at once, from the above, that to divide by a fraction, the rule is, Multiply by the denominator, and divide by the numerator. Thus 10685340=1780. It is sometimes required, in the Arithmetic of Concrete Numbers, to determine the quotient when a number, consisting of a whole number and a fraction, is divided by a whole number. As an example, let it be required to divide 8 by 12. Now 8 is the sum of 8+, and 8 is evidently equal to 40. Hence 8+, and 83+12=4+ +12, by the rule previously given for dividing a fraction. And the result, 4, may be more shortly expressed as A number consisting of a whole number and a fraction, is called a mixed number, and the fraction to which it can be reduced must always have the numerator greater than the denominator. Such a fraction is termed an improper fraction. Mixed numbers, consequently, can be reduced to improper fractions, and improper fractions to mixed numbers. Thus or, the intermediate step being omitted, when the reason of the process is understood Hence the general rule for dividing a mixed number by a whole number will be, Reduce the mixed number to an improper fraction, and divide by the whole number. It is better, though not absolutely necessary, to express the fraction in the result with as small a numerator and denominator as possible. 8+15-15--19 73÷12=59+12=58. 113÷6=56÷6=56-138-113. Examples will now be given illustrating the various points that have been explained, namely— Division of one number by another, the complete quotient being generally expressed as a whole number and a fraction, as 1937÷32= 6017; 26853÷48=55921=5597 Multiplication of a fraction or mixed number, as, × 8=24=39; 43 12=5; × 16-80-224-29; 108 × 14-1520-1520. |