whether their product, when taken in this order, be the same as at first. This method of proof proceeds on the principle, that the product of two numbers is the same in whatever order they are taken; that 4 multiplied by 3, for example, will give the same product as 3 multiplied by 4. To demonstrate this, let the units in 4 be represented by a horizontal row of physical points, and repeat this row as often as there are units in 3. Then, it is obvious, that the whole number of points will be found, either by adding three fours together, or four threes. But three fours are equal to 4 multiplied by 3, and four threes are equal 3 multiplied by 4. Hence the product of 4 by 3, is equal to the product 3 by 4; and the same may be demonstrated of any two numbers whatever. 26. Multiplication may be also proved by help of Nos. 25 and DIVISION. 28. Division is an expeditious method of finding how many times a less number can be subtracted from a greater; or, which amounts to the same thing, it is that process by which we discover how often the greater contains the less. The less of the two given numbers is called the divisor; the greater, the dividend; and the number of times that the latter contains the former, is denominated the quotient Since the quotient expresses the number of times that the dividend contains the divisor, it is manifest, that the product of the divisor and quotient must be equal to the dividend. Hence division has sometimes been defined, to be the method of finding a number, which, when multiplied by the less of two given numbers, shall produce the greater. 7 28 21 When the quotient, or number of times that the 35 dividend contains the divisor is not very great, it may be obtained by actually performing the successive subtractions. Thus, let it be required to find how often 7 is contained in 35. Here by a process of continued subtraction, we find that 7 may be taken away 5 times from 35, and that after 5 abstractions there is no remainder. Hence we infer that 35 contains 7, 5 times. 7 14 7 7 7 29. This method of finding the quotient by continued subtraction, though possible in every instance, would evidently be often very tedious and inconvenient; we have therefore recourse to another, by which the quotient may be determined in all cases, with great facility. And here it will be most convenient to conceive the quotient as that number, which, when multiplied by the divisor, will produce the dividend. Ex. 1. Let it be required to find how often 2 is 2)864 contained in 864. 432 Here we readily perceive that 800 contains 2, 400 times, because 400 multiplied by 2 are equal to 800. In the same manner 60 contains 2, 30 times, and 4 contains 2, 2 times. Hence 2 is contained 400 plus 30 plus 2 times, or 432 times in 864. 30. A simpler view might perhaps be taken of division than the preceding; for it must appear obvious to every one, that the quotient of a number divided by 2 will be the half of that number; or, if divided by 3, the third part of that number, and so on; and it is as evident, that to obtain the half, the third, or fourth part, &c. of any number, we have merely to take the half, the third, or fourth part, &c. of every collection of its units. Hence, in the preceding example, we shall have for the quotient the half of 8 hundreds, or 8 units of the third degree, which are 4, plus the half of 6 tens, or 6 units of the second degree, which are 3, plus the half of 4 units of the first degree, which are 2. Ex. 2. How often is 5 contained in 9675 ? 9 units of the fourth order, are equal to 5 plus 4, or 5 of the fourth order, and 40 of the third; these 40, together with the 6, make 46 of the third, which are equal to 45 of the third, and 10 of the second; these 10, together with the 7, are equal to 17 of the second, or 15 of the second and 20 of the first; which, together with the 5, make 25 of the first. Hence it appears, that the given number, 9675, is expressed by 5 units of the fourth order, 45 of the third, 15 of the second, and 25 of the first, thus :And by taking the fifth part of each of these collections, we obtain for the quotient 1935. This decomposition may be effected while the division is going on, by observing that the denomination of each remainder is ten times as great as that of the succeeding unit, thus: 5451525 The fifth part of 9 is 1, leaving a remainder 5)9675 of 4; this 4 being 40 of the succeeding units, will, 1935 with 6, make 46, the fifth part of which is 9, leaving a remainder of 1, which, in like manner with the 7, will make 17, the fifth part of which is 3, leaving a remainder of 2; the 2 being 20 units of the succeeding order, will, with 5, make 25, the fifth part of which is 5. This is the plan which is adopted in practice, when the divisor does not exceed 12. When the divisor is considerably great, it will be necessary that the whole numerical process appear, with the view of disburdening the memory, and insuring an accurate result. Let it be required, to find, for example, how often 35 is contained in 25760. 35)25760(7 hundreds 245 210(6 units 210 From a comparison of the divisor with the highest order of units in the dividend, we see that there can be no unit either of the fifth or fourth order in the quotient. Our next inquiry then is, what is the 35th part of 257 units of the third order, that is, of 257 hundreds; this we perceive to be 7 hundreds, which, when multiplied by 35, and the product subtracted from 257, leaves a remainder of 12 hundreds, or 120 tens; these 120 tens, together with the 6 tens, make 126 tens, the 35th part of which is 3 tens, which, when multiplied in like manner by 35, and the product subtracted from 126 tens, leaves a remainder of 21 tens, or 210 units, the 35th part of which is exactly 6 units. Thus it appears, that the 35th part of the given number, 25760, is 7 hundreds, 3 tens, and 6 units, or 736; but the 35th part of the dividend (according to what is stated at the beginning of the present number) expresses the number of times that it contains the divisor; hence 25760 contains 35, 736 times. Although the quotient in the preceding example has been obtained analytically, yet it is manifest that the mere mechanical process differs in nothing from that of the common rule the reason of which must here appear to every one. This might also be made to appear in the following manner-Suppose we wish to ascertain the quotient of 5052925 divided by 563. 563)5052925(8øøø the first partial quotient. 4504000 99p the second. 548928 70 the third. 506700 5 the fourth. 42228 8975 the sum of the partial quotients, or the whole quotient. 39418 2815 2815 If, in the above example, the ciphers be suppressed, which merely serve to indicate the rank of the digits, and one figure of the dividend be annexed to the remainder after each division, we arrive again at the common process, which is, as we now see, established on sound principle. 31. In Division, there are several abbreviations, &c. similar to those stated under Multiplication, (Nos. 22, 23, and 25,) which it would be superfluous here to exemplify. 32. Division may be proved, by observing whether the product of the divisor and quotient, added to the remainder, if there be any, be equal to the dividend. FRACTIONS. 33. A fraction is a quantity which represents a part or parts of a unit or whole. If a unit be divided into two equal parts, each of these is called a half of a unit or whole; if into three equal parts, each of them is called a third, &c. 34. A Fraction is expressed by two terms, called the Numerator and Denominator. The numerator shows of how many equal parts the fraction is composed, and the denominator indicates the number of these equal parts into which the unit is divided. Thus, for instance, if a unit be divided into seven equal parts, and five of these be taken, then these fivesevenths are expressed thus, ; the numerator being always placed above the denominator. 35. A fraction may be considered as the quotient arising from the division of the numerator by the denominator. Thus, for example, the value of the fraction is the same, whether we conceive it as denoting the fifth part of four units, or fourfifths of one unit. This may be easily shown by the help of a line. Let each of the lines AB, BC, CD, and DE, represent a unit, then will AE be four units. Divide each of these units into five equal parts; and let AF be four of these, then is AF of AB, that is, AF is of a unit. Now, since each of the five units is divided into five equal parts; the whole line AE is divided into twenty equal parts; AF, which contains four of these, is therefore the fifth part of AE, that is, of four times AB. Hence, it appears, that the fifth part of four times AB, is of the same value as four-fifths of once AB. 36. A proper fraction is one whose numerator is less than its denominator; as . 37. An improper fraction is one whose numerator is either. equal to, or greater than its denominator; as or 1. 38. A compound fraction is a fraction of a fraction; as of 3. 39. A mixed number is that which consists of a whole, or integer number, and a fraction; as 53. From the preceding definitions, the following deductions are easily made. 40. (1.) That a fraction is equal to, less or greater, than unity, according as its numerator is equal to, less or greater, than its denominator. 41. (2.) That to multiply a fraction by any number, multiply the numerator by that number, and retain the same denominator; or divide the denominator, and retain the same numerator. 42. (3.) That to divide a fraction by any number, divide the numerator by that number, and retain the same denominator; or multiply the denominator, and retain the same nu merator. Hence it follows, that if the numerator and denominator of a fraction, be either both multiplied, or both divided by the same number, the value of the fraction will remain unchanged. 43. From the preceding, it is obvious, that the same fractional value may be expressed in an endless variety of ways. It hence becomes an important object to ascertain what the lowest or least terms are, in which it can be expressed. These may be attained, by dividing the terms of the fraction continually by some small number that will divide both, until they are not any farther divisible; or by dividing them at once by their greatest common measure, that is, by the greatest number that will divide both without a remainder. 44. The method of finding the greatest common measure of two numbers depends on the two following obvious prin For an explanation of the signs +, -, X, -,=, see the beginning of Algebra. |