(c) Let x, y, and z be each prime to w: to prove that x × yxz is prime to w. Let x and x" be the prime factors of x; y', y', and y'"', the prime factors of y; and z', z'', z''', and z'''', the prime factors of x X y X 人 W 2. We then have xxyXz=x'x x'' Xy'Xy" xy"" X z' X 2" X 2" X 2"""' ; and it will be seen that, when written under this latter form, the product of x, y, and z is resolved into its prime factors-because [a] resolved into factors every one of which is a prime number. Now, as x is prime to w, neither x' nor x" is a factor of w. Again: as y is prime to w, no one of the prime factors (y', y'', y') of y is a factor of w. And as is prime to w, no one of the prime factors (z', z'', z''', z'''') of z is a factor of w. As, therefore, xxyXz and w are so related to each other that no one of the prime factors of the former number is a factor of the latter, xXyXz is [b] prime to w. 14 X 18 X 156 Perhaps this demonstration will be better understood if we substitute 14, 18, and 156 for x, y, and z, respectively; and 55 for w. We shall then have for x' and x'', 2 and 7, respectively; for y' y", and y''', 2, 3, and 3, respectively; and for z', z', 2X7X2×3×3 × 2 × 2 × 3 × 13 z''', and '''', 2, 2, 3, and 13, respectively. (II.) The second fact is easily established. Let x and y be divided by a common measure, m, and let the resulting quotients, x' and y', be prime to one another: to prove that mis the greatest common measure of x and y. 5XII 55 y am)x y x' y' y" x=amx" y=amy" m)x x=mx' y=my' mx'=amx''; my'=amy'' Now, m, if not the greatest common measure, must (Note, p. 109) be a measure of the greatest, which, consequently, must be a multiple of m. Let the greatest common measure of x and y be am, if possible, and let the division of x and y by am give the quotients " and y', respectively. Then, mx' and amx-being each equal to x-are equal to one another; my' and amy" also -being each equal to y—are equal to one another. Dividing by m, therefore, we have x'=ax", and y'=ay"; from which equality it appears that x' and y' are both measured by a. But this is impossible, x' and y' being prime to each other. So that x and y can have no greater common measure than m, which, consequently, is the greatest common measure. 4)24 36 6 9 As a practical illustration, let us take the numbers 24 and 36. When we divide these numbers by 4, one of their common measures, we find that the resulting quotients, 6 and 9, are not prime to one another, for which reason we conclude that 4 is not the greatest common measure of 24 and 36. When, however, we divide by 12, we find that the resulting quotients, 2 and 3, are prime to one another, for which reason we conclude that 12 is the greatest common measure of 24 and 36. a) w 12)24 36 2 3 x y 2 b)w' x' y' Z c)w' x'' y'' z' w' x'" y'"' z' Let us now take the four numbers w, x, y, and z, and suppose that the first three-but not the whole four-have a common measure. Writing the given numbers in a horizontal row, let us divide the first three (w, x, and y) by a common measure, a, and set down, in a second horizontal row, the resulting quotients (w', x', and y'), and also the undivided number, z. Of the numbers in the second row, let us suppose that the last threebut not the whole four-have a common measure. Let us divide the last three numbers (x', y', and 2) by a common measure, b, and set down, in a third horizontal row, along with the undivided number, w',-the resulting quotients, x, y, and z'. Of the numbers in the third row, let us suppose that not more than two-the middle twohave a common measure. Let us divide the middle pair of numbers (x" and y') by a common measure, c, and write the resulting quotients (x"" and y'"), as well as the undivided numbers, w' and z', in a fourth horizontal row. Of the numbers in the fourth row, let us suppose that every two are prime to one another; and the least common multiple of w, x, y, and z will be found to be aXbXcXw" xx" Xy"" Xz. Expressing the given numbers in terms of the divisors and the numbers in the last horizontal row, we have Now, if we can show (I.) that the least common multiple of w and x-or of axw' and axbxcxx""-is axbXcXw' Xx""'; (II.) that the least common multiple of axbXcxw'xx"" and y-or of axbxcxw' xx" and axbxcxy""-is axbXcX w'Xx""Xy""; and (III.) that the least common multiple of axbxcxwxx'"'Xy'" and z—or of a XbXcXw' ×x''' ×y'"' and bxz'-isa xbx cxw' × x'"' Xy""Xz', it will follow from what has already been established (§ 101) that axbXcxw'x''' Xy'"'Xz' is the least common multiple of w, x, y, and z. (I.) The greatest common measure of w and x—or of aל' and axbxcxx""-is a. For, when we divide w and x-or a Xw' and axbXcxx""-by a, we find the resulting quotients, w' and bxcxx", to be prime to each other. Thus, w' is prime to b, because otherwise w' and b would have a common measure, which measuring b-would measure x', y', and z (multiples of b) as well as w'; so that w', x', y', and z would have a common measure, whereas it has been assumed that not more than three of these numbers (which occur in the second horizontal row) have a common measure. Again: w' is prime to c, because otherwise w' and c would have a common measure, whichmeasuring c-would measure x" and y" (multiples of c), as well as w; so that u', x", and y" would have a common measure, whereas it has been assumed that not more than two of these numbers (which occur in the third horizontal row) have a common measure. Lastly: w' is prime to x", both numbers occurring in the last horizontal row. So that w-being prime to b, to c, and to "is (p. 117, I.) prime to bxcxx". As, therefore, the division of w and x by a gives quotient which are prime to each other, a is (p. 117, ÍI.) the greatest common measure of w and x. Consequently, the least common multiple of w and x-or of a xw' and a xbx cxx""—is (a × w') × (a × 9 × cxx")÷ a=axbxcxw'xx'". (II.) The greatest common measure of axbxcxw' × x'"' and y-or of axbxcxw'×x"" and a ×b×cxy""—is axbxc. For, when we divide the two numbers under consideration by axb xc, we find the resulting quotients, wxx'"' and y'"', to be prime to each other. Thus, as w', x'", and y'"' all occur in the last horizontal row, y'" is prime to w', and to "", and therefore to w' x x'". Consequently, the least common multiple of axb xc x w' xx'" and y, or of axbxcxwxx" and axbx cxy"",-in other words, the least common multiple ofw, x, and y,—is (axbxcx u' × x''') × (a xbxcxy"")÷axbxc=axbxcxw' ×x"" ×y'". (III.) The greatest common measure of axbxcxw' × x'"'× y" and z―or of axb× cxw' ×x'" × y''' and b×z'—is b. For, dividing the two numbers under consideration by b, we find the resulting quotients, a xcxw' × x'"' × y'"' and z', to be prime to each other. Thus, z' is prime to a, because otherwise z' and a would have a common measure, which—measuring a—would measure w, x, and y (multiples of a), as well as z-a multiple of z'; so that w, x, y, and z would have a common measure, whereas it has been assumed that not more than three of these numbers (which occur in the first horizontal row) have a com α mon measure. Again: z' is prime to c, because otherwise z' and c would have a common measure, which-measuring cwould measure x" and y' (multiples of c), as well as z'; so that x'', y'', and z' would have a common measure, whereas it has been assumed that not more than two of these numbers (which occur in the third horizontal row) have a common measure. Lastly: z' is prime to w', to x'", and to y"",-the whole four numbers occurring in the last horizontal row. So that z'being prime to a, to c, to w', to x'", and to y'"-is prime to axcxw' xx""xy"". As, therefore, the division of a xbx cxw' xxxy"" and z-or of axbxcxw' xx"" xy'" and bxz'-by b gives quotients which are prime to each other, b is the greatest common measure of axbx cxw'xx'"'xy"" and z. Consequently, the least common multiple of axbxcxw' x''' xy'"' and z, or of a xbx c × w' ×x"" × y'" and bx z',-in other words, the least common multiple of w, z, y, and z,—is (a xbx c × w' × x""xy""')x(bxz')÷b=axbxcxwxx""Xy"" xz': FRACTIONAL NUMBERS, OR FRACTIONS. 103. If a unit like one of those which we have hitherto been considering an INTEGRAL unit, let us now say, for the sake of distinction - were divided into any number of equal parts, the parts, regarded as such, would be FRACTIONAL UNITS. 104. The denomination of a fractional unit is expressed by one of that class of words called "ordinals,"*-the corresponding "cardinal" indicating how many such units there are in an integral unit. Thus, if an integral unit were divided into * There is one exception: instead of "second," we say half. 2 3 4 equal parts, each part-a "fractional unit" -would be a &c. So that a school-boy, after eating I fifth of a cake, would have 4 fifths remaining-a cake being divisible into 5 fifths; a farmer, after parting with 3 eighths of his farm, would have 5 eighths remaining—a farm being divisible into 8 eighths; and a draper, after selling 5 twelfths of a piece of cloth, would have 7 twelfths remaining the number of twelfths into which a piece of cloth is divisible being 12. 105. Two or more fractional units of the same denomination that is, two or more "halves," or two or more "thirds," or two or more "fourths," &c. (as the case may be)-constitute a FRACTIONAL NUMBER, or a FRACTION. "Three yards," "threeshillings," "three fifths," "three eighths:" in each of these instances, the number is "three," whilst the unit is-in the first case, a 66 yard;" in the second, a "shilling;" in the third, a "fifth ;" and in the fourth, an "eighth." The first two numbers are "integral;" the last two, "fractional." In practice, we cannot avoid regarding a fractional unit as a fractional "number;" just as we cannot avoid regarding an integral unit as an integral number. For the present, however, it will be well to bear in mind the distinction between a fractional "unit" and a fractional "number." Three fifths (3), four sevenths (4), five ninths (5), &c. are fractional "numbers;" whilst one fifth (3), one seventh (4), one ninth (†), &c. are fractional units.' 106. In writing a fraction,† we employ two (integral) numbers, which are placed one below the other, and separated by a horizontal line. Of these two numbers-which we speak of as the "terms" of the fraction-the upper, called the numerator, indicates how many fractional units the fraction is composed of; whilst the lower number, called the denominator, indicates how many such units there *Fourths are often called quarters. †That is, a simple fraction. (See pp. 138-9.) |