Page:Encyclopædia Britannica, Ninth Edition, v. 2.djvu/592

Rh 532 ARITHMETIC 51292 50867 dividend having five decimal places and the divisor one, the quotient must have four; a cipher is therefore prefixed to the three figures. In cases like the third example, where the dividend has fewer decimal places than the divisor, the number has to be made up in the former by ciphers, and as far as the end of the ciphers making up that number the quotient must be an integer. The quotient may be pointed at any stage after as many decimal places of the dividend have been made use of as there are in the divisor, the first figure after this being always the first decimal. This is especially to be attended to when the division does not terminate, or when a few figures only are required ; thus, 6 3 9 4 divided by 237 to two decimal places gives 269-79 ; 10 divided by 264 gives 37 87. The method of converting a vulgar fraction into a deci mal, given in 22, is in effect the division of the nume rator by the denominator, the result being expressed as a decimal. Thus, T 8 ^ = 83 -r 160 = 51875. When the quotient is re quired to a given number of &amp;lt; 635S4)272 182(428 067 places only (as three in the 254336 example), the operation may 178460 be shortened by dropping the 1271 68 last figure of the divisor at each successive multiplica tion. But this must not be commenced till the figures 425 required in the quotient are fewer than the figures in the 44 divisor, and the carriage from 44 the dropped figure is to be added in each instance. 26. We have seen that the division by means of which vulgar fractions are converted into decimals ( 22, 25) will, in certain cases, always leave a remainder. If the fraction be in its lowest terms, there must always be a remainder whenever the denominator contains any other prime number as a factor besides 2 and 5. For in adding ciphers we multiply by tens, and we introduce no other factor. It often happens that we can speedily discover what the decimal must be, so as to be able to set down any number of figures without further actual division. Unless the division terminate, the same figures will recur sooner or later ; and the same figures must recur before we use as many ciphers as there are units in the divisor. Sup posing the divisor is 17, we can only have the numbers from 1 to 1 6 as remainders ; the quotient must therefore repeat itself after all these remainders occur, if not earlier. All do occur in dividing by 1 7 ; thus, ^=1176470588235294. The fraction yV must always be greater than the decimal 41666...., however far the latter be extended; but we can make the difference as small as we choose. Thus 416666 does not differ from y^- by the millionth part of a unit. The form of the decimal 416 is used to show that the 6 is to be considered as repeated continually ; and this being understood, we can say that the decimal is equal to^. Decimals of this kind are called Recurring Decimals. They are sometimes distinguished as Repeating or Circu lating Decimals, according as one figure or more than one recur ; and as Pure or Mixed, according as the recurring figures stand alone or are preceded by non-recurring deci mals. Thus, 148 is a pure circulating decimal; 183 is a mixed repeating one. 27. To reduce a recurring decimal to a vulgar fraction, subtract the decimal figures that do not recur from the whole decimal ; set down the remainder as the numerator of the fraction, and as many nines as there are recurring figures, followed by as many ciphers as there are non recurring figures, as the denominator. The reason of proceeding in this way will appear if, taking any mixed recurring decimal, we multiply it by as many tens as there are decimal places, and again by as many tens as there are non-recurring decimals, and sub tract the second product from the first. Take, e.g., 79054 : 100000 times -79054 = 79054-054054054 .... and 100 times 79054= 79-054054054 therefore 99900 times 79054 = 78975 whence 79054 = jf |, i.e., ^. In the case of pure recurring decimals, we have no sub traction, and the denominator consists entirely of nines. Thus -3 = = ; -2T = *% = .. 28. In practical arithmetic recurring decimals are little used, except in approximations. They can be added or sub tracted readily to any number of places by extending all a place or two beyond what is required. When we have to multiply or divide by recurring decimals, we must con vert them into vulgar fractions. They may themselves be multiplied or divided by integers or ordinary decimals, though in this case, too, it is often better to use vulgar fractions, especially when exact results are desired. Poiuers and Roots of Numbers. 29. When a given Port-is number is multiplied by itself, the product multiplied again and rod by the number, and so on, the result obtained is called the second, third, fourth, &c., power of the number, according as it is the product of the number repeated two, three, four, &amp;lt;tc., times. Thus, 7x7x7 = 343 is the third power of 7; 5x5x5x5 = 625, the fourth power of 5. The term &quot; square &quot; is nearly always used instead of &quot; second power &quot; (e.g., 81 is the square of 9), and &quot; cube&quot; frequently instead of &quot; third power.&quot; The power to which a number is raised is indicated by a small figure written over the number to the right ; thus, 8 5 is the fifth power of 8. The square root of a given number is the number which, when multiplied by itself, produces the given num ber. And so, in general, whatever power one number is of another, the same root is the second of the first. Thus, 7 is the third root, or cube root, of 343 ; 5 is the fourth root of 625 ; 2 is the fifth root of 32. The sign ,J (which is really an r, from radix, a root) prefixed to a number indicates a root of it. The simple sign stands for the square root ; a figure is placed over it to denote other roots. Thus, ^100 is the square root of 100; J/256, the fourth root of 256. 30. To extract the square root of a given number, divide Extract! it into periods of two figures, by putting a point over every of RC l uaj second figure, commencing with that in the units place ; root&amp;lt; set down as the first figure of the root, the largest number whose square does not exceed the first or left-hand period ; place the square of this number under the first period, and subtract it from it ; to the remainder annex the next period; place before 582169(763 this as a trial divisor twice the root 4.9 figure; consider how often the former (omitting the right-hand figure) con- 146) Jll tains the latter, and set down the number that expresses this as the next 1523)4569 figure of the root; place also this root- 4569 figure to the right of the trial divisor; multiply by it the divisor thus completed; subtract the product from the number formed of the former remainder and the period taken down ; add another period to the remainder now found ; then double the whole root for a new trial divisor, and proceed as before. The 58 of the example being 580000, the 49 must bo 490000, and the root 700. So the 146 is 1460; the 921,