874. To find the Excess of 9's in a Number. 1. Let it be required to find the excess of 9's in 7548467. Beginning at the left hand, add the figures together, and as soon as the sum is 9 or more, reject 9 and add the remainder to the next figure, and so on. Adding 7 to 5, the sum is 12. Rejecting 9 from 12, leaves 3; and 3 added to 4 are 7, and 8 are 15. Rejecting 9 from 15, leaves 6; and 6 added to 4 are 10. Rejecting 9 from 10, leaves 1; and 1 added to 6 are 7, and 7 are 14. Finally, rejecting 9 from 14 leaves 5, the excess required. 875. Hence we derive this property of the number 9: Any number divided by 9 will leave the same remainder as the sum of its digits divided by 9. NOTES.-1. It will be observed that the excess of 9's in any two digits is always equal to the sum, or the excess in the sum, of those digits. Thus, in 15 the excess is 6, and 1+5: 6; so in 51 it is 6, and 5+1 = 6. 2. The operation of finding the excess of 9's in a number is called casting out the 9's. 2. What is the product of 746 multiplied by 475 ? 876. To prove Multiplication by Excess of 9's. Find the excess of 9's in each factor separately; then multiply these excesses together, and reject the 9's from the result; if this excess agrees with the excess of 9's in the answer, the work is right. NOTE.-The preceding is not a necessary but an incidental property of the number 9. It arises from the law of increase in the decimal notation. If the radix of the system were 8, it would belong to 7; if the radix were 12, it would belong to 11; and universally, it belongs to the number that is one less than the radix of the system of notation. CIRCULATING DECIMALS. 877. A Circulating Decimal is one in which the same figure or set of figures is continually repeated in the same order.. 878. In reducing common fractions to decimals, we find that in one class of examples the division is complete, and the quotient is an exact decimal. In another class, the same figure or set of figures is repeated again and again, and the division will not terminate, though continued indefinitely. The former are Terminate Decimals, the latter are Circulating Decimals, and the figure or figures repeated the Repetend. = Thus, = .5, = .297297+, are interminate. 879. By inspection we see that the denominators of the first class are the prime numbers 2 or 5, or are composed of the factors 2 or 5. In the second class, the denominators contain other prime factors than 2 and 5. 880. To find whether a common fraction will produce a terminate or a circulating decimal, Resolve the denominator into its prime factors. If it contains no other factors than 2 and 5, the quotient will be a terminate decimal. If it contains any other prime factors than 2 and 5, the quotient will be a circulating decimal. 881. Circulating decimals are expressed by writing the repetend once, and placing a dot over the first and last figure. Thus, the repetends .33333+ and .297297+ are written .3 and 297. 882. When the repetend begins with tenths, the called a Pure Repetend; as, .4; .297. 883. When the repetend is preceded by one or more decimal figures, it is called a Mixed Repetend; as .27; .4263i. NOTE.-The decimal figures before the repetend are called the finite part of the decimal; as 2 and 42 in the mixed repetends above. 884. Change the following fractions to terminate or circu lating decimals, and mark the repetends in each. (Art. 249.) 885. To Reduce a Pure Repetend to a Common Fraction. RULE. Take the repetend for the numerator, and make the denominator as many 9's as there are figures in the repetend. Reduce the fraction thus produced to its lowest terms. 886. To Reduce a Mixed Repetend to a Common Fraction. 14. Reduce 0.227 to a common fraction. SOLUTION.—Subtracting the finite part from the given mixed repetend, both regarded as integers, we have for the numerator 225, and for the denominator 990, that is, as many 9's as there are figures in the repetend with as many ciphers annexed as there are figures in the finite part, and the result is = = OPERATION 227 2 Given decimal. Finite part. 225 Numerator. 990 Denominator. 22, Ans. Hence, the RULE.-For the numerator, subtract the finite part from the given mixed repetend, both regarded as integers; and for the denominator, take as many 9's as there are figures in the repetend, with as many ciphers annexed as there are figures in the finite part. EXPLANATION.-Since the repetend has two figures, if we multiply the given mixed repetend by 100, and from the product subtract once the given mixed repetend, the remainder (22.5) will be 99 times the given mixed repetend; and once the mixed repetend 22.5 = 25. But 225 is the difference between 227 the given mixed repetend, regarded as an integer, and 2 the finite part of it, regarded as an integer, and 335 99 990. = = OPERATION. 0.227 × 100 = 22.7272 0.227 × 1 == .2272 0.227 × 99 = 22.5 0.227 = 28.5 45 +36= 1/8 = 22, Ans. , the same as before. 887. Circulating decimals, when reduced to common fractions, may be added, subtracted, multiplied, and divided, like other common fractions. 23. What is the sum of .5925+.4745 + .0227 ? 24. What is the difference between .6435 and .4158? SURVEYOR'S MEASURE. 888. Surveyor's Measure is used in measuring land, in laying out roads, establishing boundaries, etc. 889. The Linear Unit usually employed by surveyors is Gunter's Chain, which is 4 rods or 66 feet long, and contains 100 links. It is subdivided as in the following NOTES.-1. Gunter's chain is so called from the name of its inventor. Measurements by it are usually given in chains and hundredths of a chain. 2. In measuring roads, etc., engineers use a chain, or measuring tape, 100 feet long, each foot being divided into tenths and hundredths. 3. The mile of the table is the common land mile, which contains 5280 feet. |