SECTION V. DECIMAL FRACTIONS. 195. A Decimal Fraction is a number of tenths, hundredths, thousandths, etc. 5 196. A Decimal Fraction is usually expressed by placing'a point before the numerator and omitting the denominator; thus.5 expresses; .05 expresses 1; .005, etc. 197. The Symbol of a decimal is the period, called the decimal point, or separatrix. It indicates the decimal, and separates decimals and integers. 198. The places at the right of the decimal point are called decimal places. The first place to the right of the de cimal point is tenths, the second place is hundredths, etc. Thus, .245 expresses 2 tenths, 4 hundredths, and 5 thousandths. 199. The method of expressing decimal fractions arises from the method of notation for integers, and is a continuation of it. This beautiful law, as applied to integers and fractions, is exhibited in the following NOTATION AND NUMERATION TABLE. 200. A Decimal is a decimal fraction expressed by the method of decimal notation; as, .5, .25, etc. 2d. 3d. 4th. 5th. Hundredths. Thousandths. Hund.-thousandths. Ten-thousandths. 6th. Millionths. 8th. 9th. Billionths. 201. A Pure Decimal is one which consists of decima. figures only; as, .345. 202. A Mixed Decimal is one which consists of an integer and a decimal; as 4.35. 203. A Complex Decimal is one which contains a com、 mon fraction at the right of the decimal; as, .45}. NOTES.-1. The first treatise upon decimals was written by Stevinus, and published in 1585. 2. The decimal point, Dr. Peacock thinks, was introduced by Napier, the inventor of logarithms, in 1617, though De Morgan says that Richard Witt made as near an approach to it as Napier. EXERCISES IN NUMERATION. 1. Read the decimal .45. SOLUTION. This expresses 4 tenths and 5 hundredths, or since 4 tenths equals 40 hundredths, and 40 hundredths plus 5 hundredths equal 45 hundredths, it may also be real 45 hundredths. Hence the following rules: Rule I-Begin at tenths, and read the terms in order towards the right, giving each term its proper denomination. Rule II.-Read the decimal as a whole number, and give it the denomination of the last term at the right. NOTE. In the second method we may determine the denominator by numerating from the decimal point, and the numerator by numerating towards the decimal point. 1. Express 25 thousandths in the form of a decimal. SOLUTION.-25 thousandths equal 20 thousandths plus 5 thousandths, or 2 hundredths and 5 thousandths; hence we write the 5 in the third or thousandths place, 2 in the second or hundredths place, and fill the varant tenths place with a cipher, and we have .025. Hence the following rules: Rule I.-Place the decimal point, and then write each term so that it may express its proper denomination, using ciphers when necessary. Rule II.- Write the numerator, and ther. place the decimal point so that the right-hand term shall express the denomination of the decimal, using ciphers when necessary. NOTE. To avoid ambiguity, where integers and decimals occur in the same written number, a comma should be placed between them; thus, three hundred and seven ten-thousandths should be written .0307, but three hundred, and seven ten-thousandths, 300.0007. Express the following in decimal form: 2. Twenty-five hundredths. 3. 2 tenths and 8 hundredths. 4. 7 tenths and 9 hundredths. 5. Twenty-five thousandths. 6. 4 tenths and 7 thousandths. 7. Seven tenths and 8 thousandths. 8. Five hundred, and 25 thousandths. 9. Three tenths and 7 ten-thou sandths. 10. Seven hundredths and 9 tenthousandths. 11. Three hundred, and 78 tenthousandths. 12. Five tenths, 6 hundredths, and 7 hundred thousandths. 13. Four hundredths, seven tenthousandths, and 6 hundred-thousandths. 14. Nine hundred and sixty. nine hundred-thousandths. 15. Two tenths and three millionths. 16. Five hundredths, six thousandths, and eight millionths. 17. Thirty-five thousand, and eight millionths. 18. Ninety-three hundred and seven ten-millionths. 19. Eighteen thousand and one hundred-millionths. 20. Two million, and 6 thousand and 9 hundred-millionths. 1. Moving the decimal point one place to the right, multiphes the decimal by 10; two places, multiplies by 100; etc. For, if the point be moved one place to the right, each figure will express ten times as much as before, hence the whole decimal will be ten times as great; etc. 2. Moving the decimal point one place to the left, divides the decimal by 10; two places, divides by 100; etc. For, if the point be moved one place to the left, each figure will express 1 tenth of its previous value, hence the whole decimal will be only 1 tenth as great: eic. 3. Placing a cipher between the decimal point and the decimal, divides the decimal by 10. For, this moves each figure one place to the right in the scale, in which case they express 1 tenth as much as before, and hence the deci-· mal is only 1 tenth as great. 4. Annexing ciphers to the right of a decimal does not change its value. For, each figure retains the same place as before, and hence expresses the same value as before, and consequently the value of the decimal is unchanged. 'REDUCTION OF DECIMALS. 204. There are two cases of the reduction of decimals: 1st. To reduce decimals to common fractions. 2d. To reduce common fractions to decimals CASE I. 205. To reduce a decimal to a common fraction. 1. Reduce .45 to a common fraction. SOLUTION.-.45 expressed in the form of a common fraction is 45 which reduced to its lowest terms equals. Hence the following OPERATION. .45-12%, Ans. Rule. Write the denominator under the decimal, omitting the decimal point, and reduce the common fraction tc its lowest terms. EXAMPLES FOR PRACTICE. Reduce the following decimals to common fractions: 12. Reduce the complex decimal .23 to a common fraction SOLUTION.-.2 is 2 tenths, which, by writing OPERATION. 23 the denominator, becomes which equals or 10' 2종 중 10' .2= 10 10 , which reduced to its lowest terms equals. Therefore, etc. 206. To reduce a common fraction to a decimal. 1. Reduce to a decimal. SOLUTION. equals of 5. 5 equals 50 tenths of 50 tenths is 6 tenths and 2 tenths remaining; 2 nths equal 20 hundredths; of 20 hundredths equals 2 hundredths and 4 hundredths remaining; 4 hundredths equal 40 thousandths; of 40 thousandths is 5 thousandths. Therefore, § equals .625. OPERATION. }=} of 5 =8)5.000 .625 Rule.-I. Annex ciphers to the numerator, and divide by the denominator. II. Point off as many decimal places in the quotient as there are ciphers used. NOTES.-1. In many cases the division will not terminate; the common fraction cannot then be exactly expressed by a decimal. Such decimals are called interminate or infinite decimals. 2. The symbol + annexed to a decimal indicates that it contains other decimal terms. The symbol: annexed to a decimal indicates that the last decimal term is increased by 1. is greater than 5. This is often done when the next term 9. 18. 15 19. Ans. .875. 13. 113 256 Ans. .1875. 14. 5.0. Ans. .947368+. Ans. .44140625. Ans. 5.00125 Ans. 16.414. Ans. 8.502625 Ans. .9375. 17. 7.00. Ans. 7.00015625 |