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195. A Decimal Fraction is a number of tenths, hun dredths, thousandths, etc.
196. A Decimal Fraction is usually expressed by plac Ing a point before the numerator and omitting the denomina zor; thus.5 expresses Ø ; .05 expresses Too; .005 doo, etc.
197. The Symbol of a decimal is the period, called tbe 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 contin na tion of it. This beautiful law, as applied to integers and fractions, is exhibited in the following
Decimals 200. A Decimal is a decimal fraction expressed by tbe metbod of decimal notation; as, .5, .25, etc.
201. A Pure Decimal is one which consists of decimal 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).
NOTE8.-1. The first treatise upou decimals was written by Stevinus, and published in 1585.
2. The decimal point, Dr. Peacock thinks, was introduced by Napier, the Inventor of logarithnus, ip 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 read 45 hundredths. Hence the following rules :
Rule 1.—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 Qumerating from the decimal point, and the numerator by numerating towards the decimal point.
Read the following decimals:
13. 8.75006 4. .84 9. .7605
14. 43.30027 5. .031 10. .0576
15. 756.00279 6. .162 11. .00312
EXERCISES IN NOTATION. 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 recant 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.
Rale II.— Write the numerator, and then place the decrmal point so that the right-hand term shall express the denom ination 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 bundred, and seven ten-thousandths, 300.0007.
Express the following in decimal form: 9. Twenty-five hundredths. 13. Four hundredths, seven ten 3. 2 tenths and 8 hundredths. thousandths, and 6 hundred-thou. 4. 7 tenths and 9 hundredths. sandths. 5. Twenty-five thousandths. 14. Nine hundred and sixty. 6. 4 tenths and 7 thousandths. nine hundred-thousandths.,
7. Seven tenths and 8 thou- 15. Two tenths and three mil. sandths.
lionths. 8. Five hundred, and 25 thou- 16. Five hundredths, six tnousandths.
sandths, and eight millionths. 9. Three tenths and 7 ten-thou- 17. Thirty-five thousand, and sandths.
eight millionths. 10. Seved hundredths and 9 ten- 18. Ninety-three hundred and thousandths.
seven ten-millionths. 11. Three hundred, and 78 ten- 19. Eighteen thousand and one thousandths.
hundred-millionths. 12. Five tenths, 6 hundredths, 20. Two million, and 6 thou: and 7 buudred thousandths. sand and 9 hundred-millionths.
Express the following as decimals : 21. Po
Ans. .7. 125, 12417o. Ans. 124.07. 22. Por
Ans. .93. 26. 9671004 Ans. 967.104. 23. 1780
Ans .076. 27. 9617880 Ans. 96.0209. Ans .0302. 28. 751088%0. Ans. 75.003017.
PRINCIPLES. 1. Moving the decimal point one place to the right, multa plies 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 wholé decimal will be ten times as great; etc.
2. Moving the decimal point one place to the left, dividen 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; etc.
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.
MENTAL EXERCISES. 1. Uow many tenths in 19 in 1? in }? in f? 2. How many hundredths in 19 in $? in $? in 10? in 14? 3. How many hundredths in ? in f? in ? in off? in 3%? 4. How many halves in .5? in .50? 5ths in .2? in .49 5. How many 4ths in .25? in .75? eighths in .1259 in .375 ?
6. Express as a common fraction and reduce to lowest terms, .5; .4;.8; 25; .50 ; .75 ; .80 ; .125 ; .625.
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.
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 to, which reduced to its
.45=*=, Ans. lowest terms equals to. Hence the following
Rule.- Write the denominator under the decimal, omit ting the decimal point, and reduce the common fraction to its Lowest terms.
Reduce the following decimals to common fractions :
Ans. 94 8. .48.
Ans. 1244. 4. .125.
Ans. 16% 6. .625. Ans. . 10. 5.064.
Ans. 5 6. 3. 25. Ans. 34. 11. 17.0125.
12. Reduce the complex decimal.2% to a common fraction. SOLUTION.–24 is 24 tenths, which, by writing
10 $, which reduced to its lowest terms equals 1x.
ss=of Ana. Therefore, etc.
Reduce the following to common fractions or mixed num. bers: 18. .81. Ans. 18. 4.81.
Ans. 4% 14. .163.
Ans. . 19. 12.184. Ans. 126 16. .25%. Ans. Ps. 20. 22.034. Ans. 8235 16. .456. Ans. H. 21. 50.084.
Ans. 50 17. $0.164 Ans. $1. 22. $6.663. Ans. $64.
CASE II. 206. To reduce a common fraction to a decimah 1. Reduce fto a decimal. SOLUTION.equals 1 of 5. 5 equals 50 tenths ; OPERATION. f of 50 tenths is 6 tenths and 2 tenths remaining; 2
= of 5 tenths equal 20 hundredths; } of 20 hundredths
=8)5.000 equals 2 hundredths and 4 hundredths remaining; 4 hundredths equal 40 thousandths; 1 of 40 thousandths
.625 is 5 thousandths. Therefore, f equals .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. This is often done when the ner term is greater than 5.
Ans. .733+. 3. 4. Ans. .75. 9. H.
Ans. .9375. 4. Ans. .375. 10. 4.
Ans. .64. 5. 7. Ans. .875. 11. H.
Ans. .94126. •
Ans. .1875. 12. 18. Ans. .947368+: 7.
Ans. .3125. 13. H. Ans. .44140625.