CHAPTER VI. DECIMAL FRACTIONS.-DECIMAL OR FEDERAL MONEY. DECIMAL FRACTIONS. § 128. A Decimal Fraction is one or more 10ths, or 100ths, or 1000ths, &c., of a quantity, expressed by its numerator only with a point prefixed;-its denominator being understood to be 1 with as many Os annexed as there are figures in the numerator. Thus .3 is; the denominator being understood to be 1 with one cipher annexed, since there is one figure in the numerator. But .03 is ; the denominator being understood to be 1 with two Os annexed, because there are two figures in the numerator .03. What is expressed by .4, 4 with a point prefixed? By .05, 5 with a O and point prefixed? By .006? By .0005? By .00001 ? One tenth is how many hundredths? One hundredth is how many thousandths? One thousandth is how many ten-thousandths? The simple term decimal is sometimes used to designate a decimal fraction. $ 129. A vulgar fraction, as distinguished from a decimal, is any fraction expressed by a numerator and denominator; as, 1, 4, 70. The term fraction, used alone, commonly denotes a vulgar fraction. Notation of Decimals. § 130. The first figure on the right of the decimal point, denotes tenths; the second, hundredths; the third, thousandths; and so on, t ten-thousandths, hundred-thousandths, millionths, ten-millionths, &c. Thus in the decimal .123456, the 1 is 1 tenth, the 2 is 2 hundredths, the 3 is 3 thousandths, the 4 is 4 ten-thousandths, &c. But § 131. The first two figures on the right of the decimal point will together denote hundredths; the first three will together denote thousandths; the first four, ten-thousandths, &c. Thus in the decimal .12, the 1 being, and the 2 being, both together make 12 hundredths. 1000 Again, in .123, the 12 is, the 3 is roo, and the whole 123 thousandths. This is in accordance with the fact that the denominator understood to a decimal, is 1 with as many Os annexed as there are figures in the numerator, (§ 128). Complex or Mixed Decimals. $ 132. A complex or mixed decimal is a decimal fraction with a vulgar fraction annexed to it. 5 Thus .5 is 5 tenths; that is, 5 tenths and of 1 tenth ;= 10 253 .25 is 25 hundredths; the same as 100' The vulgar fraction annexed to a decimal, denotes its proper part of 1 tenth, or 1 hundredth, &c., according as it is annexed to tenths, or hundredths, &c.; and must not be reckoned as in a separate place of decimals. Scale of Decimals. § 133. In decimals, as in integers, ten of any lower order make one of the next higher order; or one of a higher order makes ten of the next lower order. Thus beginning at thousandths, for example, 10 thousandths make 1 hundredth; 10 hundredths make 1 tenth; 10 tenths make 1 unit. That is, the system of numbering by tens, is carried from units up through tens, hundreds, &c., and from units down through tenths, hundredths, &c. RULE XXIV. $ 134. To read a Decimal Fraction. Call the successive figures tenths, hundredths, thousandths, ten thousandths, &c., from the decimal point toward the right; then, disregarding Os next the point, read the number as if it were an integer, and add the decimal name of the last figure. EXAMPLES. 1. To read a decimal .03457. Calling the figures tenths, hundredths, &c. toward the right we find the last figure 7 to be 7 hundred thousandths. Hence the decimal is 3457 hundred-thousandths. ($131.) 2. To read the mixed number 305.004. The 4 is 4 thousandths; the whole is read, three hundred and five, and four thousandths. Note.-An ambiguity may sometimes arise in the enunciation of decimals and mixed decimal numbers. For example, .305 is three hundred and five thousandths; and 300.005 is three hundred, and five thousandths. But when read in this manner, the two are not sufficiently distinguished from each other. The ambiguity may be prevented by using the word decimal before the fraction; thus, .3 5, decimal 305 thousandths; 300.005, 300 and decimal 5 thousandths. ' EXERCISES. Read each of the following decimals and mixed numbers. § 135. Ciphers annexed to a decimal fraction do not alter the value of the decimal. Thus .3=.30=.300, &c., since 1%=1%%=100%, &c. (§ 93). § 136. Each 0 between the decimal point and the first significant decimal figure, diminishes the decimal to 1% of its value without the 0. Thus .3 is 3 tenths, .03 is 3 hundredths, and .003 is 3 thousandths; Too is 1 of 1%, and Too is 1 of 18; (§ 119). 1000 How may the integer 1 be made to denote 1 tenth ? 1 hundredth ? 1 thousandth? 1 ten-thousandth? I hundred-thousandth? How may the integer 5 be made to denote 5 hundredths? 5 thousandths? 5 hundred-thousandths? 5 millionths? § 137. To denote tenths, hundredths, &c., by decimal fractions. Prefix the decimal point to the numerator, interposing so many Os, when necessary, next the point, that-the successive figures being called tenths, hundredths, thousandths, &c., toward the right, the last figure shall have the same decimal name with the parts to be expressed. EXAMPLE. 1. To denote 54 ten-thousandths by a decimal fraction. .00 54 Prefixing the decimal point to the numerator 54, and interposing two Os next the point, we find that, when the several figures are called tenths, hundredths, &c., toward the right, the 4 is ten-thousandths,-which is the name of the parts to be expressed. EXERCISES. In the notation of both integers and decimals. Write in figures the following numbers-observing that, in the verbal expression, the integral and fractional parts are separated by a comma. 1. Fifteen hundredths. 2. Nineteen thousandths. 3. Six ten-thousandths. 4. Twenty-four thousandths. 5. Five hundred thousandths. 6. Thirty-nine millionths. 7. One hundred thousandths. 8. Ten ten-millionths. 9. Forty-nine hundredths. 10. Seventeen ten-thous'dths. 11. Fifty-two thousandths. 12. Seventy-one hundredths. 13. Eight hundred thous❜dths. 14. Ninety-one millionths. 15. One hundred thousandths. 16. Four thousand and nine, and five thousandths. 17. Twenty thousand and seven, and nineteen ten-thousandths. 18. Fifteen millions, and three hundred and two thous'dths. 19. Five hundred and four thou sand, and nine ten-thous'dths. 20. Five millions two hundred and one thousand, and three tenths. 21. Seven hundred millions, and three hundred and nine thousandths. 22. Eighteen millions three hund. and seventy-six thousand and thirty, and twelve hundredths. FEDERAL MONEY. § 138. FEDERAL MONEY, or money of the United States, is expressed in units according to the decimal scale of numeration, that is, numeration by tens. The units of Federal money are, Eagles, Dollars, Dimes, Cents, and Mills. § 139. The only denominations of Federal Money in common use, are dollars and cents ;-eagles being expressed in dollars; dimes, in cents; and smaller values in fractions of a cent. Thus, instead of 5 eagles, 4 dollars, 3 dimes, 2 cents, and 5 mills; we would say 54 dollars, 32 cents. DECIMAL NOTATION OF FEDERAL MONEY Cents being expressed only in numbers less than 100, and mills in numbers less than 10, we have the following RULE XXVI. $140. For the decimal expression of Federal Money. Regarding dollars as integers, make cents and mills decimals of a dollar, by prefixing to them the decimal point,-observing to interpose a O next the point, when the number of cents is less than 10,-and two Os next the point when only mills, or a fraction of a cent, are given. EXAMPLES. 31 cents is $.31, 31 hundredths of a dollar; 6 cents is $.064, 64 hundredths of a dollar; (§132.) 5 mills is $.005, 5 thousandths of a dollar; of a cent is $.004, 4 of 1 hundredth of a dollar; 7cts. and 5 m. is $.075, 75 thousandths of a dollar. |