DECIMAL FRACTIONS, LII. Q. When such fractions as these occur, viz. To, 180, how is a unit supposed to be divided? A. Into 10 equal parts, called tenths; and each tenth into 10 other equal parts, called hundredths, and each hundredth into 10 more equal parts, called thousandths, &c. Q. How is it customary to write such expressions? A. By taking away the denominator, and placing a comma before the numerator. 25 65 Let me see you write down, in this manner, fo, foo, 100, 525 Q. What name do you give to fractions written in this manner? Q. Why called decimal? A. From the Latin word decem, signifying ten; because they increase and decrease in a ten fold proportion, like whole numbers. まざ Q. What are all other fractions called? A. Vulgar, or Common Fractions. Q. In whole numbers, we are accustomed to call the right-hand figure, units, from which we begin to reckon, or numerate; hence it was found convenient to make the same place a starting point in decimals; and to do this, we make use of a comma; what, then, is the use of this comma? A. It merely shows where the units' place is. Q. What are the figures on the left of the comma called? A. Whole numbers. Q. What are the figures on the right of the comma called? Q. What, then, may the comma properly be called? A. Separatrix. Q. Why? A. Because it separates the decimals from the whole numbers. Q. What is the first figure at the right of the separatrix called? A. 10ths. Q. What is the second, third, fourth, &c. ? A. The second is hundredths, the third thou sandths, the fourth ten thousandths, and so on, as in the numeration of whole numbers. Let me see you write down again in the form of a decimal. Q. As the first figure at the right of the separatrix is tenths, in writing down Too, then, where must a cipher be placed? A. In the tenths' place. Let me see you write down in the form of a decimal 10. Write down 10, 180, 180. Q. How would you write down in decimals Too? A. By placing 2 ciphers at the right of the separatrix, that is, before the 7. Let me see you write it down. A. ,007. Let me see you write down 1000. Q. Why do you write 2 down with 2 ciphers before it ? A. Because in, the 2 is thousandths; consequently, the 2 must be thousandths when written down in decimals. Q. What does ,5 signify? A. fo Q. What does,05 signify? A. TOO Q. Now, as =, and as multiplying To by 10 produces , which is also equal to, how much less in value is,05 than,5? A. Ten times. Q. Why? A. Because the parts in 18 are ten times smaller than in ; and, as the numerator is the same in both expressions, consequently, the value is lessened 10 times. Q. How, then, do decimal figures decrease in value from the left towards the right? A. In a tenfold proportion. Q. What does ,50 mean? A. 5 tenths, and no hundredths. Q. What, then, is the value of a cipher at the right of decimals? Q. We have seen that ,5 is 10 times as much in value as ,05, or To what effect, then, does a cipher have placed at the left of decimals ? A. It decreases their value in a tenfold proportion. Q. Since decimals decrease from the left to the right in a tenfold proportion, how, then, must they increase from the right to the left? A. In the same proportion. Q. Since it was shown, that ,5; 25; what, then, will always be the denominator of any decimal expression? A. The figure 1, with as many ciphers placed at the right of it as there are decimal places. Let me see you write down the following decimals on your slate, and change them into a common or vulgar fraction, by placing their proper denominators under each, viz.,5,05,005,62,0225 ̊,37. Q.,25 is =, and,5 is = ; which, then, is the most in value,,25 or ,5? Q. By what, then, is the value of any decimal figures determined? A. By their distance from the units' place, or separatrix. Q. When a whole number and decimal are joined together, thus, 2,5, what is the expression called? A. A mixed number. Q. As any whole number may be reduced to tenths, hundredths, thousandths, &c. by annexing ciphers, (for multiplying by 10, 100, &c.) thus, 5 is 50 tenths, 500 hundredths, &c.; how, then, may any mixed number be read, as 25,4 ? A. 254 tenths, giving the name of the decimal to all the figures. Q. How is 25,36 read? Q. How is 5,125 read? A. 5125 thousandths. Q. What would 5125 thousandths be, written in the form of a vulgar or common fraction? A. $138. This is evident from the fact, that (an improper fraction), reduced to a mixed number again, is equal to 5,125. The pupil may learn the names of any decimal expression, as far as ten-millionths, also how to read or write decimals, from the follownig Table: 310000000 3,0000008 read 3, and 8 Ten-Millionths. 365 365,0000 000 read 365. = Exercises for the Slate. Write in decimal form 7 tenths, 42 hundredths, 62 and 25 hundredths, 7 and 426 thousandths, 24 thousandths, 3 ten-thousandths, 4 hundredths, 2 ten-thousandths, 3 millionths. Write the fractional part of the following numbers in the form of decimals, viz. 616, 1, 62100, 210, 3180, 2621000, 321880, 2100 000, 451008000, 710000000, 510000. Write the following decimal numbers in the form of vulgar or common fractions, then reduce them to their lowest terms by ¶ XXXVII.; thus, 2,5=212 in its lowest terms. Q. What money is adapted to decimal rules? A. Federal money. Q. What is the money unit? Q. How is it so adapted? A. As 10 dimes make a dollar, and 10 cents a dime, &c., dimes are 10ths of a dollar, cents are 100ths, and mills are 1000ths of a dollar. Q. How are 3 dollars 2 dimes 4 cents and 5 mills written? A. $3,245. ADDITION OF DECIMALS. LIII. Q. As we have seen that decimals increase from right to left in the same proportion as units, tens, hundreds, &c., how, then, may all the operations of decimals be performed? A. As in whole numbers. 1. A merchant bought 5 825 62 Note.-The only difficulty which ever arises, consists in determining where the decimal point ought to be placed. This will be noticed in its proper place. barrels of rice at one time for $27, at another of a barrel for $4,255, at another 278 of a barrel for $10, and at another of a barrel for $2; how many barrels did he buy in all? and what did they cost him? Ισσσ 627 Barrels. OPERATION. Dollars. As we have seen that decimals cor 27,825 respond with the de 4,255 nominations of Fed 0,72 2,627 Ans. 6,988 barrels, for $ 35,427 cimals down, placing dimes under dimes, cents under cents, &c., that is, tenths under tenths, hundredths under hundredths, &c., and add them up as in Addition of Federal Money. From these illustrations we derive the following RULE. Q. How are the numbers to be written down? A. Tenths under tenths, hundredths under hundredths, and so on. Q. How do you proceed to add? |