Operation. 300 24 60 60 12 can be taken from 15 once and 3 remains. If we add one cipher to the remainder and divide by 12, there will still be a remainder of 6, we therefore add 2 ciphers which is the same as multiplying the remainder by 100. (See page 45.) Each of the 3 remaining dollars is now divided into 100 parts, and as many times as 12 can be taken from the 300, so many hundredth parts each man will have in addition to the 1 dollar expressed by the first figure in the quotient. The 25 at the right of the comma, are 25 hundredths and the denominator 100 understood. 25 is of 100, which is equal to the vulgar fraction which would have been formed had we set the divisor under the remainder, therefore the answer is $11, or $1,25. If we add three ciphers to the remainder of any question, the remainder will be multiplied by 1000, and each unit of the remainder in that case, would be divided into 1000 equal parts, and that part of the quotient obtained from the remainder after the 3 ciphers are annexed, must be thousandth parts of an unit. In the same manner it might be shown that the more ciphers we annex to the remainder, the less parts of an unit the quotient figures obtained from the remainder, will express. When we annexed one cipher, the fraction was tenths; when we annexed two ciphers, the fraction was hundredths, Let 21 be divided by 12. Operation. 12) 21 (1,7 5 1 2 90 8 4 60 60 In this example we annex one cipher to the remainder 9, which reduces it to tenths, and 7 the quotient figure found by dividing the 90 tenths, is 7 tenths. The next remainder 6, is 6 tenths, and by annexing a cipher to it, it divides each of the tenths in the 6 divided into 10 equal parts; one of these small parts must therefore be a tenth of 1 tenth. 5 the right hand figure of the quotient, shows how many of these small parts contained in the CO, there are in the quotient; 5 must therefore be 5 tenths of one of the parts contained in 7,-or in other words, 10 parts as large as those contained in the 5, are equal to 1 of the parts contained in the 7, or equal to 1 tenth of an unit. Annexing the ciphers one at a time is the same thing as though we had annexed them both to 9 and then brought down, the right hand one to 6. In the preceding question, we annexed two ciphers and found the two decimal figures in the quotient to be hundredths; but we have annexed two ciphers in this, therefore, 75 when read together, are hundredths. 9 the first remainder written over the divisor, makes a vulgar fraction equal to 3, showing what part of another unit belongs in the quotient, and 75 being of 100, 75 hundredths denote the same-thing. The whole quotient is 17, or 1,75, or it is equal to 12, which may be read one, seven tenths and five hundredths, or one, and seventy-five hundredths, for 7 is the same part of 10 that 70 is of 100. From what has been said, it appears that every time we annex a cipher to a remainder, the next quotient figure implies parts of an unit, only 1 tenth part as large as those expressed by the figure at the left hand of it. It also appears, that the denominator which is generally understood, will always have as many ciphers in it as the number of ciphers annexed to the remainder, and this number will equal the number of decimal places in the numerator. The denominator of a decimal fraction equals an unit. If any thing be divided into 10 equal parts, one of the parts is a tenth of the same thing, and any number of those parts less than 10, would be represented by one figure which would stand in the first decimal place; but the denominator of one decimal place is 10, which shows how many of those parts would be required to make a whole. The same might be shown when the denominator is 100, 1000, &c. An unit is considered as the dividing point between whole numbers and decimals. Whole numbers increase towards the left from unity in a ten fold proportion; decimals decrease towards the right in the same proportion. From the foregoing explanation we obtain the following general principles. 1. Decimal fractions properly belong to the right hand of whole numbers. 2. The denominator to a decimal fraction is 1 with as many ciphers to the right hand of it, as the numerator has places of figures. 3. 10 in any one place of a decimal, is equal to 1 in the next left hand place. 4. All operations in decimals may be performed as with whole numbers. 5. It is not necessary to write the denominator to a decimal fraction. Changing Vulgar Fractions to Decimals. It has been seen in the operations of the questions in the explanation of decimals, that annexing ciphers to a remainder and dividing by the divisor, the quotient figures thus obtained, are decimals expressing the same part of an unit as the vulgar fraction form ed by writing the divisor under the remainder. As the numerator of any vulgar fraction represents a remainder, and the denominator the divisor, annexing ciphers to the numerator of a vulgar fraction and dividing by the denominator, the quotient figures, if there be no remainder, will be a decimal expressing the same part of the same unit as the vulgar fraction. Method of reading Decimal Fractions. When a decimal fraction stands alone, the numerator is read as a whole number, and the denominator repeated afterwards. ,9 is read nine tenths; ,26 twenty-six hundredths;,129 one hundred and twenty-nine thousandths, &c. But when a decimal is written in connexion with a whole number, the whole number is first read and then the decimal as before. 2,6 is read two and six tenths; 325,35, three hundred and twenty-five and thirty-five hundredths, &c. The following table exhibits the names of the decimal places to millionths. The same numbers as those in the table, are set on the same line at the left hand with the decimals in the form of vulgar fractions. TABLE. Hundredths. Tenths. Ten thousandths. 56 10.9 126 5 6 0 0 7 0 470 4013 1 0 1 0 1 000125 Ciphers written at the left hand of decimals, lessen their value, but if written at the right hand do not alter it. ,8 or ,80 is 8 tenths; but ,08 is 8 hun. dredths, and ,008 is 8 thousandths.* RULE For changing Vulgar Fractions to Decimals. Annex ciphers to the numerator and divide by the denominator. 1. What decimal fraction is equal to ? 100÷4,25 the answer. 2. Change to a decimal. 3. Change to a decimal. 4. Change to a decimal. 8 Ans. ,75. Ans. 12. Ans.,2. Ans. 56. Ans. 625. of a dollar for a pen knife; how many tenths of a dollar did he give? 8. Change to a decimal. 0 260 9. Change to a decimal. 65 375 Operation. 3 7 5) 6 5 0 (1 7 3 3 3 3 3. 375 Ans. 2. Ans.,4. 2750 2625 1250 1125 1250 1 12.5 1.25 The word decimal is derived from the Latin word decem, which signifies ten. Decimal Fractions were invented by Regiomontanus, in 1464; but did not come into general use until the last part of the 16th century. |