The benefits arising are many and great: we can then make calculations with ease, by carrying at every ten as in whole numbers. For in all decimal operations, except Duodecimals, we carry at every ten. KEY TO CARD No. 27. LESSON 8. What is the decimal of of a pound? 12)1.000(.08333 .40* 36 4 Remarks. In the first place we fix the point and annex three ciphers to the 1 thus, 1.000. Then say, how many times 12 in 100? 8 times and 4 remain.Bring down the other cipher makes 40. How many times 12 in 40? 3 times and 4 remain. Then annex a cipher to the remainder and mark it with an asterisk thus, 40*. Then proceed. How many times 12 in 40? 3 times and 4 remain. Annex another cipher to the remainder and mark it as before. Then how many times 12 in 40 again? 3 times and 4 remain; and in this manner 4 would remain continually without end. We will therefore call this decimal a surd,† that is, a number which cannot be reduced to exactness. Let this decimal be marked with the sign of more, thus, .08333+4. But whence came this 0 on the left side of the 8? I will explain it ; we had three places of decimals annexed to the 1; then we added two more marked * Ciphers added for Decimals. ↑ A decimal that comes to a close without a 'remainder, as in Lesson 4, may be called a finite decimal. with a star or asterisk; the whole make five decimal places in the dividend. Now, the rule says, "If there be a deficiency of decimal places in the quotient, supply the defect by prefixing ciphers on the left." Let us count the decimals in the quotient, beginning at the right: 3, 3, 3, 8: here are only four, but we ought to have five; because there are five decimal places in the dividend. Then prefix a cipher on the left of 8 in the quotient, and place the separatrix or point on the left of the cipher. To read this decimal, call the point units: ther proceed to the right; "Tens, hundreds, thousands, tens of thousands, hundreds of thousands." "Eight thousand three hundred and thirty-three; One hundred thousandths." Answer in figures, .08333+4 of a pound, or form of a vulgar fraction, thus, 8888+4. 8333 KEY TO CARD No. 27. LESSON 9. What is the decimal of $? 15)8.00000(53333+5|| 75 . 50 45 .50 45 .50 45 .50 45 .5 Remarks. This decimal is also a surd. The quotients of such fractions must be continued in proportion to the magnitude of the integer from which they proceed; that is to say, of a pound is a sum of greater value than than of a farthing; and as decimals decrease towards the right, it is more needful to extend the decimals of a pound, than of an inferior denomination. Five places of decimals are Ilcommonly sufficient for mi nuteness in most cases of importance; but for the sake of curiosity in some peculiar calculations, the operator may extend decimals to what length he please. NOTE. The remainder of a surd must be added in proof. Copy the following on your slates, having reference at the same time, to the Table on this page. .1 .12 .123 .1234 One, tenth. Twelve, hundredths. One hundred and twenty-three, Thou- One thousand two hundred and thirty-four, .12345 Twelve thousand three hundred and fortyfive, One hundred thousandths. .123456 One hundred twenty-three thousand four hundred and fifty-six, Millionths. Decimals may be read collectively or singly. Collectively, thus: One hundred twenty-three thousand four hundred and fifty-six, Millionths. Singly, thus: One tenth, two hundredths, three thousandths, four ten thousandths, five hundred thousandths, six millionths. decimal figures of the sum to be added. Begin at the right of the sum total, and count as many towards the left for decimals. Rule 2. Subtraction. Let the decimal places in the remainder, be equal to the greatest number of decimals in either of the two numbers above singly considered. Rule 3. Multiplication. The number of decimal places in the product, must be equal to the whole of those in the two factors; and if figures are wanting in the product, prefix ciphers to the left, to supply the defect. Rule 4. Division. The number of decimal places in the divisor and quotient counted together, must be equal to those in the dividend, NOTE 1. If decimal places be wanting in the dividend, annex so many ciphers as you please. 2. If there be decimal places wanting in the quotient, supply the defect by prefixing ciphers. These Rules shall be more fully explained, when we work by them separately. CASE 2. To find the integral value of a decimal. RULE. Multiply the decimal by the denominations of the integer, and point off the decimal parts to the right; the left will be integers; as in lesson 7. KEY TO CARD No. 27. By what rule do we point off decimals in multiplication? By counting the decimal places in both factors; that is in the multiplicand and multiplier: Then begin at the right of the product and count so many towards the left as were in the two factors; there place the point. The figures on the left are integers. See Rule 3, Page 167. NOTE. If there be not so many figures in the product as are in the two factors, supply the defect by prefixing ciphers to the left hand. EXAMPLE. Multiply .004|| Here are three decimals in the up.8 per factor, and there is one in the low By er; therefore we must have four in Product, .0032 the product. REMARKS. It is of importance to know the place of the decimal point; because decimals decrease in a tenfold proportion towards the right, as whole numbers increase towards the left. In the above example, if the cipher had not been prefixed, the number would read .032 "thirty-two thousandths." Whereas the true number is "thirtytwo ten thousandths.” LESSON 11. What is the value of .5 of a dollar in cents? .5X100 100 LESSON 13. How many cents in .875 of a dollar. .875 100 Answer, 50.0 cents. LESSON 12, What number of shillings are in .75 of a pound? it 20 Answer, 15.00s 87.500 Answer, 87.5 cents. In this result or product may be observed that the fraction is five hundred thousandths; but that is the same as fifty hundredths, or five tenths. |