tenths of thousandths, and five hundredths of thousandths, and read it both ways. Write six tenths, no hundredths, no thousandths, and five tenths of thousandths, and read it both ways. Write six thousand four hundred and thirty-six, tenths of thousandths, and tell how many tenths, hundredths, and thousandths there are. Write four hundred and seventy-nine thousandths, and tell how many tenths, and hundredths there are. Write five hundred and six thousandths, and tell how many tenths there are. Write five hundred and ninety-six hundredths of thousandths, and read it both ways. From the above it appears, that in decimals, the order next to the separatrix is tenths; the second order from the separatrix is hundredths; the third order is thousandths ; the fourth order is tenths of thousandths ; the fifth order is hundredths of thousandths, &c. Decimals are often written with whole numbers. Thus, 2,5. 36,349. Whole numbers and decimals together, are called mix. ed decimals. When decimals are written without whole numbers, they are called pure decimals. Write twenty-four whole numbers, and twenty-four hundredths. Two hundred whole numbers, and five tenths. RULE FOR READING DECIMALS. Read the numerator, as if it were whole numbers, and then add the name of the denominator ; or, Read the number of each separate order, and follow it with the name of the order in which it stands. Read the following decimals both ways. ,11. ,020. ,5005. ,32568.,0505 ,521. ,43002. In decimals what is the first order, at the right of the separatrix ? What is the second order? What is the fourth order? What is the third ? The fifth? What are mixed and pure decimals? What are the rules for reading decimals? Give examples of both methods, 24,690. 6,40043. 6,4000. 69,964. 86,0092. 2,002. 16,00020. In writing decimals from the dictation of the teacher, the pupil needs to understand the two methods very clearly. Thus for example, he may have this decimal, ,00205, dictated in two ways, viz. : 205 hundredths of thousandths, or 2 thousandths, and 5 hundredths of thousandths. In the first mode of dictation, he must write the 205 as if it were whole numbers, and then prefix ciphers to make the figures of the numerator equal to the ciphers of the denominator. In the second mode of dictation, he must put a cipher in each order which is not mentioned; viz. : in the orders tenths, hundredths, and tenths of thousandths, and a 2 in the order of thousandths, and a 5 in the order of hun. dredths of thousandths. Let the pupil write the following in both methods of dictation. 8 hundredths, 6 tenths of thousandths; or 806 tenths of thousandths. 2 tenths, 4 tenths of thousandths; or 2004 tenths of thousandths. 2 thousandths, 5 tenths of thousandths ; or 25 tenths of thousandths 3 hundredths, 6 thousandths, 5 tenths of thousandths ; or 365 tenths of thousandths. RULE FOR WRITING DECIMALS. Write the numerator as if it were whole numbers, and then prefix a separatrix. If the figures of the decimal do not equal in number the ciphers of the denominator, prefix ciphers to make them equal, before placing the separatrix; or, Write each order separately, placing ciphers in the orders omitted. Write the following: What are the rules for writing decimals? Give examples of both methods. 2. Two tenths, five thousandths, six tenths of thou. sandths. Here the order of hundredths is omitted, and has a cipher put in it. 3. Two hundred and four hundredths of thousandths. 4. Two thousandths ; four hundredths of thousandths. What orders are omitted? 5. Sixteen tenths of thousandths. 6. One thousandth, six tenths of thousandths. What orders are omitted ? 7. Four hundred and five thousandths. What orders are omitted ? 8. Four tenths, five thousandths. What orders are omitted ? 9. Three hundred and sixty-five tenths of thousandths. What order has a cipher placed in it? 10. Four hundredths, five tenths of thousandths. What orders are omitted ? 11. Twenty-six thousand, nine hundred and forty-six hundredths of thousandths. 12. Two tenths, six hundredths, nine thousandths, four tenths of thousandths, six hundredths of thousandths. In mixed decimals, it will be seen, that the orders are reckoned from the separatrix, both ways. Thus in 98423,46795, the first order at the right of the separatrix is tenths, and the first order at the left is units. What is the second order at the right, and the second or. der at the left of the separatrix ? What is the third order at the right, and at the left of the separatrix ? What is the fourth order at the right, and at the left of the separatrix ? What is the fifth order at the right, and at the left of the separatrix ? If you have the decimal ,2, and place a cipher at the right, thus ,20, what does it become? Is the value alter. ed ? How is it altered ? Ans. The parts are made ten times smaller, and there are ten times more of them, so that the value remains the same. If you place a cipher at the left of , 2, thus, ,02, what does it become? How much smaller is a hundredth than a tenth ? How much smaller does it make a decimal to prefix a cipher to it? If you put two ciphers at the right of ,2, what effect is produced? If you put them at the left of it, what effect is produced ? The following principle is exhibited above : Ciphers placed at the right of decimals, change their names, but not their value. Ciphers placed at the left of decimals, diminish their va. lue ten times for every cipher thus prefixed. Prefix a cipher to ,91 and read it. Annex a cipher to ,91 and read it. Prefix a cipher to ,20 and read it. Annex a cipher to ,20 and read it. SIGNS AND ABBREVIATIONS USED IN ARITHMETIC. The following signs are used instead of the words they represent. + signifies plus or added to. signifies minus or lessened by. X signifies multiplied by. + signifies divided by. signifies equals. E. signifies Eagles. $ signifies Dollars. d. signifies Dimes. cts. signifies cents. m. signifies mills. ADDITION. Addition is uniting several numbers in one. What effect is produced by placing ciphers at the right of decimals ? What effect have ciphers when placed at the left of decimals? Give examples of the abbreviations used in Arithmetic. What is addition? There are four different processes of addition. The first is Simple Addition, in which ten units of one order make one unit of the next higher order. Thus, ten units make one of the order of tens; ten tens make one of the order of hundreds ; ten hundreds make one of the order of thousands, &c. The second is Decimal Addition, in which decimal frac. tions are added to each other. Thus, ,5,50 ,505 are ad. ded together. The third is Compound Addition, in which other num. bers besides ten, make units of higher orders. Thus, four units of the order of farthings, make one unit of the order of pence. Twelve units of the order of pence, make one of the shilling order. Twenty of the shilling order, make one of the pound order, &c. The fourth is the Addition of Vulgar Fractions, in which vulgar fractions are added to each other. Thus and į are added to each other. SIMPLE ADDITION. If 8 units are added to 9 units, how many are there of the order of tens? Write the 8 under the 9, and draw a line under. Place the units of the answer, under the figures added, and set the 1 ten before them. If 13 apples are added to 25 apples, how many are there in the whole ? Write the units under units, and tens under tens. Add the units first, and place the answer under the unit co. lumn. Then add the tens in the same way. Add 12 cents to 5 cents. Describe the four kinds of addition. |