We have already considered Numeration and Notation, Addition, Subtraction, Multiplication, and Division of Integers, and are now prepared to proceed with Fractions. A Fraction (7) is a number expressing one or more of the equal parts of a unit; as five tenths, one half. Fractions with reference to the mode of expressing them are of two kinds, namely, Decimal and Common. 112. Decimals are those fractions which express one or more of the parts of a unit which has been divided into tenths, hundredths, thousandths, &c. Decimals are expressed in accordance with the Principles of Arabic Numeration and Notation of Integers (41). The term decimal is derived from the Latin decem, which signifies 10. A number consisting of an Integer and Decimal is a Mixed Number (8). By referring to the picture on page 80, you will notice a ball resting on the keystone (units) of the arch. This represents the Decimal Point, and is in decimals what ones of units are in integers,—the First Place. 113. The Decimal Point is used to distinguish Decimals from Integers. When the point is placed between figures it is read and; thus, 2.5 is read "2 and 5 tenths." It may also be seen in the picture that places equally distant from units have corresponding names, with the exception of the decimal termination ths. Thus, tens,-tenths; hundreds,-hundredths; &c. 114. The Value which any figure in a decimal expresses is determined by the place it occupies. The Places are Point, Tenths, Hundredths, Thousandths, Ten-thou sandths, &c. The greatest number of tenths expressed by one figure is 9. 10 tenths = 1. EXERCISES. 1. Read .6; 6.6; .9; 9.9; 18.7; 23.5. 2. Write eight tenths; 12 and 3 tenths; 745 and 4 tenths 116. Hundredths is the Third Place or Order. The greatest number of hundredths expressed by two figures is 99. 100 hundredths = 1. EXERCISES. 1. Read .24; 5.47; .88; 475.15. 2. Read 42.09; .75; 1876.93; .19. 3. Read 14.14; 23.23; 44.44; .07. 4. Read 111.11; 213.02; 400.04; 500.05. 5. Write one hundred and five hundredths. 6. Write fifty and forty-one hundredths. 7. Write seventy-six hundredths; 10 hundredths. 8. Write nine hundred ninety-nine and ninety-nine hundredths. 9. Write three thousand forty and forty thousand three hundred-thousandths. D A The greatest number of thousandths expressed by three figures is 999. 1000 thousandths = 1. EXERCISES. 1. Read .359; 4000.004; 35.035. 2. Read 485.485; 75.075; 19000.019. 3. Read .011; 408.008; 143.043. 4. Read 4279.279; 11.048; .099. 5. Write 325 thousandths; 4 thousandths. 6. Write 97 and 12 thousandths; 803 thousandths. 7. Write 4 and 4 thousandths; 47 thousandths. 8. Write 11245 and 45 thousandths; 9 thousandths. When the right-hand figure of a decimal is hundredths (116), the decimal is read as hundredths; when the right-hand figure is thousandths (117) the decimal is read as thousandths; &c. Hence, 118. RULE FOR NUMERATON. Read the figures of a decimal the same as the figures of an integer, adding the name of the right-hand order of the decimal. 119. RULE FOR NOTATION.-Write the given decimal as a whole number. Prefix ciphers, if necessary, to fill the required number of places. To the left write the decimal point. EXAMPLE.-Read 123456.72345. By the preceding method we are required to perform three operations in order to read the example given: FIRST.-To read the integer-123 THOUSAND 456. 345. SECOND. To read the decimal as an integer—AND 72 THOUSAND THIRD. To ascertain the name of the right-hand decimal place by beginning at 7 and saying tenths, hundredths, thousandths, ten-thousandths, HUNDRED-THOUSANDTHS. SECOND METHOD. 120. The Decimal Point (.) occupies that place in a decimal number which gives name to the decimal. = 72 345 72 thousand 345 hundred-thousandths. By the Second method the integer is read as in the First method. But in reading the decimal as an integer we notice that the point is in hundreds' order of thousands' period and know instantly that HUNDREDTHOUSANDTHS is the name of the decimal. Hence, 121. RULE FOR NUMERATION.-Read the figures of a decimal the same as the figures of an integer, adding the name of the place in the decimal occupied by the point, counting from the right as in integers. 122. RULE FOR NOTATION. -Write the given decimal as a whole number, prefixing ciphers, if necessary, to cause the point to stand in the place which gives the name of the decimal. Write the following in figures: 1. Two hundred fifty-six ten-thousandths. 3. Fourteen hundred-thousandths. 4. Eight thousand ninety-nine billionths. 5. One million three ten-millionths. 6. Seven hundred forty thousand one millionths. 7. Ninety-eight and one hundred three thousand four ten-millionths. 8. Four thousand and four hundred-thousandths. 10. Four thousand and four hundred thousandths. 13. Six thousand and six hundred thousandths. 14. Three hundred one billionths. 15. Ninety-three million and 93 millionths. 16. Eleven billion and eleven hundred-millionths. 17. Five tenths; Fifty hundredths; Five hundred thou sandths; Five thousand ten-thousandths. |