One. 236. The Notation of Decimals is an extension of the Notation of Integers. (Art. 36.) The number is read, Six hundred thirty-eight millions, four hundred twenty-five thousands, six hundred seventy-two, and thirty-two million six hundred seventy-two thousand, five hundred forty-five hundred-millionths. The scale of decrease of decimal orders may be illustrated by the following diagram: 237. The value of each figure in decimals, as well as in integers, is determined by the place it occupies, counting from units. Thus, a figure in the first place on the right of the decimal point, has ten times the value of the same figure in the next lower order, or hundredths place, and only one-tenth the value it would have in units place. 9th. 238. Orders equally distant on the right and left from units place, have corresponding names. Thus, tenths correspond to tens, hundredths to hundreds, etc. Hence, 239. The Numerator of a decimal fraction, when written alone, must contain as many figures as there are ciphers in its denominator. If it has not significant figures enough, the deficiency must be supplied by prefixing ciphers. Thus, 15 expressed decimally is .05; 100 is .005, etc. Hence, 240. To write decimals, we have the following RULE. Write tenths in the first decimal place, hundredths in the second, thousandths in the third, etc. Write the following fractions decimally: (Art. 239.) 13. Forty-two hundredths. 16. 43 ten-thousandths. 14. Twenty-one thousandths. 17. 65 hundred-thousandths. EXPLANATION.-Beginning at units, we say, " units, tenths, hundredths, thousandths," etc., to the lowest order, which is millionths. We now read the significant figures as if integers, and pronounce the name millionths. Ans. Four hundred twenty-seven millionths. Hence, the RULE. Read the significant figures of the decimal as integers, and give it the name of the lowest order. NOTES.-1. In mixed decimals, read the integral part as if it stood alone, then read the decimal. Or, having read the integral part, pronounce the word "decimal," then read the decimal figures as if integers. 2. In reading mixed decimals, the word "and" should not be used except between integers and decimals. 243. 1. How many tenths in 1? How many hundredths? How many thousandths? 2. How many tenths in 2? In 5? In 6? 3. How many hundredths in 3? In 4? In 7? ILLUSTRATION OF PRINCIPLES. 244. Since the orders of decimals decrease from left to right by Tens, it follows: 1o. Prefixing a cipher to a decimal, diminishes its value 10 times, and reduces it to the next lower order. 2°. Removing a cipher from the left of a decimal, increases its value 10 times, and reduces it to the next higher order. Thus, .005 = 1000; but .05 = etc. 3°. Annexing a cipher to a decimal, or removing one from its right, does not change its value. WRITTEN EXERCISES. 245. To Reduce Decimals to a Common Denominator. OPERATION. 1. Reduce .5, .42, and .006, to a common denominator. ANALYSIS. The lowest order of the given decimals is thousandths. Annexing ciphers to decimals does not change their value. The fractions are .500, .420, and .006, Ans. (Art. 244, 3°.) Hence, the .50.500 .42 = 0.420 .006 0.006 RULE.-Annex to each as many ciphers as may be re quired to make their decimal places equal. 2. Reduce .20, 2.0004, and 7.008, to a c. d. 3. Reduce 2 tenths, 6 hundredths, and 8 thousandths, to a common denominator. 4. Reduce .03, .125, .7, and .2362, to a c. d. ORAL EXERCISES. 246. 1. Reduce .5 to a common fraction. ANALYSIS.-0.5 = , and reduced to its lowest terms, equals, Ans. 2. How many halves in .50? In .500? 3. How many fifths in .4? In .6? 4. How many fourths in .25? Fifths in .20 ? 5. How many tenths in .40? In .60? 6. How many twentieths in .60? In .80? WRITTEN EXERCISES. 247. To Reduce Decimals to Common Fractions. 1. Reduce .68 to a common fraction. SOLUTION.-The 68 denominator of .68 is 100. Therefore, .68 = 100 or (Art. 235.) Hence, the RULE.-Erase the decimal point, write the numerator over its denominator, and reduce the fraction to its lowest terms. (Art. 185.) 2. Reduce .333 to a common fraction in the lowest terms. 248. 1. How many tenths in ? ANALYSIS.-Since there are 10 tenths in 1, in 1 half there is 1 half of 10 tenths, or 5 tenths, Ans. 2. How many hundredths in ? How many thousandths? 3. How many decimal places are required to express tenths? To express hundredths? Thousandths? (Art. 239.) 4. In how many tenths? In ? In ? 5. In how many hundredths? In ? 6. How many hundredths in? In? In ? 7. How many thousandths in? In? In? WRITTEN EXERCISES. 249. To Reduce Common Fractions to Decimals. 1. Reduce to a decimal fraction. Since we can OPERATION. 8) 3.000 .375, Ans. ANALYSIS. of 1 equals of 3. not divide 3 by 8, we reduce it to tenths by annexing a cipher. (Art. 244, 3°.) Now of 30 tenths is 3 tenths and 6 tenths over. 6 tenths 60 hundredths, and of 60 hundredths 7 hundredths and 4 hundredths over. But 4 hundredths = 40 thousandths, and of 40 thousandths sandths. Therefore - .375. Hence, the - 5 thou RULE.--Annex ciphers to the numerator and divide by the denominator. From the right of the quotient point off as many decimal figures as there are ciphers annexed. |