COMPLEX FRACTIONS.

186. A Complex Fraction is one whose numerator is a fraction or a mixed number.

incorrectly, called complex fractions. They are expressions which indicate division; the part above the line being the dividend, and the part below the divisor.

348

1. Reduce to a simple fraction.

EXPLANATION.-This expression means that is to be divided by . Inverting the divisor and multiplying, we have, which equals o

EXPLANATION.-The expression indicates that of fis to be divided by + }. Simplify the dividend and the divisor by performing the multiplication indicated in the former and the addition in the latter, and we have f divided by 7. Inverting the divisor and multiplying, we have }}.

Reduce the following expressions to their simplest form :

DECIMAL FRACTIONS.

187. A Decimal Fraction is a fraction whose denominator is ten or some power of ten-as tenths, hundredths, thousandths, etc. A decimal fraction is generally expressed by placing a period, called the decimal point, before the numerator and omitting the denominator. Thus, .7 expresses; .07 expresses ; .007, 1ōbō; etc.

188. The Decimal Point always indicates the decimal fraction and separates it from the whole number.

189. The Decimal Places are the places to the right of the decimal point. The first place to the right of the decimal point is tenths, the second place is hundredths, the third, thousandths, etc. Thus, .2 represents ; .23 represents; .234, 3, etc.

190. The relation of decimal fractions to the unit and to one

another may be seen in the following:

NUMERATION TABLE.

1. The number is read 5 billion 555 million 555 thousand 555, and 555 million 555 thousand 555 billionths.

2. The orders of decimals below billionths are ten-billionths, hundredbillionths, trillionths, ten-trillionths, hundred-trillionths, quadrillionths, etc.

191. A Decimal is a decimal fraction expressed by the method of decimal notation.

Thus, .5, .75, and .075 are decimals.

192. A Pure Decimal consists of decimal figures only.

Thus, .425 is a pure decimal.

193. A Mixed Decimal consists of an integer and a decimal. Thus, 3.5 is a mixed decimal.

194. A Complex Decimal contains a fraction at the right of the decimal.

Thus, .33 is a complex decimal.

EXERCISES IN NUMERATION.

195. 1. Read the expression 3.425.

The whole expression is read, 3 and 425 thousandths.

RULE.-Read the decimal as a whole number and give it the name of the last place at the right.

1. First determine the numerator by numerating towards the decimal point from right to left; then determine the denominator by numerating towards the right from the decimal point.

2. In reading, the word “and” should be used only to separate the whole number from the decimal.

EXERCISES IN NOTATION.

196. 1. Express decimally thirty-five thousandths.

EXPLANATION.-Sinc› thousandths occupy the third place, three figures are necessary to express the decimal. We write the number 35, and prefix a cipher to cause the figures to occupy their proper place. Hence, the decimal is written .035.

RULE. Write the numerator as a whole number, prefix ciphers if necessary, so that the right hand figure will occupy the order named, and place the decimal point before tenths.

Avoid using the word "and" in reading whole numbers. It should be used only in reading a mixed decimal. Thus, 125.6 is read one hundred twenty-five and six tenths. Keeping this in mind will aid in writing decimals.

Express decimally :

2. Four tenths.

Seven tenths. Two hundredths. Five hundredths. Nine hundredths.

3. Twelve hundredths. Fourteen hundredths. Six thousandths. Two hundred seven thousandths. Fifteen ten-thousandths.

4. Two hundred six thousandths. Three hundred seventy-two ten thousandths.

5. Five hundred twenty-five millionths. Thirty-six ten-thousandths. Twenty-one hundred-thousandths.

6. Seventy-five, and five tenths. Thirty-two, and thirty-two thousandths. Three hundred forty, and five thousandths.

7. One thousand, and one thousandth. Two thousand sixtythree, and twenty-six hundred thousandths. Four hundred fifty, and six hundred eighty-one millionths.

8. 32 ten-thousandths. 123 millionths. 406 thousandths. 2147 hundred-thousandths. 6347 ten-millionths. 4004 millionths.

9. 34572 hundred-thousandths.

23465 ten-millionths. 47 bil

lionths. 1035 hundred-millionths. 5 ten-billionths.

26. How do the fractions in numbers 18, 19, 20, and 21 compare in value? How do the decimals compare in value? How does annexing ciphers to a decimal effect it? Give the reason.

27. How do the fractions in numbers 22, 23, 24, and 25 compare in value? How then do the decimals compare in value? How does prefixing a cipher to a decimal effect it? Prefixing two ciphers? Three ciphers?

30. Read the numbers expressed in number 28. What is the effect of moving the decimal point one place to the right? Two places to the right? Three places?

31. Read the numbers expressed in number 29. What is the effect of moving the decimal point one place to the left? Two places to the left? Three places?

197. PRINCIPLES.-1. Annexing ciphers to a decimal does not change its value.

2. A decimal cipher prefixed to a decimal, divides the decimal by ten; two decimal ciphers prefixed to the decimal, divide it by a hundred; etc.

3. Moving the decimal point one place to the right, multiplies the decimal by 10; two places, multiplies by 100; three places, multiplies by 1000; etc.

4. Moving the decimal point one place to the left, divides the decimal by 10; two places, divides by 100; three places, divides by 1000; etc.

198. The currency of the United States is based on the decimal system of notation; and the cents, mills, etc., may be read as decimals of a dollar.

Thus, $5.25 may be read five and twenty-five hundredths dollars.