Decimal fractions, for the sake of brevity, are usually called decimals.

196. Since tenths are equal to ten times as many hundredths, and hundredths are equal to ten times as many thousandths, thousandths to ten times as many ten-thousandths, etc., it is evident that decimals have the same law of increase and decrease as integers, and that the denominator may therefore be indicated by the position of the figures.

According to the decimal system of notation, figures decrease in tenfold ratio in passing from left to right; therefore a figure at the right of units will express tenths, at the right of tenths, hundredths, at the right of hundredths, thousandths, etc., as is exhibited by the following expressions:

From this mode of expressing decimal fractions the following principles are deduced:

197. PRINCIPLES.-1. Decimals conform to the same principles of notation as integers.

2. Each decimal cipher prefixed to a decimal diminishes its value tenfold, since it removes each figure one place to the right. 3. Annexing ciphers to a decimal does not alter its value, since it does not change the place of any figure of the decimal.

4. The denominator of a decimal, when expressed, is 1 with as many ciphers annexed as there are orders in the decimal.

The Decimal Point is a period placed before the decimal. Thus, .6 represents ; .54 represents 100

54

The decimal point is also called the Separatrix, since it is also used to separate integers from decimals.

198. A Pure Decimal Number is one which consists of decimals only; as .387.

199. A Mixed Decimal Number is one which consists of an integer and a decimal; as 46.3, which is equal to 46,3%.

200. A Complex Decimal is one which has a common fraction at the right of the decimal; as .33, which is

By examining this table it will be seen that tenths occupy the first decimal place, hundredths the second, thousandths the third, ten-thousandths the fourth, hundred-thousandths the fifth, millionths the sixth, etc. Hence,

The place occupied by any order of decimals is one less than that occupied by the corresponding order of integers.

What decimal place is occupied by hundredths? Tenths? Hundred-millionths? Thousandths? Ten-thousandths? Tenmillionths? Millionths? Billionths? Hundred-thousandths?

EXAMPLES IN NUMERATION.

202. 1. Read the decimal 4.246.

ANALYSIS.-The figures of the decimal express 2 tenths, 4 hundredths and 6 thousandths, which, reduced to equivalent fractions having a common denominator, become 200 thousandths, 40 thousandths and 6 thousandths, or 246 thousandths.

The whole expression is read 4 and 246 thousandths.

RULE. Read the decimal as an integral number and give it the denomination of the right-hand figure.

EXAMPLES IN NOTATION.

203. 1. Express decimally forty-three thousandths.

ANALYSIS. Since 43 thousandths are equal to 4 hundredths and 3 thousandths, we write 3 in thousandths' place and 4 in hundredths' place, and as there are no tenths, 0 in tenths' place. Hence, forty-three thousandths.043.

RULE. Write the numerator of the decimal, prefix ciphers if necessary to indicate the denominator, and place the decimal point before tenths.

Express decimally:

2. Eight tenths. Nine tenths. Five tenths.

3. Two hundredths. Eight hundredths. Six hundredths. 4. Six thousandths. Four-hundred-two thousandths.

5. Nine ten-thousandths. Eight hundred ten-thousandths. 6. Seventeen hundredths. Fifteen hundred-thousandths. 7. Forty-eight thousandths. Five hundred ten-millionths. 8. Ninety-three ten-thousandths. Ninety-three billionths. 9. Fifty-one hundred-thousandths. Fifty-one thousandths. 10. 107 millionths. 306 ten-millionths.

11. 3259 hundred-thousandths.

4268 hundred-millionths.

12. 429 ten-millionths. 3842 hundred-thousandths.

13. 4300 billionths. 38496 hundred-billionths.

14. 85 hundred-millionths. 85 hundred-billionths. 15. Six thousand ten-thousandths. Five ten-billionths. 16. Five and six-tenths. Eight and nine ten-thousandths. 17. Eighty ten-thousandths. Forty hundred-thousandths.. Express decimally:

In reading expressions of United States currency, the cents, mills, etc., may be read as decimals of a dollar. Thus, $4.7235 may be read 4 dollars 7235 cents, or $4,7235.

REDUCTION.

CASE I.

10000•

204. To reduce dissimilar decimals to similar decimals.

1. How many tenths of an apple are there in 1 apple? How many hundredths in 10 apples? How many thousandths? 2. How many hundredths are there in 6 tenths? How many thousandths? How many ten-thousandths?

3. Express 6. hundredths as thousandths. As ten-thousandths. As hundred-thousandths. As millionths.

4. Express 8 thousandths as ten-thousandths. As hundred-thousandths. As millionths.

205. PRINCIPLE.-Annexing ciphers to a decimal does not alter its value.

WRITTEN EXERCISES.

1. Reduce .5, .36, .406 and 3.3109 to similar fractions.

PROCESS.

.5

= .5000

.36

= .3600 .406 = .4060 3.3109

3.3109

ANALYSIS.-The lowest order of decimals in the given numbers is ten-thou sandths, and to reduce the decimals to similar decimals, we must change them all to ten-thousandths, or to other decimals having an equal number of places. Since annexing ciphers to a decimal

does not alter its value, we give to each number four decimal places by annexing ciphers, and this renders them similar.

RULE.-Give to all the given decimals the same number of decimal places by annexing ciphers.