units' place, we obtain decimal orders, each of which, as in the case of integers, invests its figure with a value ten times as great as the next order on its right.

Decimals are therefore expressed according to the same system as integers. Hence they may be written beside integers (with the separatrix to separate them), and may be added, subtracted, multiplied, and divided, in the same way as integers.

In the expression 2222.2222, each 2, whether integral or decimal, has a value ten times as great as the 2 next on its right. The integral twos represent collections of units; the decimal twos, parts of a unit.

174. TABLE.-The names of the places on the right of the units' place resemble those on the left. They may be learned from the following Table :

ORDERS OF INTEGERS.

ORDERS OF DECIMALS.

Observe that in going from tens to hundreds, thousands, &c., we pass to higher orders; but in going from tenths to hundredths, thousandths, &c., we pass to lower orders.

Observe that two figures are required to express tens (10), one to express tenths (.1); three for hundreds, two for hundredths; and generally, one less figure for a decimal order than for an order of integers of similar

name.

175. GENERAL PRINCIPLES.-As in whole numbers, so in decimals, we give a figure a certain value by writing it in a certain place. Thus we express

Where, then, may decimals be written, and how may they be added, &c. ?—174. What resemblance may be noticed in the names of the decimal orders? Name the orders of decimals, going to the right from the decimal point. Name the orders of integers, going to the left. In which case do we pass to higher orders, and in which to lower? How many figures are required, to express tens? To express tenths? To express hundreds? To express hundredths? What general principle is deduced from this?-175. How do we give a decimal figure a certain value? Givo examples.

by writing 9 in the place of tenths

.9;

Too by writing 9 in the place of hundredths .09;

Too by writing 9 in the place of thousandths .009, &c.

176. From the above examples we see that vacant decimal places on the left must be filled with naughts. By leaving out the naughts in nine hundredths and nine thousandths, as written above, we would change them to nine tenths.

177. We also see that to express a decimal we must use as many figures as there are naughts in its denominator. There is one naught in 10; one decimal figure expresses tenths. There are two naughts in 100; two decimal figures express hundredths, &c.

178. It follows that every decimal has for its denominator 1 with as many naughts as there are figures in the numerator.

179. A naught prefixed to a whole number does not change its value; every naught annexed multiplies it by 10. With decimals it is not so.

A naught prefixed to a decimal (on the right of the separatrix) throws its figures one place to the right, and thus divides it by 10: .3 is ten times as great as .03.

A naught annexed to a decimal does not change its value, because denominator as well as numerator is multiplied by 10. .3.30 (10%).

30

180. RULE.-To express a decimal in figures, write its numerator as a whole number. If it contains fewer figures than the denominator contains naughts, prefix naughts to supply the deficiency. Finally, prefix the decimal point.

EXAMPLE.-Write forty-two millionths as a decimal.

176. How must vacant decimal places on the left be filled?-177. How many figures must we use, to express a decimal ?-178. What does every decimal have for its denominator?-179. What is the effect of prefixing a naught to a whole number? To a decimal? What is the effect of annexing a naught to a whole number? To a decimal?-180. Recite the rule for expressing a decimal in figures. Give examples.

Write the numerator as a whole number, 42. The denominator contains six naughts; hence, as the numerator contains but two figures, we must prefix to it four naughts. Ans. .000042.

So, four and 357 millionths,

Ten and nineteen ten-thousandths,
Twenty and eighty-nine billionths,

EXERCISE.

4.000357 10.0019 20.000000089

1. How many figures are required, to express thousandths ($177)? To express millionths? Billionths? Hundredths? Tenthousandths? Hundred-trillionths? Ten-millionths?

2. Give the denominators of the following decimals:- .001; .00001; .19; 4.1; .000003; 15.62; .3333; 5.162.

3. Write the following as decimals, letting the decimal points range in line:-37 thousandths; 3 hundredths; 48 millionths; 95 hundred-millionths; 490 hundred-thousandths; 1240 ten-millionths; 10000004 hundred-millionths; 96 billionths; 9301 hundred-millionths; 2711 trillionths.

4. Eight hundred and forty-one thousand ten-millionths. 5. Eighty-thousand,* four hundred and two millionths. 6. Seventy-one million three thousand and four billionths. 7. Eight hundred and ninety-six thousand hundred-millionths. 8. Forty-nine thousand,* and seven hundred-thousandths. 9. Sixty billion and fourteen thousand trillionths. 10. Eight hundred million and ninety-nine ten-billionths. 11. Seventeen thousand and forty-one ten-trillionths. 12. ; ; THO; TO 400; 180; 168550.

3241

6357

Numeration of Decimals.

181. RULE.-Read the numerator first as a whole number, then name the denominator, as in common fractions. .09 is read Nine hundredths.

.090018

Ninety-thousand and eighteen millionths. 70.000000401 Seventy, and four hundred and one billionths.

181. Recite the rule for reading decimals.

The comma is here used to show that what precedes it is whole number.

Ex.-Add 9.0421, .42386, 881, .03, and 23.5945.

That we may unite things of the same kind, we write the numbers down with their decimal points ranging perpendicularly in line, which brings figures of the same order in the same column. Add as in whole numbers, and place a decimal point in the result under the points in the numbers added.

RULE.-1. Write the numbers with their decimal points ranging in line.

9.0421

.42386

881.

.03

23.5945

Ans. 914.09046

Add as in whole numbers. Place the decimal point in the result under the points in the numbers added. 2. Prove by adding from the top downward.

1. Add .123, .11496, 4.01, .06784, and 9.0342.

Ans. 13.35.

2. What is the value of .4397+219.31+3.00067+.043851 +5675.159+99.0004759+.15006342 ? Ans. 5997.10376032.

3. 8.8+450.329+.988927+87.71+.9+.272073.

4. .999999+9.887706+.07809+88.199+.4.

5. 7.71+.853 +9.6+96.96+.096+960+.54.

6. 105.501+0.105+8.648301+.19+.776655432+.3.

7. Find the sum of 2063 millionths; 3064 ten-thousandths; 99 hundredths; 500, and 6009 hundred-thousandths; seven, and 12 millionths; and 863003 billionths. Ans. 508.359428003.

182. Set down the given example in addition of decimals, and perform the operation. Give the rule.

8. Add five thousandths; nineteen, and eighteen millionths; five hundred and twenty hundred-thousandths; forty, and seven tenths; 87 hundredths; 919 ten-thousandths. Ans. 60.672118. 9. Required the sum of nineteen tenths; four hundred, and two hundredths; ninety-three thousandths; one hundred-thousandth. 10. Add, as decimals, 8180; 101000; 100000; 10000; 21000000

497

5617

183. Subtraction of Decimals.

EXAMPLE.-From 4.19 subtract .000001. Write the subtrahend under the minuend, with the decimal points in line. Write naughts in the vacant places of the minuend (or supply them mentally), and subtract as in whole numbers. Place a decimal point in the remainder under the other points.

4.190000

.000001

Ans. 4.189999

RULE.-1. Write the subtrahend under the minuend, with their decimal points ranging in line. Subtract as in whole numbers. Place the decimal point in the remainder under the other decimal points.

2. Prove by adding subtrahend and remainder.

5. Subtract 47.99999 from 831.012.

6. From .8754321 take .0006.

7. From 9.3 take the sum of .47 and 2.961.

8. Subtrahend, .88637; minuend, 312.42; mainder.

Ans. 783.01201.
Ans. .8748321.

Ans. 5.869. required, the re

Ans. 999.995.

9. From one thousand take five thousandths. 10. Take 11 hundred-thousandths from 117 thousandths. 11. From three million and one millionth, subtract one tenth. 12. Find the value of 2.4+.009+.73 — 1.8.

13. From eight and three tenths take eighty-four hundredths.

183. Set down the given exampie in subtraction of decimals. Perform the operation. Give the rule.