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10 45 100 125 1000 1047 10000
.1047. This manner of writing decimal fractions is a mere language, and is used to avoid the inconvenience of writing the denominators. The denominator, however, of every decimal fraction is always understood. It is a unit 1 with as many ciphers annexed as there are places of figures in the numerator.
The place next to the decimal point is called tenths; the next place to the right, the place of hundredths; the next, the place of thousandths; and so on, for places further to the right, according to the following table.
DECIMAL NUMERATION TABLE
163. Let us now write and numerate the following decimals.
163. Does the value of a figure depend upon the place which it occupies? How does the value change from the left towards the right? What do ten parts of any one place inake? How do they increase from the right towards the left? How may whole numbers be joined with decimals? What is a number called when composed partly of whole numbers and partly of decimals?
4 100 4
.4 Four hundredths,
.04 Four thousandths,
.004 Four ten-thousandths,
.0004 Four hundred thousandths,
.00004 Four millionths,
.000004 Four ten-millionths,
.0 0 0 0 0 0 4. Here we see, that the same figure expresses different values, according to the place which it occupies. But of * is equal to Tôo=.04 to of Ιοοο
1000000=.000004 of 1000000
=.0000004. Therefore the value of the parts of a unit, expressed by the different figures in passing from the left to the right, diminishes in a tenfold proportion.
Hence, ten of the parts in any one of the places, are equal to one of the parts in the place next to the left; that is, ten thousandths make one hundredth, ten hundredths make one-tenth, and ten-tenths a unit 1.
This law of increase from the right hand towards the left, is the same as in whole numbers; therefore,
Whole numbers and decimal fractions may be united by placing the decimal point between them. Thus, Whole numbers.
@ Tens of millions.
Tens of thousandths.
Tens of millionths.
UNITS. o Tenths.
8 3 6 3
A number composed partly of a whole number and partly of a decimal, is called a mixed number.
Write the following numbers in figures, and numerate them. 1. Forty-one, and three-tenths.
41.3. 2. Sixteen, and three millionths.
16.000003. 3. Five, and nine hundredths.
5.09. 4. Sixty-five, and fifteen thousandths. 65.015. 5. Eighty, and three millionths.
80.000003. 6. Two, and three hundred millionths. 7. Four hundred and ninety-two thousandths. 8. Three thousand, and twenty-one ten thousandths. 9. Forty-seven, and twenty-one ten thousandths. 10. Fifteen hundred and three millionths. 11. Thirty-nine, and six hundred and forty thousandths. 12. Three thousand, eight hundred and forty millionths. 13. Six hundred and fifty thousandths. 14. Fifty thousand, and four hundredths. 15. Six hundred, and eighteen ten thousandths. 16. Three millionths. 17. Thirty-nine hundred thousandths.
164. The denominations of Federal Money will cor. respond to the decimal division, if we regard 1 dollar as the unit. For, the dimes are tenths of the dollar, the cents are hundredths of the dollar, and the mills, being tenths of the cent, are thousandths of the dollar.
1. Express $16, 3 dimes 8 cents and 9 mills deci. mally,
Ans. $16.389. 2. Express $95, 8 dimes 9 cents 5 mills decimally.
Ans. 3. Express $107, 9 dimes 6 cents 8 mills decimally.
164. If the denominations of Federal Money be expressed decimally what is the unit? What part of a dollar is 1 dime? What part of a dime is a cent? What part of a cent is a mill? What part of a dollar is 1 cent? 1 mill?
4. Express $47 and 25 cents decimally.
Ans. $47.25. 5. Express $39,39 cents and 7 mills decimally.
Ans. $39.397. 6. Express $12 and 3 mills decimally. Ans. 7. Express $147 and 4 cents decimally.
Ans. $147.04. 8. Express $148, 4 mills decimally. Ans. $148.004. 9. Express four dollars, six mills decimally.
165. A cipher is annexed to a number, when it is placed on the right of it. If ciphers be annexed to the numerator of a decimal fraction, the same number of ciphers must also be annexed to the denominator ; for there must be as many ciphers in the denominator as there are places of figures in the numerator (ART. 162). The numerator and denominator will therefore have been multiplied by the same number, and consequently the value of the fraction will not be changed (Art. 122). Hence,
Annering ciphers to a decimal fraction does not alter its value.
We may take as an example, .3=1. If now we annex a cipher to the numerator, we must, at the same time, annex one to the denominator, which gives,
.30 1900 by annexing one cipher,
= .3000= 3.000, all of which are equal to %=.3. Also, .5=to=.50==.500= 1000%. Also, .8=.80=.800=.8000=.80000.
166. Prefixing a cipher is placing it on the left of a number. If ciphers be prefixed to the numerator of a decimal fraction, the same number of ciphers must be annexed to the denominator. Now, the numerator will
165. When is a cipher annexed to a number? Does the annexing of ciphers to a decimal alter its value? Why not? What does three-tenths become by annexing a cipher? What by annexing two ciphers? Three ciphers? What does .8 become by annexing a cipher? By annexing two ciphers ? By annexing three ciphers ?
remain unchanged while the denominator will be increased ten times for every cipher which is annexed, and the value of the fraction will be decreased in the same proportion (Art. 120). Hence,
Prefixing ciphers to a decimal fraction diminishes its value ten times for every cipher prefixed. Take, for example, the fraction .2=lo.
.02 102 by prefixing one cipher,
100% by prefixing two ciphers,
-10000 in which the fraction is diminished ten times for every cipher prefixed.
ADDITION OF DECIMAL FRACTIONS.
167. It must be recollected that only like parts of the same unit can be added together, and therefore in setting down the numbers for addition the figures occupying places of the same value must be placed directly under each other.
The addition of decimal fractions is then made in the same manner as that of whole numbers.
For example, add 37.04, 704.3, and .0376 together.
In this example, we place the tenths under tenths, the hundredths under 37.04 hundredths, and this brings the decimal 704.3 points and the like parts of the unit
.0376 directly under each other. We then
741.3776 add as in whole numbers.
166. When is a cipher prefixed to a number? When prefixed to a decimal, does it increase the numerator? Does it increase the denominator? What effect then has it on the value of the fraction ? What do .5 become by prefixing a cipher? By prefixing two ciphers ? By prefixing three? What do .07 become by prefixing a cipher ? By prefixing two? By prefixing three ? By prefixing four ?
167. What parts of unity may be added together? How do you set down the numbers for addition? How will the decimal points fall? How do you then add ? How many decimal places do you point off in the sum ?