Thus it appears, that from any given place in whole numbers to any given place in decimals, is a regular descending series, formed by a uniform divisor. The right-hand place is the quotient of the left divided by 10. The first decimal place is the quotient of the unit's place divided by 10, and is called the tenth's place. The decimal point, therefore, occupies a position between the unit's place and its quotient. The second place is the quotient of the tenth's place divided by 10, or the unit's place divided by 100, and is called the hundredth's place. Thus decimal fractions, like whole numbers, have a local value, and are subject to the same law of increase from the right hand towards the left. As 1, in the place of tens, is equal to 10 in the unit's place, so 1 in the place of units is equal to 10 in the place of tenths. From this circumstance, we may know the value of those parts of the unit contained in the numerator, although the denominator be not expressed. This property of a decimal fraction also distinguishes it from a vulgar fraction, for there is no place on either side of the unit where the numerator of a vulgar fraction can be placed, which will give name to the fraction; its denominator must, therefore, always be expressed.

Although ciphers placed at the right hand of a decimal fraction do not affect its value, yet, placed at the left, they diminish it in a tenfold proportion, by removing the significant figure so much farther from the unit's place. Thus, .5 .05 .005 express different values, viz.-.5 is .05 is 10%, .005 is 1000 Write denominators to the following decimals: .5; .25; .026; .3245; .56783; .789024.

05

Write the following without their denominators. 1. Twenty-five hundredths.

2. Four hundred and fifty-two thousandths.

3. Five hundred and sixty ten thousandths. 4. Sixty-two hundred thousandths.

5. Forty-five millionths.

6. Eighty-seven billionths.

Ans. .25.

Ans. .452.

7. Ninety-eight trillionths.

8. Twenty-five, and four thousandths.

As whole numbers are written, units under units, tens under tens, from right to left, so decimals are written tenths under tenths, from left to right.

EXAMPLES.

1. Write 2 tenths; 3 hundredths; 4 thousandths; 6 ten thousandths.

.2

.03

.004
.0006

2. Write twenty-nine thousandths; three hundred and fourteen thousandths; five ten thousandths, and sixty-seven millionths.

.029

.314

.0005

.000067

3. Write five tenths; five hundredths; fifty thousandths, and forty-nine; one hundred thousandths, and sixteen thousandths.

4. Write forty-five and five tenths; six hundred and fortyfive and four thousandths; twenty-nine and four thousandths; sixty-seven and forty-seven thousandths.

5. Write four hundred and fifty-three, and fifty-seven ten thousandths; five thousand and five hundredths; twenty-four and three millionths; thirty-six and eighty-two billionths.

ADDITION OF DECIMALS.

Art. 89.-1. Write one hundred and one tenth; twenty and two hundredths; five units and five thousandths, and add them together.

Operation.

100.1

20.02

5.005

125.125 Ans.

As whole numbers can only be added by writing them in their proper places and uniting those of the same name; so decimals, when written tenths in the place of tenths, hundredths in the place of hundredths, etc., are added by uniting those

of the same name or denomination. The amount, both in decimals and whole numbers, takes its name from the lowest, or right-hand place of the numbers added: thus, 1 hundred, 2 tens and 5 units, when added, are read 125 units; and one tenth, 2 hundredths and five thousandths, when added, are read, 125 thousandths.

Decimal fractions may also be added and illustrated in the same manner as vulgar fractions.

2. Add two and five tenths; four and six hundredths; seven and three thousandths.

Then

2.5-25, and 4.06-106, and 7.003-7003.

25 × 100-2500, and 106×10=1060.

OBS.-Multiplying the terms of a fraction by the same quantity does not alter its value. (See Art. 61.)

From the foregoing it is evident, that decimal fractions are reduced to a common denominator by writing tenths in the place of tenths, and hundredths in the place of hundredths, and supposing those decimal places, which are deficient, to be supplied by ciphers.

Applying the decimal point to the amount, is equivalent to dividing it by its own denominator, which we have seen is the denominator of the lowest of the given decimals, or that decimal whose denominator is the largest. But the decimal places in the numerator of a decimal fraction, are equal to the number of ciphers in its denominator, the denominator being understood; therefore, addition of decimals may be performed by the following

QUESTIONS.-16. How is the first decimal place produced? 17. The second, third, &c.? 18. How are decimals to be added, written? 19. From what does the amount take its name? 20. Applying the decimal point is equal to what? 21. How are decimal fractions reduced to a common denominator?

RULE.

Place the numbers, tenths under tenths, hundredths under hundredths, etc.; or, so that the decimal points may stand directly under each other. Add as in whole numbers; observing to point off as many places for decimals in the amount, as will be equal to the greatest number of decimals in any of the given numbers.

4. Add thirty-five and four tenths; five hundred twentynine and seven millionths; sixty-nine, four hundred and sixtythree thousandths; two hundred, sixteen and two hundredths; seventy-seven, nine hundred and two tenths.

Ans. 1827.083007.

5. Add forty-nine and sixty-seven hundredths; six hundred seventy-nine, two hundred seventy-five thousandths; one thousand four hundred, fifty-five thousandths, nine hundred and ninety-nine millionths.

6. Add 249.39; 6712.9123; 6.3219; 2739.235; 5.671; 723.2674; 926.679; 72.601.

7. Add .7+9.2+.321+279.+4.67+349.2+3.956.

8. Purchased of one man 325.5 lbs. of beef; of another, 175.75; of another, 178.028; what was the amount?

9. I receive of A. $183.25; of B. $138.89; of C. $372.218; of D. $88.99; of E. $137.29; what is the amount of the whole ?

Add $59.67; $158.355; $375.752; $167.375;

10. $567.756.

SUBTRACTION OF DECIMALS.

Art. 90.-1. From three and two tenths, take one and five tenths.

Operation.

3.2

1.5

1.7 Ans.

Because five tenths cannot be taken from two tenths, we borrow 1 from the unit's place, which, reduced to tenths, equals 10 tenths; 18+2=12, then 12-==.7. Lastly, 1 from 2, and 1 remains. Again, three and two tenths is the quotient of 32 divided by 10, (see definition of a mixed number, Art. 54;) therefore, 3.2=32, and 1.5-15; then 32-15=17=1.7 Ans., as before. Pointing off the remainder is dividing it by its own denominator. Hence the

RULE.

Write the numbers and point the result, as in Addition of Decimals, and subtract as in whole numbers.

6. From two hundred and sixty-nine and three tenths, take fifty-seven and thirty-nine hundredths.

Ans. 211.91.

7. Take twenty-four thousandths from nine hundredths.

Ans. .066.

8. Take sixty-five millionths from five tenths. 9. From three hundred seventy-five thousand and three tenths, take two hundred forty-nine and thirty-nine one hundred thousandths. Ans. 374751.29961.

10. From 361.2 take 276.75.

11. From 456.35 take 27.356.

12. From 5678.0002 take 3980.96715.

QUESTIONS.-22. How are decimals to be subtracted, written? 23. How can five tenths be taken from two tenths? 24. What is done with the unit borrowed?