a single repetend, and a point over the first and last figures of a compound repetend.

The first place, next after the decimal mark, is 10th parts, the second is 100th parts, the third is 1000th parts, and so on, decreasing toward the right by Icths, or in creasing toward the left by roths, the same as whole or integral numbers do. As in the following

∞ Hundreds of thousands.

∞ Tens of thousands. Thousands.

∞Hundreds.

∞ Tens.

Hundred thousandth parts.
&c.
Millionth parts.

∞ Ten thousandth parts.
∞ Hundredth parts.
∞ Thousandth parts.
∞ Tenth parts.
∞ Units,

88888888888

Cyphers on the right of decimals do not alter their value.

Meri mar

But cyphers before decimal figures, and after the separating point, diminish the value in a tenfold proportion for

So that, in any mixed or fractional number, if the separating point be moved one, two, three, &c. places to the right hand, every figure will be 10, 100, 1000, &c. times greater than before.

But if the point be moved toward the left hand, then every figure will be diminished in the same manner, or the whole quantity will be divided by 10, 100, 1000, &ς.

ADDITION

ADDITION of DECIMALS.

RULE.

1. Set the numbers under each other according to the value of their places, as in whole numbers, or so that the decimal points may stand each directly under the pre ceding.

2. Then add as in whole numbers, placing the decimal point, in the sum, directly under the other points.

2. What is the sum of 276, 39 213, 72014.9, 417, 5032 and 2214 298? Ans. 79993.411. 3. What is the sum of 014, 9816, 32, 15914, 72913

and 0047? Ans. 2.20857. 4. What is the sum of 27:148, 918.73, 14016, 294304,

7138 and 2217 ? Ans. 309488 2918. 5. Required the sum of 312.984, 213918, 2700 42, 3.153, 27.2 and 58106.

1

Ans. 3646 2088.

SUBTRACTION of DECIMALS,

RULE.

1. Set the less number under the greater in the same manner as in addition.

2. Then subtract as in whole numbers, and place the decimal point in the remainder directly under the other points.

EXAMPLES.

4. What is the difference between 91713 and 407?

Ans. 315-287.

5. What is the difference between 16.37 and 800 135,?

Ans. 783.765.

MULTIPLICATION of DECIMALS.

RULE.*

1. Set down the factors under each other, and multiply them as in whole numbers.

2. And from the product, toward the right hand, point off as many figures for decimals, as there are decimal places in both the factors. But if there be not so many figures in the product as there ought to be decimals, prefix the proper number of cyphers to supply the defect.

EXAMPLES.

* To prove the truth of the rule, let 9776 and 823 be the to be multiplied; now these are equivalent to and

numbers

823

823

0776

18045648 by the nature of notation, and consisting of as many places, as there are cyphers, that is, of as many places as are in both the numbers;

EXAMPLES.

(1) 91.78 38

9178

73424

27534

34.96818

2. What is the product of 520.3 and 417?

Ans. 216.9651.

3. What is the product of 516 and 21 ? Ans. 1083.6. 4. What is the product of 217 and 0431?

Ans. 0093527.

Ans. 0004641.

5. What is the product of 051 and 0091 ?

NOTE. When decimals are to be multiplied by ic, or 100, or 1000, &c. that is, by I with any number of cyphers, it is done by only moving the decimal point so many places further to the right hand, as there are cyphers in the said multiplier; subjoining cyphers if there be not so many figures.

When the product would contain several more decimals than are neccessary for the purpose in hand, the work may be much contracted, and only the proper number of decimals retained.

RULE.

1. Set the unit figure of the multiplier under such decimal place of the multiplicand as you intend the last of your product product shall be, writing the other figures of the multiplier in an inverted order.

2. Then, in multiplying, reject all the figures in the multiplicand, which are on the right of the figure you are multiplying by; setting down the products so that their right-hand figures may fall each in a straight line under the preceding; and carrying to such right-hand figures from the product of the two preceding figures in the multiplicand thus, viz. I from 5 to 14, 2 from 15 to 24, 3 from 25 to 34, &c. inclusively; and the sum of the lines will be the product to the number of decimals required, and will commonly be the nearest unit in the last figure.

EXAMPLES.

1. Multiply 27 14986 by 92 41035, so as to retain only

four places of decimals in the product.

2. Multiply 480 14936 by 2'72416, retaining four deci

mals in the product.

Ans. 13080036.

Ans. 341 80097.

3. Multiply 73 8429753 by 4.628754, retaining five decimals in the product.

4. Multiply 8634-875 by 843 7527, retaining only the integers in the product.

1

Ans. 7285699.

DIVISION

L