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To find the value of a decimal in terms of the inferior denominations. RULE.-Multiply the decimal by the nun.ber of parts in the next lower denomination; and cut off as many places for a remainder, to the right hand, as there are places in the given decimal.

Multiply that remainder by the parts in the next lower denomination again, cutting off for another remainder as before.

Proceed in the same manner through all the parts of the integer; then the several denominations separated on the left hand, will make up the answer. Note. This operation is the same as Reduction Descending in whole numbers.

EXAMPLES.

1. Required to find the value of 775 pounds sterling.

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To reduce integers or decimals to equivalent decimals of higher denominations,

RULE Divide by the number of parts in the next higher denomination; continuing the operation to as many higher denominations as may be necessary, the same as in Reduction Ascending of whole numbers.

EXAMPLES.

1. Reduce 1 dwt. to the decimal of a pound troy

20

1 dwt.

12 0.05 oz.

0.004166, &c. lb. answer.

2. Reduce 9d. to the decimal of a pound.
3. Reduce 7 dr. to the decimal of a pound avoird.
4. Reduce 26d. to the decimal of a £.

5. Reduce 2-15 lb. to the decimal of a cwt.

6. Reduce 24 yards to the decimal of a mile.

7. Reduce 056 poles to the decimal of an acre.

8. Reduce 1-2 pints of wine to the decimal of a hhd. 9. Reduce 14 minutes to the decimal of a day. 10. Reduce 21 pints to the decimal of a peck.

Ans. 03757. Ans. 02734375 lb. Ans. 0010833, &c. £. Ans. 019196 + cwt. Ans. 013636, &c. miles. Ans. 00035 ac. Ans. 00238+ lihd. Ans. 009722, &c. da. Ans. 013125 pec.

NOTE. When there are several numbers, to be reduced all to the decimal of the highest.

Set the given numbers directly under each other, for dividends, proceeding orderly from the lowest denomination to the highest.

Opposite to each dividend, on the left hand, set such a number for a divisor as will bring it to the next higher name; drawing a perpendicular line between all the divisors and dividends.

Begin at the uppermost, and perform all the divisions; only observing to set the quotient of each division, as decimal parts, on the right hand of the dividend next below it; so shall the last quotient be the decimal required.

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RULE. I'repare the terms by reducing the vulgar fractions to decimals, any compound numbers either to decimals of the higher denominations, or to integers of the lower, also the first and third terms to the same name: then multiply and divide as in whole numbers.

Note. Any of the convenient examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals.-The following example, which is the first in Vulgar Fractions, is wrought here to show the method.

If 3 of a yard of velvet cost 27, what willyd. cost?

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DUODECIMALS, or CROSS MULTIPLICATION, is a rule made use of by workmen and artificers, in computing the contents of their works.

Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents.

RULE.-Set down the two dimensions, to be multiplied together, one under the other, so that feet stand under feet, inches under inches, &c.

Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet.

In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting however what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as these are of a foot.

Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many.

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INVOLUTION is the raising of Powers from any given number, as a root. A Power is a quantity produced by multiplying any given number, called the Root, a certain number of times continually by itself. Thus,

2= 2X2=

2 X 2 X 2 =

2 × 2 × 2 × 2 =

2 is the root, or first power of 2.

4 is the 2d power, or square of 2.

8 is the 3d power, or cube of 2. 16 is the 4th power of 2, &c.

And in this manner may be calculated the following Table of the first nine powers of the first nine numbers.

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7 49 343 2401
864 512 4096 32768 262144
9 81 729 6561 59049

16807

117649

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The Index or Exponent of a Power, is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the same. So 1 is the index or exponent of the 1st power or root, 2 of the 2d power or square, 3 of the 3d power or cube, 4 of the 4th power, and

so on.

Powers, that are to be raised, are usually denoted by placing the index above the root or first power.

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When two or more powers are multiplied together, their product will be that power whose index is the sum of the exponents of the factors or powers multi.

Note. Any of the convenient examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals.—The following example, which is the first in Vulgar Fractions, is wrought here to show the method.

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DUODECIMALS, or CROSS MULTIPLICATION, is a rule made use of by workmen and artificers, in computing the contents of their works.

Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents.

RULE.-Set down the two dimensions, to be multiplied together, one under the other, so that feet stand under feet, inches under inches, &c.

Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet.

In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting however what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as these are of a foot.

Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many.

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