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2. Multiply 728-68425 by 68-543, so as to have 4 places of decimals in the product.

3. Divide 16-538426 by 28.6765, so as to have 3 places of decimals in the quotient.

4. Divide 310.5 by 41.2363455, so as to have 4 places of decimals in the quotient.

METHOD 2.

437. By the use of reciprocal divisors. It has been shown by the note to paragraph 359, page 138, that fractions are reci procal when, by their multiplication into themselves, they produce unity. When fractions are reciprocal, one of them must represent a quantity more, and the other less, than unity.

If two fractions, that are reciprocal, be reduced to decimals, then these decimal expressions multiplied into themselves will produce unity, as = 1·25, and 8, then

1, and 1.25 X 8 = 1.000. then 25 X 2 = 1, and 25 ×

364 2.5
1.25

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•8)3642.5

455 3:125

Again, 2525, and 25·04, 041·00. It has also been shown by the above-mentioned note, that to divide by any quantity is the same as to multiply by its reciprocal; consequently, to multiply by any quantity is the same as to divide by its reciprocal. See operations.

182 1 2 5 437 10 0

4553.1 2 5

The numbers which are most suitable for this method of contraction are those which are here exhibited, together with their reciprocal decimals:

Numbers

25 125 250 625 1250

Reciprocal divisors ⚫04 •008 004 0016 0008

and are reciprocals; if the terms of one of these fractions be inverted, it becomes the other; so if any of the above reciprocal divisors be converted into a vulgar fraction, its terms inverted and reduced to a whole number, it becomes the corresponding reciprocal; thus 008= is, when inverted, 1000 = 125.

8

8 1000

438 Immediately connected with this, is a short mode of multiplying whole numbers; 125 is the of 1000, therefore if we annex three cyphers to the multiplicand, and divide by 8, we shall have the result.

439. As multiplication can be shortened by reciprocal divisors, so division can be shortened by reciprocal multipliers.

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440. By proportioning approximate products. This method will be easily understood by a few examples.

Multiply 295-6420 by 612.5 and by 5875.

29564.20 by removing the deci

6 mal two spaces to the

+3695·525 = 12 is the of 100.

29564.20

6

right, we multiply by

177385.20 100.

177385.20

-3695-525

173689.675

68436842.5

+684368 425

do.

10 do.

-684368 425

181080 725

Multiply 68436-8425 by 1010 and 990. 68436842 5 multiplied 1000 times

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Division by Approximate Divisors.

441. The chief use to which division by approximate divi sors can be applied is, when the given number is near 100, 1000, &c. It will be best understood from a few examples.

Divide 6881624.75 by 1001, and 6867875.25 by 999, so as to have two places of decimals.

6881-62475

6 881-
6+

6874.75

6867.87525

68678
68

6874.75

Note. Moving the decimal is equivalent to dividing by 1000.

Divide 69121210 925 by 1010, and 67752474.075 by 990.

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1. How are finite decimals multiplied? How are they divided?

2. How is the pointing in multiplication and division of decimals regulated?

3. Should the number of decimal places in the product not amount to the number in the multiplicand and multiplier taken together, what must be done?

4. Should the decimal places in the quotient not amount in number to the difference in number between those in the divisor and those in the dividend, what must be done?

5. How are simple repetends multiplied and divided?

6. What difference is there in the manner of carrying in the multiplication and division of finite, and in the multiplication and division of repeating decimals?

7. How do we multiply by repetends?

8. How do we divide by repetends?

9. By what short method can we multiply by a number composed of nines, or of nines and cyphers? By what short method can we divide by such numbers?

10. How are repetends multiplied and divided by repetends?

11. What are approximating decimals?

12. Do approximating decimals approach sufficiently near the truth for all common occurrences of business?

13. Let the student give examples in Addition, Subtraction, Multiplication, and Division, by approximating decimals.

14. By what modes may multiplication and division of decimals be contracted.

15. Let the student give examples in these different modes of contraction.

CHAPTER XVI.

OF POWERS AND THEIR INDICES OR
EXPONENTS.

443. The continued product of unity by any number is called the powers of that number: thus, 2. 4. 8. 16. 32. are the first, second, third, fourth, and fifth powers of 2. The indix or exponent of any power is a small figure written to the right, thus 22, and indicates the number of multiplications used in producing the power.

444. In the following table the lower line exhibits the continued products or powers of the number 2, and the upper line the corresponding exponents :-

Exponents,
Powers,

0. 1. 2. 3. 4. 5. 6. 7. 8.

1. 2. 4. 8. 16. 32. 64. 128. 256.

445. If we add together any two exponents in the above table, we shall find that the power indicated by the exponent of their sum is the product of the multiplication of the two powers indicated by the exponents so added together. Thus, the sum produced by the addition of the exponents 2 and 4 is 6; under the exponent 6 stands the power 64, which is the product of the multiplication of the powers 4 and 16, indicated by the exponents 2 and 4. This will be clearly seen by the following examples:

Exponents, 2 + 4 = 6
Powers, 4 X 1664

3+5=8

8 × 32

256

The difference of the exponents of any two powers will be the index of the quotient, arising from the division of one of these powers by the other. Thus the difference between 2 and 8 is 6; under 6, in the above table, stands 64 256÷4. From this it will be seen, that adding the exponents of two different powers of a number, is the same as multiplying the two powers of that number indicated by these exponents; and

that subtracting one exponent from another is equivalent to dividing their corresponding powers one by the other.

446. When unity is continually divided by any number, 2 for instance, the quotients arising, as 1, 1, 1, TO T

&c. are

also called powers of 2; but as they are formed by a process contrary to the former, their indices are distinguished from the former indices by having the sign (—) placed before them, and are called negative indices. These powers placed under their indices are exhibited in the following table, and have the saine relative properties as the former.

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Exponents, 0.
Powers, 1. 2.

I 4.

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1. 1. 10. 12. 64. 718. 236.

447. In all operations the effect of negative exponents is contrary to that of positive ones. Thus if a negative and positive power have to be multiplied together, we may take the difference of their exponents for the exponent of the product required.

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448. Hence a negative power and a positive power, whose exponents are the same number, are reciprocals, for when multiplied into each other they produce unity; and the exponents subtracted leave 0, which is the exponent of unity.

449. When we have numbers to multiply into themselves a certain number of times, their products are called powers of that number, as has just been shown. Thus 2 X 2 = 4 is the second power of 2, or the square of 2, and may be represented by 22. 2 X2 X 2 = 8, is the third power of 2, or the cube of 2, and may be represented by 23. 2 × 2 × 2 × 2 may be represented by 21, and so on. Now if we would multiply these two last quantities, viz., 23 and 24, we have only to add their exponents. Thus 23 X 21 — 23 + + — 27 = 8 × 16 = 2 × 2 × 2 × 2 × 2 × 2 × 2 = 128. 22 × 2, or, which is the same, 22 × 21 = 23 = 2 × 2 × 2 = 4 × 28. To divide 2+ by 23 is to subtract their exponents, or to add to 21 the negative third power of 2. Thus 2+23 =2432 168, or 2+ 23 24 3 = 2 = 16 × } = 5 = 2.

450. As the continued products of any number are called powers of that number, so the number by whose continued product a power is produced, is called the root of that power. Thus, 4 is the square or second power of 2, because 2 x 2 = 4. 8 is the cube of 2, because 2 × 2 × 2 = 8; and 2 is the square root of 4; because 2 multiplied by itself, or 2 squared,

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