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OPERATION.

reserved in the last product added, are 36 tens, or 3 hundreds and 6 tens; we write the 6 tens in the product in tens' place, and reserve the 3 hundreds to add to the next product; 8 times 3 hundreds are 24 hundreds, and the 3 hundreds reserved in the last product added, are 27 hundreds, which being written in the product, each figure in the place of its order, gives, for the entire product, 2768. 2. Multiply 758 by 346.

ANALYSIS. In this example the multiplicand is 758

to be taken 346 times, which may be done by 346 taking the multiplicand separately as many times 4548

as there are units expressed by each figure of the 3032

multiplier. 758 multiplied by 6 units is 4548 2274 units, (II); 758 multiplied by 4 tens is 3032 tens, 262268

(II), which we write with its lowest order in tens'

place, or under the figure used as a multiplier; 758 multiplied by 3 hundreds is 2274 hundreds, (II), which we write with its lowest order in hundreds’ place. Since the sum of these products must be the entire product of the given numbers, (III), we add the results, and obtain 262268, the answer.

Notes.-1. When the multiplier contains two or more figures, the several results obtained by multiplying by each figure are called partial products.

2. When there are ciphers between the significant figures of the multiplier, pass over them, and multiply by the significant figures only.

83. From these principles and illustrations we deduce the following general

RULE. I. Write the multiplier under the multiplicand, placing units of the same order under each other.

II. Multiply the multiplicand by each figure of the multiplier successively, beginning with the unit figure, and write the first figure of each partial product under the figure of the multiplier used, writing down and carrying as in addition,

III. If there are partial products, add them, and their sum will be the product required

Note. - The multiplier denotes simply the number of times the multiplicand is to be taken; hence, in the analysis of a problem, the multiplier must be con. sidered as abstract, though the multiplicand may be either abstract or concrete.

84. PROOF. There are two principal methods of proving multiplication

OPERATION.

Partial

1st. By varying the partial products.

Invert the order of the factors; that is, multiply the multiplier by the multiplicand; if the product is the same as the first result, the work is correct.

2d. By excess of 9's.

85. The illustration of this method depends upon the following principles :

I. If the excess of 9's be subtracted from a number, the remainder will be a number having no excess of O's.

II. If a number having no excess of 9's be multiplied by any number, the product will have no excess of 9’s. 1. Let it be required to multiply 473 by 138.

ANALYSIS

The excess of 473 = 468 + 5

9's in 473 is 5, and 473 = 468 138 = 135 + 3

+ 5, of which the first part, 468 x 135 = 63180

468, contains no excess of 9's, 5 x 135

675 (I). The excess of 9's in products. 468 X 3 1404 138 is 3, and 138 135 + 3, of 5 X 3

15 which the first part, 135, conEntire product,

65274

tains no excess of 9's, (I).

Multiplying both parts of the multiplicand by each part of the multiplier, we have four partial products, of which the first three have no excess of 9's, because each contains a factor having no excess of 9's, (II). Therefore, the excess of 9's in the entire product must be the same as the excess of 9's in the last partial product, 15, which we find to be 1+5=6. The same may be shown of any two numbers. Hence, to prove multiplication by excess of 9's,

Find the excess of 9's in each of the two factors, and multiply them together; if the excess of 9's in this product is equal to the excess of 9's in the product of the factors, the work is supposed to be right.

Note.-If the excess of 9's in either factor is 1, the excess of 9's in the product will be 0, (II).

EXAMPLES FOR PRACTICE. (1.) (2.)

(3.)

(4.) Multiply 475 3172

9827

7198 Ву

9
14
84

216 Prod. 4275 44408 825468 1554768

5. Multiply 31416 by 175.

Ans. 5497800 6. Multiply 40930 by 779.

Ans, 31884470. 7. Multiply 46481 by 936. 8. Multiply 15607 by 3094. 9. Multiply 281216 by 978.

Ans. 275029240 10. Multiply 30204 by 4267.

Ans. 128,880,468. 11. What is the product of 4414 X 2341 ?

Ans. 10,403,404. 12. What is the product of 4567 x 9009 ?

Ans. 41,144,103. 13. What is the product of 2778588 x 9867?

Ans. 27,416,327,796. 14. What is the product of 7060504 x 30204 ?

Ans. 213,255,462,816. 15. What will be the cost of building 276 miles of railroad at $61320 per mile?

Ans. $16,924,320. 16. If it require 125 tons of iron rail for one mile of railroad, how many tons will be required for 196 miles?

17. A merchant tailor bought 36 pieces of broadcloth, each piece containing 47 yards, at 7 dollars a yard; how much did he pay for the whole ?

Ans. $11,844. 18. The railroads in the State of New York, in operation in 1858, amounted to 2590 miles in length, and their average cost was about $52916 per mile; what was the total cost of the railroads in New York ?

Ans. $137,052,440. 19. The Illinois Central Railroad is 700 miles long, and cost $45210 per

what was its total cost? 20. The salary of a member of Congress is $3000, and in 1860 there were 303 members; how much did they all receive ?

21. Th. United States contain an area of 2988892 square miles, and in 1850 they contained 8 inhabitants to each square mile; what was their entire population ?

Ans. 23,911,136. 22. Great Britain and Ireland have an area of 118949 square miles, and in 1850 they contained a population of 232 to the square mile; what was their entire population ? Ans. 27,596,168.

23. The national debt of France amounts to $32 for each indi.

mile;

vidual, ana the population in 1850 was 35781628; what was the entire debt of France ?

Ans. 1,145,012,096.

POWERS OF NUMBERS.

86. We have learned (15) that a power is the product arising from multiplying a number by itself, or repeating it any number of times as a factor; (16), that a root is a factor repeated to produce a power; and (40) an index or exponent is the number indicating the power to which a number is to be raised.

87. The First Power of any number is the number itself, or the root; thus, 2, 3, 5, are first powers or roots.

88. The Second Power, or Square, of a number is the product arising from using the number two times as a factor; thus, 22 -= 2 X 2= 4; 52= 5 x 5= 25.

89. The Third Power, or Cube, of a number is the product arising from using the number three times as a factor; thus, 43= 4 X 4 X 4= 64.

90. The higher powers are named in the order of their numbers, as Fourth Power, Fifth Power, Sixth Power, etc. 91. 1. What is the third power or cube of 23 ?

ANALYSIS. We multiply 23 23 x 23 x 23

by 23, and the product by 23;

and, since 23 has been taken 3 times as a factor, the last product, 12167, must be the third power or cube of 23. Hence,

RULE. Multiply the number by itself as many times, less 1, as there are units in the exponent of the required power.

Note.-The process of producing any required power of a number by multiplication is called Involution.

OPERATION.

= 12167

EXAMPLES FOR PRACTICE.

Ans. 5184. Ans. 248832.

1. What is the square of 72 ?
2. What is the fifth power of 12?
3. What is the cube of 25 ?
4. What is the seventh power of 7?

Ans. 823543.

5. What is the fourth power of 19 ? Ans. 130321. 6. Required the sixth power of 3.

Ans. 729. 7. Find the powers indicated in the following exprussions : 95, 118, 182, 1254, 786", 946, 1004, 175, 251.5 8. Multiply 88 by 15%.

Ans. 115200. 9. What is the product of 252 x 34? 10. 73 x 200 41 x 112, and how many ? Ans. 37.624.

GENERAL PRINCIPLES OF MULTIPICATION.

92. There are certain general principles of multiplication, of use in various contractions and applications which occur in subsequent portions of this work. These relate, 1st, to changing the factors by addition or subtraction; 2d, to the use of successive factors in continued multiplication.

CHANGING THE FACTORS BY ADDITION OR SUBTRACTION.

93. The product is equal to either factor taken as many times as there are units in the other factor. (82, I). Hence,

I: Adding 1 to either factor, adds the other factor to the product.

II. Subtracting 1 from either factor, subtracts the other factor from the product. Hence,

III. Adding any number to either fuctor, INCREASES the product by as many times the other factor as there are units in the number added ; and SUBTRACTING any number from either factor, DIMINISHES the product by as many times the other factor as there are units in the number subtracted.

CONTINUED MULTIPLICATION. 94. A Continued Multiplication is the process of finding the product of three or more factors, by multiplying the first by the second, this result by the third, and so on.

95. To show the nature of continued multiplication, we observe:

1st. If any number, as 17, be multiplied by any other number, as 3, the result will be 3 times 17; if this result be multiplied by

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