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multiply 697 by 3, we remark that since 697 is to be repeated 3 times, it may be done by writing it down 3 times, and then addling, thus :

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And it is obvious that all questions of multiplication may be performed by addition.

Hence, multiplication is sometimes defined as being a coscise way of performing several additions.

NOTE.-When a zero or 0 occurs in the multiplier, we may observe that its pro duct must remain 0, since nothing repeated any number of times is still nothing.

PROOF OF MULTIPLICATION.

17. If we interchange the multiplier and multiplicand, and then multiply, we shall obtain the same product if the work is right. (See Art. 15.)

As in addition, these two results may be alike, and still the work may be wrong, since mistakes may occur in both operations. As good proof as any, is to carefully repeat the multiplication.

When 0 is multiplied by any number, what is the result? How is multiplication wmecimes defined. How may multiplication be proved? Is this method infallible 1 Why not? What is as good proof as any other ?

CASE L

18. When the multiplier consists of only one figure.

From what has already been done, we deduce this

RULE.

Place the multiplier under the unit figure of the muitipricand. Draw a horizontal line underneath.

Then multiply each figure of the multiplicand by the mub tiplier, observing to carry one for every ten, as in addition.

When the multiplier consists of but one figure, how do you proceed? What rulo to you observe in carrying ?

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CASE I. 19. When the multiplier consists of more than one figure

RULE

1. Place the multiplier under the multiplicand, so that units may stand under units, tens under tens, hundreds under hundreds, fc.

II. Multiply successively by each figure of the multiplier, as in Case I., observing to place the right-hand figure of each partial product directly under the figure multiplied by.

III. Then add together these partial products, and the sum will be the total product sought.

When the multiplier consists of more than one figure, how do you write it? How do you then multiply? How do you add up?

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4. Multiply 12345 by 12.

Ans. 148140. 5. Multiply 23456 by 11

Ans. 258016. 6. Multiply 34567 by 13.

Ans. 449371. 7. Multiply 780056 by 21. Ans. 16381176. 8. Multiply 6503456 by 234. Ans. 1521808704. 9. Multiply 3471032 by 70056. Ans. 243166617792. 10. Multiply 1240578 by 302014.

Ans. 374671924092.

11. Multiply 235678 by 753465.

Ans. 177575124270. 12. Multiply 98610275 by 35789.

Ans. 3529163131975.

CASE III.

20. When the multiplier, or multiplicand, or both, have one or more ciphers at the right.

We know from what has been said, (Art. 4,) tinat multiplying by 10 is the same as annexing a cipher to the right of the figure or sum to be multiplied ; multiplying by 100 is the same as annexing two ciphers to the right of the figure or sum to be multiplied, &c.

Hence we deduce this

RULE.

Multiply by the significant figures,(as in Case II.) and to the product annex as many ciphers as there are in both multiplier and multiplicand.

When there are ciphers at the right of the multiplier, or multiplicand, or both, bow do you proceed ?

EXAMPLES

1. Multiply 365 by 10.

Ans. 3650 2. Multiply 12040 by 100.

Ans. 1204000. 3 Multiply 204500 by 3000. Ans. 613500000. 4. Multiply 7003000 by 240000.

Ans. 1680720000000. 5 Multiply 307210000 by 3780000.

Ans. 1161253800000000.

CASE IV.

21. When the multiplier is a composite number.

A composite number is one which may be produced by multip.ying two or more numbers together. Thus: 35 is a composite number, which may be produced by multiply ing 5 and 7 together.

The 5 and 7 are called the factors or component parts of 35.

The factors of 12, are 3 and 4, or 2 and 6.
Suppose we wish to multiply 48 by 35.

If we first multiply 48 by 5, we find 240 for the product; if now we multiply this product by 7, we obtain 1680, which is evidently the same as 35 times 48.

Hence we infer this

RULE. Multiply the sum given by one of the factors, and this product by another factor, and so on, until all the factors are u sed. The last product will be the one sought.

EXAMPLES

-; 1. Multiply 365 by 28. ... The factors of 28 are 4 and 7. Hence we have this

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