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When the multiplier is a number consisting of several figures, after we have found the product of the multiplicand by the first figure of the multiplier, we suppose the multiplier to be divided into parts, and after the same manner, find the product of the multiplicand by the second figure of the multiplier; but as the figure, by which we are multiplying, stands in the place of tens, the product must be ten times its number; and, therefore, the first figure in this product must be written in the place of tens, or, which is the same, directly under the figure by which we are multiplying. And proceeding in the same manner with all the figures of the multiplier, separately, it is evident, we shall multiply all the parts of the multiplicand by all the parts of the multiplier; therefore, the sum of these several products will be the whole product required.

If there are ciphers between the significant figures of the multiplier, write them in a line with the product of the next siguificant figure, directly under their places in the multiplier. As, in Example 2d.

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* The reason of the method of proof by multiplication, depends upon this proposition, that if two numbers are to be multiplied together, either of them may be made the multiplier or the multiplicand, and the product will be the same. Frample. 6×4—24, and 4x6=24.

CASE II.

When there are ciphers on the right hand of either the multiplicand or multiplier, or both.

RULE. Neglect those ciphers; then place the significant figures under one another, and multiply by them only; add them together, as before directed, and place to the right hand as many ciphers as there are in both the factors.

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4. What is the value of a farm of 600 acres, at 20 dollars an acre!

5. Multiply 50.750.000 by 75.000.

Ans. 12.000 dollars

Ans. 3.806.250.000.000.

39. How do you proceed when any of the right hand figures of the multiplicand or multiplier are ciphers?

CASE III.

To multiply by 10, 100, 1000, &c.

RULE.-Set down the multiplicand underneath, and join the ciphers in the multiplier to the right hand of them.*

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When the multiplier is a composite number, (or exactly equal to the product of any two figures in the multiplication table

RULE.-Multiply first by one of those figures, and that product by the other, the last product will be the answer.†

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* This is evident from the nature of numbers, since every cipher annexed to the right of a number increases the value of that number in a tenfold proportion.

†The reason of this rule is obvious; for any number multiplied by the component parts of another, must give the same product as if it were multiplied by that number at once.

40. What is the rule when the multiplier is 10, 100, 1900, &c. ?41. When the multiplier is a composite number, what is your rule?

D

CASE V.

To multiply by 9, 99, 999, &c.

RULE.-Annex as many ciphers to the right of the multiplicand as there are figures in your multiplier, and from the number thus produced, subtract the given multiplicand, and the remainder will be the product.

EXAMPLES.

1. Multiply 5384976 by 9999.

53849760000

There being four 9s in the given multiplier, 5384976 add four ciphers (0000) at the right hand; then write the multiplicand underneath, and subtract ac

53844375024 cording to the rule.

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V. DIVISION.

DIVISION is the method of finding how many times a less number is contained in a greater; or dividing a quantity given, into any number of parts assigned; and is a concise way of performing several subtractions.

There are four parts to be noted in Division:

1. The Dividend, or number given to be divided.

2. The Divisor, or number given to divide by.

3. The Quotient, or answer to the question, which shows how often the divisor is contained in the dividend.

4. The Remainder (which is always less than the divisor, and of the same name with the dividend) is very uncertain, as there is sometimes a remainder, and sometimes none.

SIMPLE DIVISION

Is the dividing of one number by another, without regard to their values; as 56, divided by 8, produces 7 in the quotient : That is, 8 is contained 7 times in 56.

GENERAL RULE.

1. Draw a curve line on each side of the dividend, and write the divisor at the left hand.

2. Take the same number of the first left hand figures of the dividend that there are in the divisor, if they be equal to, or greater than the divisor; but if they be less than the divisor, take one more; find the number of times the divisor is contained in them, and write a figure representing the number at the right hand of the dividend, which will be the first figure of the quotient.

3. Multiply the divisor by this quotient figure, and write the product under that part of the dividend taken.

4. Subtract this product from that part of the dividend taken, and bring down the next figure of the dividend, and place it at the right hand of the remainder: then find a quotient figure, multiply and subtract as before directed; proceed in the same manner until all the figures in the dividend are brought down and divided.

5. When a figure has been brought down and placed at the right hand of the remainder, if the number be less than the divisor, write a cipher in the quotient, and bring down another figure.

43. What is Division ?--44. How many parts are there in Division, and what are they?- -45. What is Simple Division?! -46. What is your rule ?

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