Εικόνες σελίδας
PDF
Ηλεκτρ. έκδοση
[blocks in formation]

one,

The class may now take the first four exercises, and form them into with 6 figures in width and 6 in depth, and proceed in like manner with the remainder, thus increasing the exercises both in width and depth to the end, always remembering that the result of each addition is not to be written till the student has it complete in his mind.

Repeat all the above exercises in addition, commencing each at the left instead of the right. For example, in No. 5, say: sixteen hundred ; a hundred and ten, seventeen hundred and ten; twenty-six, seventeen hundred and thirty-six.

2. Subtraction by Inspection.

Find the differences of the following numbers, observing that the subtrahend is placed sometimes above and sometimes below. The whole result should be ascertained before any part of it is written. subtraction in the ordinary method.

Prove by

[blocks in formation]

12. 796346 149287

13. 9712645
2793487

14. 287638472
639724576

15. 8296384 5149427

Repeat the above exercises, by throwing two into one, proving as before; and repeat them once more, performing the subtraction by adding the complement of the subtrahend.

Find the differences of the following pairs of numbers, by the addition of the complement, proving by adding the result to the double subtrahend.

[blocks in formation]

Repeat the first ten exercises immediately above, first by doubling two horizontally, and again by doubling each pair vertically, placing the four numbers of the minuend together, and also the four of the subtrahend, connecting each set by a brace.

3. Multiplication by Inspection.

Find the products of the following factors, not writing them till every figure of the result is attained. Prove by multiplication in the ordinary

[blocks in formation]

Method of Operation.-[To be read slowly by the teacher, the class keeping their eye on the figures.] No. 1. Beginning on the left, twelve thousand; sixteen hundred, thirteen thousand, six hundred; two hundred, thirteen thousand, eight hundred; thirteen thousand, eight hundred and eight.

No. 2. Thirty thousand; fifteen hundred, thirty-one thousand, five hundred; two hundred and fifty, thirty-one thousand, seven hundred and fifty; thirty-one thousand, seven hundred and seventy.

[blocks in formation]

1. Repeat the above eighteen exercises, omitting the words in italics : that is, throwing the partial product at once into the general result without naming it.

2. Repeat the same eighteen exercises, with an additional figure to each multiplier: that is, 1 ten to each, then 2, 3, 4, and 5 tens to each. Method of operating with two figures in the multiplier. No. 1, with 14. First by 10, then by 4. 10. Thirty-four thousand, five hundred and twenty. 4. Twelve thousand, forty-six thousand, five hundred and twenty; sixteen hundred, forty-eight thousand, one hundred and twenty; two hundred and eight, forty-eight thousand, three hundred and twentyeight.

No. 2, with 35. First by 30, then by 5 [5=1o]. 30. A hundred and eighty-nine thousand; sixteen hundred and twenty, a hundred and ninety thousand, six hundred and twenty. 5. Thirty-one thousand, seven hundred, and seventy, two hundred and twenty-two thousand, three hundred, and ninety.

No. 3, with 26. First by 20, then by 6. 20. A hundred and sixtyniñe thousand, eight hundred and forty. 6. Fifty thousand, four hundred, two hundred and twenty thousand, two hundred and forty; five hundred and forty; two hundred and twenty thousand, seven hundred and eighty; twelve; two hundred and twenty thousand, seven hundred and ninety-two.

No. 5, with 38. 3. A hundred and forty-seven thousand; a thousand and eighty, a hundred and forty-eight thousand and eighty. 8. Thirtytwo thousand; a hundred and eighty thousand and eighty; seventy-two hundred; a hundred and eighty-seven thousand, two hundred and eighty; two hundred and forty; a hundred and eighty-seven thousand, five hundred and twenty; forty-eight; a hundred and eighty-seven thousand, five hundred and sixty-eight.

3. Repeat the 18 exercises with the figures of the multiplicand in reverse order, and with 6, and 7, and 8, and 9, in the ten's place of the multiplier.

[blocks in formation]

The teacher may extend these exercises as far as he may find it profitable to the class. Some pupils have accomplished the multiplication of 9 figures in the one factor, and 5 in the other, after a very short practice. To others they come hard. But all will be highly benefited by their use.

4. Division by Inspection.

Find the quotients in the following exercises by inspection, beginning at the left, not writing them till the whole quotient is attained. Prove by multiplication by inspection.

1. 2)653,492

2. 4)134,684 3. 7)179,426 4. 5)286,942

[blocks in formation]

Repeat each of the above 11 exercises, with an additional figure for tens in the divisor, namely, 1, 2, 3, 4. Then repeat again, with the figures in each dividend reversed, with an additional figure, 6, 7, 8, 9 tens in the divisor. Prove as above.

The teacher can extend this practice by new exercises, increasing the number of figures, both in the divisor and the dividend, till the class has acquired sufficient dexterity.

5. Evolution by Inspection.

a. Extraction of the Square Root by Inspection.

Find the nearest square root of each of the following numbers: 576 ; 1296; 1458; 3975; 2482; 9176; 7056; 2209; 20736; 53824. Prove by involution by inspection.

b. Extraction of the Cube Root by Inspection.

Find the nearest cube root of the following numbers by inspection; and prove by involution by inspection: 13824; 46656; 53824; 592,704; 638,576; 2,985,984; 12,487,168.

When a perfect cube does not exceed 1,000,000, its root may be found almost at a glance, as follows: Fill up the table of squares and cubes in p. 164, if not already done; and then take notice that the unit's figure in the cubes of 2 and of 3, and also in those of their complements (8 and 7), is the same as the complements of their respective roots. Observe, also, that the unit's figure in the cubes of all the other digits (1, 4, 5, 6, 9) is the same as that of their root. Thus :

[merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][ocr errors][merged small][ocr errors][merged small][merged small]

From the above table it is evident that it only requires a knowledge of the cube of each of the digits, to determine the root of any perfect cube not exceeding 1,000,000, by a mere glance at the second period and at the unit's figure of the power.

Exercises for the Slate or Black-board.

1. What is the cube root of 262,144? Ans. 6 is the greatest root in the second period, the unit's figure is 4: 64, therefore, is the root.

2. What is the cube root of 389,017? Ans. 7 being the greatest cube in 389, and 3 the complement of 7, the cube root is 73.

3. Find the cube roots of the following perfect powers by a glance: 54,872; 884,736; 185,193; 474,552; 5832; 15,625; 59,319. Prove by involution by inspection.

The extraction of the square root by this method requires more attention; but for that very reason is more useful as a mental exercise. The roots of perfect squares which do not exceed 10,000 may be ascertained thus. By an examination of the squares of the digits, it will be perceived that every perfect square ending in 5 has 5 for the unit's figure of its root; and that the squares of 1, 2, 3, 4, and of their complements, 9, 8, 7, 6, have for unit's figure 1, 4, 9, 6, respectively. Thus, having determined the ten's figure of the root by a glance at the second period, we know that if the unit's figure of the square be

[merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][merged small][ocr errors][merged small][ocr errors][ocr errors][ocr errors][merged small]

Exercises for the Slate and Black-board.

Ans. The greatest square in the remainder small compared with The whole root, of course, is 72. Ans. The greatest square in 12 great in proportion to 3, must be is 36.

1. What is the square root of 5184? second period being that of 7, and the 7, the unit must be 2 rather than 8. 2. What is the square root of 1296? is that of 3, and the remainder being 6 rather than 4. The whole root, then, 3. What is the square root of 2025? Ans. The greatest square in 20 is 4, and as the unit's figure in the square is 5, that of the root must be 5 also. The whole root, then, is 45.

4. Find the square roots of 1024; 3136; 784; 4225; 2116; 3249; 6561; 2401. Prove by involution by inspection.

SYNOPSIS,

OR, RECAPITULATION OF PRINCIPLES DEVELOPED IN THE PRECEDING PAGES. I. THERE are only two operations in arithmetic, increase and decrease. A number may be increased by one or more additions. It may be dimin

« ΠροηγούμενηΣυνέχεια »