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15. All intervening Places to be filled.

(a.) When, as is usually the case, the places are only marked by the figures occupying them, every place between any given figure and the point must be occupied by some one of the digits, for otherwise it will be difficult or impossible to tell the place or denomination of the given figure.

Illustration. The 3 of the number 3 24 may mean 3 thousands, or 3 ten-thousands; but when the space between the 3 and 2 is filled, the denomination of the 3 is at once apparent. Thus, in 3024, or 3124, or 3724, the 3 represents 3 thousands; but in 30024, 31024, 32324, or 37824, it represents 3 ten-thousands.

(b.) The digit to be used in any intervening place is determined by the number of units to be represented of the denomination of that place. If there are none, then zero should be used; if then 1; if two, then 2; &c.

one,

Illustration. In order that 9 may represent 9 ten-millions, it must be written in the eighth place, at the left of the point, and hence there must be seven figures to the right of it. If we wish to express only 9 tenmillions, or 90 millions, the intervening places must be filled by zeroes, thus, 90,000,000; but if we wish to express 90 millions, 3 thousand, 8 hundred, and 7, we write 9 in the eighth place, as before, 3 in the fourth, 8 in the third, 7 in the first, and zeros in all the intermediate places; thus, 90,003,807.

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(a.) In writing numbers by figures we may begin at the left, and write in each successive period the figures necessary to express the required number of units of the denomination of that period; or we may begin at the right, and write in each successive place the figures expressing the required number of units of the denomination of that place, taking care to write a zero in each place otherwise unoccupied.

(b.) In writing numbers be careful to distinguish the decimal point from the commas used to separate the periods. The former should be a dot, so carefully made that it cannot be mistaken for a comma, while the latter should be made with equal care. No number has more than one decimal point.

(c.) Accuracy in reading and writing numbers is of the greatest importance. If the numbers used in solving a problem are copied incor rectly, the results obtained will of necessity be wrong; and if the book from which the problem was obtained is not at hand, or if the facts and transactions on which the problem was based are forgotten, it will be very difficult, and usually impossible, tc make the necessary correction.

1. Represent by figures five hundred and twenty-seven thousand, four hundred and eighty-nine.

2. Eight thousand, four hundred and seven.

3. Eighty-five thousand.

4. Eighty-five thousand and one.

5. Eighty-five thousand and thirty-one.

6. Nine million, eight hundred and fifty-six thousand, seven hundred and twenty-one.

7. Twelve million, twelve thousand, and twelve.

8. Four billion, eight hundred seventy-six million, five hundred and four thousand, three hundred and one.

9. Four billion, eight hundred and four million, eight tundred and four thousand, eight hundred and four.

10. Thirty-seven million, eight hundred and fifty-nine thousand.

11. Thirty-seven billion, eight hundred and fifty-nine million.

12. Thirty-seven billion, eight hundred and fifty-nine thousand.

13. Thirty-seven million, eight hundred and fifty-nine.

14. Forty billion, three hundred and forty million, four hundred and eighty-seven thousand, five hundred and nine.

15. Five billion, eight hundred and seventy-six thousand, seven hundred and forty-six.

16. Seventy-five trillion, eight hundred and seventy-six billion, four hundred and eighty-two million, four hundred and seventy-six thousand, three hundred and twenty-seven.

17. Four trillion, seven hundred and sixty-four billion, eight hundred and twenty-one million, six hundred and seventeen thousand, four hundred and fifty-one.

18. Seven hundred and twenty-five trillion, eight hundred and seventy-six billion, four hundred and three million, eight hundred and fifty thousand, four hundred.

19. Three hundred and six trillion, eighteen billion, four hundred million, three thousand, four hundred and seventy-five. 20. Three trillion, three hundred and ninety-nine billion, three hundred and ninety-nine million, three hundred thousand, four hundred and three.

21. Eighty-seven trillion, five hundred and four billion, three hundred million, seven thousand, six hundred and seventy-five. 22. Seventy quadrillion, seven hundred and seven billion, seven thousand and seven.

23. Eighty-seven quintillion, eight trillion, seven hundred billion, eight hundred and seventy thousand, and eighty-seven.

24. Three hundred and fifty-four sextillion, four hundred and seven quintillion, two hundred and nine quadrillion, nine hundred and seventeen trillion, seven hundred billion, eightysix million, seven thousand, eight hundred and fifty-two.

25. Seven hundred and seven quintillion, two hundred and six thousand.

26. Five hundred and seventy quadrillion, five hundred and seventy.

27. Eight sextillion, eight trillion, eight thousand, and eight.

17. Any Combination of Figures may be read as though alone.

(a.) Combinations of figures, wherever placed, can be read as though they stood alone, if the name of the place of the right hand figure be given after reading the figures.

For instance, 347 always stands for, and may be read as, three hundred and forty-seven of the denomination of the place occupied by the 7.

(b.) To illustrate this still further, we have written opposite each of the following numbers the value expressed by 347 in that number.

1. 347,241,
2. 347,100,

3. 4,134,721,

4. 23,476,258,675,

347 thousands.

347 thousands.

347 hundreds.

347 ten-millions.

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(c.) Let the pupil give the value expressed by the 409 in each of the following numbers, and also the value expressed

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1. What is the greatest number of tens that can be taken from 546,372?

Ans. 54637 tens.

2. What is the greatest number of ten-thousands that can be taken from 53,075,423,697 ?

Ans. 5307542 ten-thousands.

What is the greatest number of hundreds that can be

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11. What is the greatest number of thousands that can be taken from each of the above numbers? of tens? of millions?* of billions? of hundred thousands? of ten millions?

12. Express the value of 457869 in hundreds and units. Ans. 457869 equals 4578 hundreds and 69 units.

Express the value of each of the following in hundreds and units:

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* If any number is less than a million, as 2876, the answer may be, 2876 is less than a million, and therefore no millions can be taken from it.

21. Express the value of each of the above in tens and units. In ten-thousands and units.

22. Express as much of the value of each of the above as you can in ten-thousands; as much of the remainder as you can in hundreds, and the rest in units.

Model of Answer. 8796784879 ten-thousands, 67 hundreds, and 84 units.

23. Express as much of the value of each of the above as you can in ten-millions, as much of the remainder as you can in thousands, and the rest in units.

19. Comparison of the Values represented by the same Figure in different Places.

(a.) Since, as we have seen, the figure in any place represents ten times the value it would represent if written one place farther towards the right, one hundred times the value it would represent if written two places farther towards the right, &c., it follows that it must represent one* tenth of the value it would represent if written one place farther towards the left, one one-* hundredth of the value it would represent if written two places farther towards the left, &c.

NOTE. The expression "Numbers increase from right to left in a tenfold ratio," is not a true statement of the fact.

A unit of one decimal denomination always bears the same ratio to a unit of the next higher that 1 does to 10; but the ratio which the value of a figure of one decimal denomination bears to the value of a figure of the next higher is as 1 to 10 only when the figures are alike. For instance, in 22 the value of the left hand figure is ten times that of the right hand figure, while in 25 it is only four times, and in 91 it is ninety times.

Even when the figures are alike the ratio of increase is not tenfold. We increase a number by adding to it. To increase a number once, one addition must be made to it. To increase it twice, two additions must be made, &c. A number increased by once itself will produce twice it

* If the explanations on page 3d are not sufficient to enable the pupil to understand the meaning and use of the fractional expressions contained in this section, he should study the first part of the section on fractions before going farther.

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