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By this table it will be seen thật 2 in the first placé denotes simply 2 units, that 3 in the second place denotes as many tens as there are simple units in the figure, or 3 tens; that 2 in the third place, denotes as many hundreds as there are units in the figure, or 2 hundreds ; apd so on. Hence to read any number, we have only to observe the following
RULE. To the simple value of each figure join the name of its place, beginning at the left hand and reading the figures in their order towards the right.
The figures in the above table would read, three sextillions, four hundred fifty-six quintillions, seven hundred fifty-four quadrillions, three hundred seventy-eight trillions, four hun: dred sixty-four billions, nine hundred seventy-four millions, three hundred one thousand, two hundred tbirty-two.
75. In reading very large numbers it is often convenient to divide them into periods of three figures each, as in the follow
532, 123, 410, 864, 232, 012, 345, 862,051, 234, 525,411, 243,673, By this table it will be seen that any number, however large, after dividing it into periods, and knowing the names of the periods, can be read with the same ease as one consisting of three figures only; for the same names, (hundreds, tens, units) are repeated in every period, and we have only to join to these, successively, the names of the periods. The first, or right hand period, is read, six hundred seventy-three-units, the second, two hundred forty-three thousands, the third, four hun. dred eleven millions, and so on.
76. The foregoing is according to the French numeration, which, on account of its simplicity, is now generally adopted in English books. In the older Arithmetics, and in the two former editions of this work, a period is made 10 consist of șix figures, and these were subdivided into half periods, as in the following
What is seen by the first numera. 1 table? tion table ?
What names are repeated in every What is the rule for reading num- i period ? bers?
What is the difference between the How are large numbers sometimes French and English methods of divided ?
numeration ? What is learned from the second |
TABLE III. Periods. 1 Sextill. Quintill. Quadrill. Trill. Billions, Millions. Vuits.
These two methods agree for the nine first places; but beyond this the places take different names. Five billions, for example, in the former method, is read five thousand millions in the latter. The principles of notation are, notwithstanding, the same in both throughout, the difference consisting only in the enunciation.
EXAMPLES FOR PRACTICE. Write the following in figures: 1 Enumerate, or write the following Eight. Seventeen. Ninety-three. in words : 'Three hundred sixty. Five thou
91 97890112 sand four hundred and seven. Thir
65 1 64351234 ty thousand fifty-nine. Seven
137111055 millions. Sixty-four billions. One
20401 8900000000 hundred nine quadrillions, one hun
60735 39000010010 dred nine millions, one hundred nine
123456 2:22000222002 thousand, one hundred and nine,
77. Numbers are called simple, when their units are all of the same kind, as men, or dollars, &c.
ANALYSIS. 78. 1. How many cents are 3 cents and 4'cents ? Here are two collections of cents, and it is proposed to find how large a collection both these will inake, if put together. The child may not be able to answer the question at once; but having learned how to form numbers by the successive addition of unity (2, 72.) he will perceive, that he can get the answer correctly, either by adding a unit to four three times, or a unit to three four times (7). In this way he must 'proceed, till, by practice, the results arising from the addition of small numbers are committed to meniory, and then he will be able tions which involve such additions almost instantaneously. But when the numbers are large, or numerous, it will be found most convenient to write them down before performing the addition.
What is numeration ?
| When are numbers called simple? What is meant by Analysis?
2. A boy gave 36 cents for a book, and 23 cents for a slate, how many cents did he give for both?
Here the first number is made up of 3 tens and.6 units, and the second of 2 tens and 3 units. Now if we add the 3 units of one with the 6 units of the other, their sum is 9 units, and the 2 tens of one added to the 3 legs of the other, their sum is 5 tens. These two results taken together, are 5 tens and 9 units, or 59, which is the number of cents given for the book and slate. The common way of performing the above operation is
36 cents. I to write the numbers under one another, so th
hand. Then begin at the bottom of the right hand column, Ans.59 cents. I and add together the figures in that column, thus-3 and 6 are 9, and write the 9 directly under the colunin. Proceeding to the column of tens, we say, 2 and 3 are 5, and write the 5 directly under the column of tens. Then will the 5 tens and 9 units each stand in its proper place in the answer, making 59.
3. If a man travel 25 miles the first day, 30 the next, and 33 the next, how far will he travel in the three days ?
Ans. 88 miles. 79. 4. A man bought a pair of horses for 216 dollars, a sleigh for 84 dollars, and a harness for 63 dollars, what did they all cost him?
216 dolls. / Here we write down the numbers as before, and begin 84 dolls. I with the right hand column-3 and 4 are 7, and 6 are 63 dolls. | 13; hut 13 are 1 ten and 3 units; we therefore wrile
I the 3 under the column of units, and carry the 1 ten to Ans. 363 dolls. I the column of tens, saying, 1 to 6 are 7, and 8 are 15, and I are 16. But 16 tens are 1 hundred and 6 lens; we therefore write the 6 under the column of tens, and carry the l into the column of hursreds, saying, 1 to 2 are 3, which we write down in the place of hundreds, and the work is done. From what precedes the scholar will be able to understand the following definition and rule.
SIMPLE ADDITION. 80. Simple Addition is the uniting together of several simple numbers into one whole or total number, called the sum, or amount.
RULE. 81. Write the numbers to be added under one another, with units under units, tens under tens, and so on, and draw a line below them. Bégin at the bottom and add up the figures in the right hand column :-if the sum be less than ten, write it below the line at the foot of the column ; if it be ten, or an exact nuinber of tens, write a cipher, and carry the tens to the next column; or if it be more than ten, and not an exact number of tens, write down the excess of tens and carry the less as above. Proceed in the same way with the columns of tens, hundreds, &c. always remembering, that ten units of apy one order, are just equal to one unit of the next higher order.
What is the process by which a | What is simple addition ?
child would add two numbers to. How are the numbers to be written gether?
PROOF. 82. Begin at the top and reckon each column downwards, and if their amounts agree with the former, the operation is supposed to have been rightly performed.
Note.-No method of proving an arithmetical operation, will demonstrate the work to be correct; but as we should not be likely to cominit errors in both operations, which should exactly balance each other, the proof renders the correctness of the operation highly probable.
QUESTIONS FOR PRACTICE. 5. According to the census / 8. How many dollars are of 1820, Windsor contained | 2565 dollars, 7009 dollars, and 2956 inhabitants, Middlebury, 796 dollars when added togeth2535, Montpelier 2308, and | er? Ans. 10370 dolls. Burlington, 2111, bow many
1 9. In a certain town there inhabitants were there in those
are 8 schools, the number of four towns ?
scholars in the first is 24, in Operation. 2956 Windsor.
the second 32, in the third 28,
in the fourth 36, in the fifth 26, 2535 Middlebury.
in the sixth 27, in the seventh 2308 Montpelier.
40, and in the eighth 38; how 2111 Burlington.
many scholars in all the 9910 Total.
10. Sir Isaac Newton was 9910 Proof. born in the year 1642, and was 6. A man has three fields, 1 85 years old when he died ; in. one contains 12 acres, another |
| what year did he die ? 23 acres, and the other 47 !
Ans. 1727. acres ; how many acres are
11. I have 100 bushels of there in the whole ? Ans. 82.
wheat, worth 125 dollars, 150 7. A person killed an ox, i bushels of rye, worth 90 dolthe meat of which weighed 642 | lars, and 90° bushels of corn. pounds, the hide 105 pounds, i worth 45 dollars, how many and the tallow 92 pounds; į bushels have I, and what is it what did they all weigh?
worth? Ans. 340 bush. Ads. 839.
worth 260 dolls.
Where do you begin the addition? number of tens,what is to be done? If the amount of the column be less What is the sign of addition ?
, what is to be done with What is the sign of equality ? it?
Explain the reason of carrying the If the amount be just ten, or an ex- i tens?
act number of tens, what is to be | How is addition proved? done?
Does the proof demonstrate the op it be over ten, and not an exact | eration to be right?
12. A man killed 4 bogs, one was the whole number of inweighed 371 pounds, one 510 habitants at that time? pounds, one 472 pounds, and
Ans. 9637999. the other 396 pounds; what | 17. It is 38 miles from Burdid they all weigh?
lington to Montpelier, 47 from Ans. 1749 pounds. | Montpelier to Woodstock, and 13. The difference between | 14 from Woodstock to Windtwo numbers is 5, and the least sor; how far is it from Burlingnumber is 7; what is the ton to Windsor? Ans. 99 miles. greater?
Ans. 12. 18. How many days in a 14. The difference between common year, there being in two numbers is 1448, and the Jan. 31 days, in Feb. 28, in - least number is 2575; what is March 31, in April 30, in May the greater? Ans. 4023. 31, in June 30, in July 31, in
15. There are three bags of August 31, in Sept. 30, in Oct. moner, one.contains 6462 dolls. i 31, in Nov. 30, and in Dec. 31 one 8224 dolls. and the other days?
Ans. 365. 5749 dolls. how many dollars 19. A person being asked in the three bags ?
his age, said that he was 9 . Ans. 20435 dolls. I years old when his youngest 16. According to the census brother was born, that his broof the United States in 1820, ther was 27 years old when his there were 3995053 free white | eldest son was born, and that males, 3866657 free white fe- his son was 16 years old ; what males, and 1776289 persons of was the person's age ? every other description; what I
Ans. 52 years. 20. 21.
22. 23213 2424612 8192735 9.876987 16423 1234 567 214268 7986698 21230 7654321 1541320 4343434 90418 2112710
ANALYSIS. 83. We have seen that Addition is an operation, by which several numbers are united into one sum. Now it frequently happens that the numbers to be added are all equal, in which case the operation may be abridged by a process called Multiplication.