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11. Mental Arithmetic treats of performing arithmetical operations without the aid of written characters.
12. Written Arithmetic treats of performing arithmeti. cal operations with written characters.
13. Arithmetical Language is the method of expressing numbers.
14. Arithmetical Language is of two kinds, Oral and Written. The former is called Numeration and the latter is called Notation.
NOTE.-A number is really the horo many of the collection instead of the collection ; but the definition given, which is a modification of Euclid's, is Nmpler and sufficiently accurate.
NUMERATION. 15. Numeration is the method of naming numbers, and of reading them when expressed by characters. It is the oral expression of numbers.
16. Since it would require too many words to give each number a separate name, numbers are named according to the following simple principle:
Principle.- We name a few of the first numbers, and then form groups or collections, name these groups, and use the names of the first numbers to number these groups.
17. A single thing is named one : one and one more are gamed two; two and one more, three; three and one more, four; and thus we obtain the simple names,
One, two, three, four, five, six, seven, eight, nine, ten.
18. Now, regarding the collection ten as a single thing, we might count one and ten, two and ten, three and ten, etc., as far as ten and ten, which we would call two tens. By this principle were obtained the following numbers:
Eleven, twelve, thirteen, fourteen, fifteen, sixteen, seventeen, eighteen, nineteen, twenty.
19. Proceeding in the same way, we would have two tens and one, two tens and two, two tens and three, etc. By thir principle were obtained the following numbers:
Twenty-one, twenty-two, twenty-three, twenty-four, twenty five, twenty-six, twenty-seven, twenty-eight, twenty-nine.
20. Continuing in the same manner, we would have three. tens, four-tens, five-tens, etc. By this principle were de rived the following ordinary names:
Thirty, forty, fifty, sixty, seventy, eighty, ninety
21. A group of ten tens is called a hundred; a group of ter. hundreds, a thousand; the next group receiving a new game consists of a thousand thousands, called a million ; the next group of a thousand millions, called a billion, etc.
22. After a thousand, the two intermediate groups between those having a distinct name, are numbered by tens and hundreds, as ten thousand and hundred thousand.
NOTES.–1. The above shows the principle by which the numbers were named. The names, however, were not derived from the particular expressions given, but originated in the Saxon language.
2. Eleven is from the Saxon endlefen, or Gothic ainlif (ain, one, and lif, ten); twelve is from the Saxon twelif, or Gothic tvalif (tva, two, and lif, ten). Some have supposed that eleven meant one left after ten, and twelve, two lef" after ten.
3. Twenty is from the Saxon twentig (twegen, two, and tig, a ten); thirty 18 from the Saxon thritig (thri, three, and tig, a ten), etc.
4. Hundred is a primitive word; thousand is from the Saxon thusend, or Gothic thusundi, (thus, ten, and hund, hundred); million, billion, etc., are from the Latin.
NOTATION. 23. Notation is the method of writing numbers. Numbers may
be written in three ways: 1st. By words, or common language. 2d. By figures, called the Arabic Method. 3d. By letters, called the Roman Method. NOTE.—The method by words is that of ordinary written language and Deeds no explanation.
ARABIC NOTATION. 24. The Arabic System of Notation is the method of expressing numbers by characters called figures.
25. In this system numbers are expressed according to the following principle:
Principle.- We employ characters to represent the first nine numbers, and then use these characters to number the groups, the group numbered being indicated by the position of the character.
NAMES AND VALUES.
26. Figures.-Figures are characters used in expressing numbers. There are ten figures ased, as follows: 1, 2, 3, 4, 5, 6, 7,
one, two, three, four, ive, sli, seven, eight, bine, cipher or zero 27. By the combination of these figures all numbers may be expressed; hence they are appropriately called the alpha bet of arithmetic.
28. Combination.—These figures are combined according to the following principle:
1. A figure standing alone, or in the first place at the right of other figures, expresses UNITS or ONES.
2. A figure standing in the second place, counting from tho right, expresses TENS; in the third place, AUNDREDS ; in the fourth place, THOUSANDS, etc.; thus, 10 is 1 ten, or ten.
100 is 1 hundred. 20 2 tens, or twenty.
200 2 hundred. 30 3 tens, or thirty.
300 “ 3 hundred. 40 4 tens, or forty.
400 4 hundred. 50 5 tens, or fifty.
1000 1 thousand. 60 6 cens, or sixty.
thousand. 90 9 tens, or ninety. 4000 4 thousand.
29. The name of each of the first twenty-one places is represented by the following
30. Periods.-For convenience in writing and reading aumbers, the figures are arranged in periods of three places each, as shown in the table. The first three places constitate the first, or units period ; the second three places constitute the second, or thousands period, etc.
1. Required the names of the following places : First; third ; second ; sixth ; fourth ; eighth; tenth; ninth; twelfth ; fifth ; seventh ; eleventh ; thirteenth ; seventeenth ; fourteenth ; sixteenth ; eighteenth ; fifteenth ; nineteenth ; twenty-first; twentieth.
2. Required the places of the foilowing:
Teng ; hundreds ; thousands ; millions ; ten-thousands ; hundred thousands ; ten-millions ; billions ; hundred-millions ; hundred-bil. lions ; units ; ten-billions ; trillions ; quadrillions ; hundred-quintillions ; ten-trillions ; ten-quintillions ; hundred-quadrillions ; quintillions; hundred-trillions ; ten-quadrillions. 3. Required the names of the following periods: 1. First.
5. Fourth. 2. Third.
6. Seventh. 4. Required the period and place of the following:
Thousands ; millions ; ten-thousands ; hundred-millions ; billions ; hundred-trillions ; trillions ; ten-trillions ; quadrillions ; ten-quadrillions ; hundred-trillions ; quintillions ; hundred-quintillions ; hundred thousands ; ten-millions.
31. The combination of figures to express a number forms a numerical expression. Thus, 25 is a numerical expression which denotes the same as the common word twenty-five.
32. The different figures of a numerical expression are called terms. Terms are also used to indicate the numbers represented by the figures.
NOTE.—The use of the word term, to indicate both the figures and the numbers represented by them, enables us to avoid the error of using the word figure for the word number.
EXERCISES IN NUMERATION. 33. The pupils are now prepared to learn to read numbers when expressed by figures. From the preceding explana tions we have the following rule for numeration :
Rule.-I. Begin at the right hand, and separate the numerical expression into periods of three figures each.
II. Then begin at the left hand and read each period in succession, giving the name of each period except the last.
NOTB.-The name of the last period is usually omitted, because it to understood.
1. What number is expressed by 6325478?
SOLUTION. Separating the numerical expression OPERATION. into periods of three figures each, beginning at the 6,325,478 right hand, we have 6,325,478. The third period is 6 millions, the second period is 325 thousands, and the first is 478 units ; hence the number is 6 millions, 325 thousands, 478.
Read the following numerical expressions : 2. 3876 | 10. 468217 | 18.
80305072 8. 2185 11. 654879 19.
65073058 4. 3072 | 12.
484378513 6. 5678 13. 1234567 | 21.
123456789 6. 12630 14.
854327031 7. 70851 | 15. 8507032 23.
80735468579 8. 32468 | 16. 54372568 24. 20650708462067
507035 17. 87072135 25. 798653013285678521 NOTE.-After pupils are familiar with reading by dividing into periods, the division may be omitted or performed mentally.
EXERCISES IN NOTATION.
34. Having learned to read numerical expressions, we are now prepared to write them. From the principles which have been explained, we derive tbe following rule:
Rule.--I. Begin at the left and write the hundreds, tens, and units of each period in their proper
order. II. When there are vacant places, fill them with ciphers.
1. Express in figures the number four thousand three hundred and four.
SOLUTION.–We write the 4 thousands in the 4th place, 3 hundreds in the 3d place, a cipher in the 2d 4,304 place, there being no tens, and 4 units in the 1st place, and we have 4304.
Express the following numbers in tigures: 2. One hundred and six.
8. Three hundred and ifty. 3. nne hundred and ten. 4. iwo hundred and forty.
9. Four hundred and twenty 5. Two hundred and sixty-five. eight. 6. Two hundred and nine.
10. Seven hundred and eighty. 7. Three hundred and twelve four.