THE

NORMAL

WRITTEN ARITHMETIC.

SECTION I.

ARITHMETICAL LANGUAGE.

1. Arithmetic is the science of numbers and the art of computing with them.

2. A Unit is a single thing or one. A thing is a concrete unit; one is an abstract unit.

3. A Number is a unit or a collection of units. Numbers are concrete and abstract.

4. A Concrete Number is one in which the kind of unit is named; as, two yards, five books.

5. An Abstract Number is one in which the kind of unit is not named; as, two, four, etc.

6. Similar Numbers are those in which the units are alike; as, two boys and four boys.

7. Dissimilar Numbers are those in which the units are unlike; as, two boys and four books.

8. A Problem is a question requiring some unknown result from that which is known.

9. A Solution of a problem is a process of obtaining the required result.

10. A Rule is a statement of the method of solving a problem.

11. Mental Arithmetic treats of performing arithmetical operations without the aid of written characters.

12. Written Arithmetic treats of performing arithmetical 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 how many of the collection instead of the collection; but the definition given, which is a modification of Euclid's, is simpler 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 named 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 this principle were obtained the following numbers.

Twenty-one, tu enty-two, twenty-three, twenty-four, twentyfive, twenty-six, twenty-seven, twenty-eight, twenty-nine.

20. Continuing in the same manner, we would have threetens, four-tens, five-tens, etc. By this principle were derived the following ordinary names:

Thirty, forty, fifty, sixty, seventy, eighty, ninety.

21. A group of ten tens is called a hundred; a group of ten hundreds, a thousand; the next group receiving a new name 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 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 left after ten.

3. Twenty is from the Saxon twentig (twegen, two, and tig, a ten); thirty is 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 needs 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

26. Figures.-Figures are characters used in expressing There are ten figures used, as follows:

numbers.

FIGURES.

AND VALUES.

0.

naught,

1, 2, 3, 4, 5, 6, 7, 8, 9, NAMES .} one, two, three, four, five, six, seven, eight, nine, 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, countiny from the right, expresses TENS; in the third place, HUNDREDS; in the fourth place, THOUSANDS, etc.; thus,

29. The name of each of the first twenty-one places is represented by the following

30. Periods. For convenience in writing and reading numbers, the figures are arranged in periods of three places each, as shown in the table. The first three places constitute 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 following:

Tens; hundreds; thousands; millions; ten-thousands; hundredthousands; ten-millions; billions; hundred-millions; hundred-billions; units; ten-billions; trillions; quadrillions; hundred-quintil lions; ten-trillions; ten-quintillions; hundred-quadrillions; quintillions; hundred-trillions; ten-quadrillions.

3. Required the names of the following periods:

1. First.
2. Third.

3. Second.

4. Fifth.

5. Fourth.

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 explanations 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.