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TO TEACHERS.

Teachers will notice that the author's Graded Series of arithmetics consists of three parts. Having completed the Primary Arithmetic, or Part I., which is an introductory course in the science for begin ners, the pupil is prepared to take up the present volume, Part II. Part II. begins at the beginning of arithmetic and extends as far as Percentage. It embraces Numeration and Notation, Fundamenta. and Derivative Operations, Fractions, Decimals and their applications, Denominate Numbers, and Practical Measurements. After completing Part II., the pupil will be prepared to take up Part III., which begins at Percentage and finishes the course in arithmetic. These two parts are also bound in one volume for those who prefer the book in that form.

THE

NORMAL

UNION 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:

1

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. principle were obtained the following numbers:

By this

Eleven, twelve, thirteen, fourteen, fifteen, sixteen, seven

teen, 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. principle were obtained the following numbers:

By this

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 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 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 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 peeds nc 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 numbers. There are ten figures used, as follows: FIGURES.. 1, 2, 3, 4, 5, 6, 7, 8, 9,

0.

naught,

NAMES one, two, three, four, five, six, seven, eight, nine, cipher or zero.

AND VALUES.

27. By the combination of these figures all numbers may be expressed; hence they are appropriately called the alphabet 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 the right, expresses TENS; in the third place, HUNDREDS; in the fourth place, THOUSANDS, etc.; thus,

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

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