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ELEMENTS OF ARITHMETIC.

INTRODUCTORY DEFINITIONS.

1. Arithmetic is the science of numbers, and also the art of computation.

2. Number is the result of the comparison of a quantity with unity.

3. Quantity is anything that can be increased, diminished, or measured; as, the length of a road, the surface of a body, the weight of an article.

4. A Unit is a quantity with which we compare others of the same kind.

5. The comparison of quantity with unity produces three kinds of numbers: Integers, Fractions, and Mixed Numbers.

6. An Integer is a number which contains its unit. an exact number of times: as 12, 15; 6 boys, 4 apples.

7. A Fraction is a number which is less than a unit; as,,, J.

8. A Mixed Number consists of an integer and a fraction. Thus, 14, 24, 41.

9. According to the nature of their unit, numbers are divided into two classes; viz., Abstract and Concrete.

10. An Abstract Number is a number the nature of whose unit is not determined; as, 16, 425, 7840.

11. A Concrete Number is a number the nature of whose unit is determined; as, 16 men, 425 days, 7840 dollars.

TERMS EMPLOYED.

1. Arithmetical Operations are the divers changes to which numbers are subject.

There are four fundamental operations: Addition, Subtraction, Multiplication, and Division.

2. The Proof of an operation is a second operation made in order to verify the correctness of the first. The proof does not give an absolute certainty, but only a great probability, that the operation has been correctly performed.

3. A Problem is a question to be solved.

4. The Resolution of a problem consists in finding the unknown quantities by means of the known.

The resolution of a problem comprises both the solution and the calculation, or operation.

5. The Solution is the series of operations necessary to arrive at the required result.

6. The Calculation is the performance of the operations indicated by the solution.

NOTE. The calculation refers exclusively to abstract numbers or those considered as such.

7. A Principle is a fundamental truth from which others are derived.

8. A Rule is a statement of the method of obtaining a desired result.

NUMERATION AND NOTATION.

12. Numeration is the method of reading numbers expressed by characters.

13. Notation is the method of writing numbers. 14. Numbers may be represented as follows:

I. By words; as, one, two, three.

II. By figures, called the Arabic Method; as, 1, 2, 3. III. By letters, called the Roman Method; as, I, V, X, C.

15. In the Arabic Method, numbers are expressed by the following ten

Figures: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. Names: One, Two, Three, Four, Five, Six, Seven, Eight, Nine, Naught.

16. The first nine figures are called significant because they represent a value. But the tenth, by itself, represents nothing. It is only an auxiliary figure; its office being to hold the place of any order whatever, when there are no units of that order in the number.

17. Each of the first nine numbers expresses simple units, or units of the first order.

18. The number which follows the ninth is called ten. It is represented by writing the figure 1 with a naught after it; thus, 10.

19. Ten is the unit of the second order, and is equal to ten units of the first order.

20. We count by tens as we count by simple units,

saying one ten, two tens, three tens, nine tens. custom has replaced these words by the following:

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But

60. 70.

80.

90.

NOTE.-The "ty" in these words signifies ten.

21. The names of the numbers included between two consecutive tens, are formed by joining to the name of the first of these tens, the name of each of the first nine numbers, saying:

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The highest number expressed by two figures, being ninety-nine, 99.

22. But instead of saying ten and one, ten and two, ten and three,....ten and nine, custom has adopted the

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NOTE. The "teen" in the words thirteen, etc., to nineteen, means ten. So that, strictly speaking, thirteen means three and ten; fourteen, four and ten, etc.

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