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THE FOUNDATIONS

OF

HIGHER ARITHMETIC

CHAPTER I

BASIC PRINCIPLES: NOTATION AND NUMERATION

1. Basic Principles.—Every science rests upon certain basic principles, or primary truths, as a foundation. On taking up the study of a science, the learner should become familiar with the basic principles at the beginning, since their use is to guide him in reasoning from the known, or given, to the related unknown, or required. In mathematical science, the basic principles are in the form of definitions, theorems and axioms.

GENERAL DEFINITIONS

2. A definition (Latin definire, to bound or limit) is such a description of any thing as will distinguish it from all other things.

3. Quantity (Latin quantus, how much) is any thing which can be increased or diminished; it embraces number and magnitude.

4. Science (Latin scire, to know) is knowledge properly organized.

5. Mathematics is the science of quantity.

6. Arithmetic (Greek arithmos, number) is the science of numbers.

7. A unit (Latin unus, one) is one thing, or a group of things regarded as a single thing.

A unit is any standard of reference employed in counting any collection of objects, or in measuring any magnitude.-J. C. GLASHAN.

8. Number (Latin numerare, to count) is the ratio of one quantity to another quantity of the same kind taken as a unit.

“Number in the strict sense is the measure of quantity. It definitely measures a given quantity by denoting how many units of measurement make up the quantity.”*

9. A concrete number (Latin concrescere, to grow together) is a number applied to some particular unit; as,

4 quarts, 7 dollars, etc.

NOTE.- In the strict sense, number is always abstract. It is a ratio, and answers the question, how many ? In the examples above, “4” and “y” tell how many, and“

“quarts” and “dollars” tell of what. The idea of number is expressed by “4” and “7," not by “quarts” and “dollars."

When the unit is named, however, it is convenient and customary to speak of a number as concrete, to distinguish from pure number, which is always abstract.

10. An abstract number (Latin abstrahere, to draw away) is a number used without reference to any particular unit; as,

4, 7, 39, etc.

* The Psychology of Number, by McLellan and Dewey, p. 93.

11. A principle (Latin principium, a beginning, or origin) is a general truth.

12. A problem (Greek problema, a question) is a question offered for solution.

13. A solution (Latin solvere, to loosen) is a clear statement showing how the result is obtained.

14. An axiom (Greek axioma, a requisite) is a selfevident truth.

The following are the axioms most frequently used in mathematics:

1. The whole is greater than any of its parts.
2. The whole is equal to the sum of all its parts.
3. If equals are added to equals, the sums are equal.
4. If equals are subtracted from equals, the remain-

ders are equal.
5. If equals are added to unequals, the sums are

unequal. 6. If equals are subtracted from unequals, the re

mainders are unequal. 7. If equals are multiplied by the same number, the

products are equal. 8. If equals are divided by the same number, the

quotients are equal.
9. If things are equal to the same thing, they are

equal to each other.
10. The same parts of equals are equal.
11. Equal powers of equals are equal.
12. Equal roots of equals are numerically equal.

15. Notation (Latin notare, to mark) is the art of representing numbers by means of symbols. It is intimately connected with numeration, which is generally defined as the art of reading numbers. The ability to read numbers is usually implied in the ability to write them. At the present time in this country, only two methods of notation are in general use. The first to be considered is the

ROMAN NOTATION

16. In the Roman Notation, seven capital letters are used. This method is rarely followed, except in numbering chapters and divisions of books, and on the dials of clocks and watches.

Letters: I, V, X, L, C, D, M.

Values: 1, 5, 10, 50, 100, 500, 1000. 17. The following principles are followed in combining the letters: 1. Repeating a letter repeats its value; as,

II=2; XX=20; XXX=30. 2. When a letter is placed before another of greater

value, its value is taken from that of the greater;

as,

IV=4; IX=9; XL=40. 3. When a letter is placed after another of greater value, its value is added; as,

VI=6; XI=11; LX=60. 4. When a letter is placed between two letters of

greater value, its value is taken from the following letter, not added to the preceding; as,

XIV=14, not 16.

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