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mathematicians serve further to humanize the subject. No attempt has been made to give a connected account of the development of algebra even in outline; these notes will serve their purpose if they create a desire to read some standard work on the history of mathematics.

Other aids which teachers will appreciate are the inductive developments, the cross references, illustrative problems, methods of testing results, topical and logical arrangement, definite classification, the frequent reviews, and the sum maries of the theoretic chapters.

The authors wish to express their gratitude for the assistance rendered by those who read the manuscript and the proof sheets. For valuable constructive suggestions in preparing the manuscript, they are indebted to Mr. Allen H. Knapp, of the Central High School, Springfield, Mass., and Mr. Julius J. H. Hayn, of the Masten Park High School, Buffalo, N. Y.; while for efficient aid in reading the proofs, they owe much to Mr. Matthew R. McCann, of the English High School, Worcester, Mass., and Mr. William H. Wentworth, of the Northwestern High School, Detroit, Mich. For the portraits of famous mathematicians reproduced in this volume, they are indebted to the generosity of Professor David Eugene Smith, of Teachers College, Columbia University, New York City, who placed at their disposal his unique collection.

THE AUTHORS.

A HIGH SCHOOL ALGEBRA

PART TWO

CHAPTER I

REVIEW AND EXTENSION OF PROCESSES

POSITIVE AND NEGATIVE NUMBERS

1. Algebra is concerned with the study of numbers. The number of objects in any set (for example, the number of books on a shelf) is found by counting. Such numbers are called whole numbers or integers; also primitive or absolute numbers.

In arithmetic, numbers are usually represented by means of the numerals, 0, 1, 2, 3 ... 9, according to a system known as the decimal notation. In algebra, numbers are represented by numerals and also by letters, either singly or in combinations.

2. Graphical Representation. The natural integers may be represented by equidistant points of a straight line, thus :

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3. Addition. If two sets of objects are united into a single set (for example, the books on two shelves placed on a single shelf), the number of objects in the single set is called the sum of the numbers of objects in the two original sets. The process of finding the sum is called addition.

The sign, +, between two number symbols indicates that the numbers are to be added. In the simplest instances the sum is found by counting.

Thus, to find 5+7, we first count 5, and then count 7 more of the number words next following (six, seven, eight, etc.). The number word with which we end (twelve) names the sum.

4. Graphical Representation. The sum of two integers may be represented graphically thus:

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Theoretically, the sum of two integers can in every instance be found by counting. But it is not necessary or desirable to do so when either (or both) of the numbers is larger than nine. In this case, the properties of the decimal notation, as learned in arithmetic, enable us to abridge the process of counting.

5. Commutative Law of Addition. If two sets of objects are to be united into a single set, the number of objects in the result is obviously the same whether the objects of the second set are united with those of the first, or those of the first united with those of the second.

For example, the number of books is the same whether those on the first shelf be placed on the second, or those on the second be placed on the first.

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This fact is called the commutative law of addition.

The letters a and b are here used to stand for integers, but the law will apply when they stand for any algebraic numbers.

6. Graphical Representation. The commutative law may be represented graphically thus :

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7. Addition of Two or More Whole Numbers. If more than two sets of objects are united into a single set, the number of objects in the resulting set is called the sum of the number of objects in the original sets, and the process of finding the sum is called addition. As in the case of two numbers, the sum of three or more numbers may be found by counting in the simplest instances, and for larger numbers, the process may be abridged by use of the properties of the decimal notation.

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