First Principles of Algebra - 1912

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Preface

In writing the First Principles of Algebra the authors have had constantly before them two chief aims:

(1) To provide a gradual and natural introduction to the symbols and processes of algebra.

(2) To give vital purpose to the study of algebra by using it to do interesting and valuable things.

Each of these aims leads to the same order of topics, which, however, differs somewhat from the conventional order.

If it is admitted that there should be a gradually increasing complexity of forms to be manipulated, it follows that factoring and complicated work in fractions have no proper place in the first half year. This book is arranged so that factoring may begin with the second semester and complicated fractions may come still later. Simple fractions are treated in Chapter V.

The pupil is introduced to the algebraic notation by recalling and stating in terms of letters certain rules of arithmetic with which he is already familiar. The simplicity of the algebraic formulas, compared with the arithmetical statement of rules known to the pupil, cannot fail to impress him with the usefulness and power of the subject which h is about to study. This impression will be deepened when, in Chapter VI [Literal Equations and Their Uses], rules which caused considerable trouble in arithmetic are derived with the utmost ease by algebraic processes.

[snip]

For the development of skill in algebraic manipulation it is not sufficient to solve a certain number of exercises when an operation is first introduced. To fix each operation in the learner's mind, there must be recurring drills extending over a considerable period of time. These are amply provided for in this book. The fundamental operations on integral and simple fractional expressions, the solution of simple equations, and the representation of given conditions in algebraic symbols are constantly reviewed in the numerous lists of "drill exercises," many of which may be solved mentally. Factoring is practised almost daily throughout the second half year.

The principles of algebra used in the Elementary Course are enunciated in a small number of short rules--eighteen in all. The purpose of these rules is to furnish, in simple form, a codification of those operations of algebra which require special emphasis. Such a codification has several important advantages:

By constant reference to these few fundamental statements they become an organic, and hence a permanent, part of the learner's mental equipment.

By their systematic use he is made to realize that the processes of algebra, which seem so multifarious and heterogeneous, are, in reality, few and simple.

Such a body of principles furnishes a ready means for the correction of erroneous notions, a constant incitement to effective review, and a definite basis upon which to proceed at each stage of progress.

The authors gratefully acknowledge the receipt of many helpful suggestions from teachers who have used their High School Algebra

H. E. Slaught.
N. J. Lennes.
Chicago and New York,
April, 1912

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Chapter VI: Literal Equations and Their Uses

102. Some of the advantages of algebra over arithmetic in solving problems have been pointed out in the preceding chapters. For instance, the brevity and simplicity of statement secured through the use of letters to represent numbers; the translation of problems into equations; and the clear and logical solution of these equations, step by step.

Another advantaged is set forth in the present chapter; namely, the opportunity offered in Algebra to summarize the solution of a whole class of problems by solving what is called a literal equation, thus obtaining a formula which may be used in solving other problems.

For example, in arithmetic we solved many problems obtaining the interest when the principal, rate, and time were given. We now see that all of these can be summarized in the one literal equation

i = prt.

[snip]

104. In arithmetic a problem is said to be solved when a numerical answer is obtained which satisfies the conditions given. The solutions thus far found in algebra have, for the most part, been of this sort.

It is customary, however, to say that a problem has been solved in the algebraic sense when a formula is found which gives complete directions for deriving the numerical answer.

Thus, p = i/rt is a solution for the principal since it states precisely how to find the principal in terms of interest, rate, and time.

105. It is thus seen that from the literal equation i=prt we obtain the complete solution of every problem which calls for any one of these four numbers in terms of the other three.

In modern times machines are extensively used for computation. The algebraic solution of a literal equation gets the problem ready for the computing machine, that is, it gets the formula which the computer must use.

First Principles of Algebra: Elementary Course
by H.E. Slaught Associate Professor of Mathematics in the University of Chicago
N.J. Lennes Instructor in Mathematics in Columbia University
Boston ALlyn and Bacon 1912
pp. pp. iii - v; 92-93

One hundred years ago, mathematicians wrote math textbooks with the advice of math teachers. Today parents have to organize grass roots political movements to lobby for the inclusion of mathematicians in the design of national standards.


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