The New Student's Reference Work/Algebra

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Al'gebra is a branch of mathematics which deals chiefly with “functions” or general values instead of special values as in arithmetic.  The ancient Egyptians practised simple equations, an example being this:  “Its whole added to its seventh gives 19, how much is it?”  In other words, they solved the equation \frac{x}{7} + x = 19.  The Greeks added something to algebra; thus Euclid, about 300 B. C., knew that (a + b)² = a² + b² + 2ab.  Other steps in advance were made in Alexandria and Persia.  But algebra was only used as a help to arithmetic until Viéte or Vieta, a Frenchman, in 1591 made of it an independent science.  As to the usefulness of algebra, it can only be said that it is needful to all advanced work in mathematics.  Its value to the professional man or workman may not be great, except that it is well for every man to know a little of each of the branches of truth.

The teaching of algebra might well follow the historical order; and begin with simple equations as did the Egyptians.  For here algebra is of obvious use in making the problems of arithmetic more simple.  Let one ask the following “catch” question:  “A goose weighs six pounds and half its own weight, what is the weight of the goose?”  The answer is seldom given rightly without setting x for the weight of the goose, thus: x = 6 + \frac{x}{2} which gives the answer 12 pounds.  It is better to begin, however, with practical questions. The most important modern change in the teaching of algebra has been brought about by Professor Chrystal, who has called attention to the nature of general functions as the real object of study in this science.  A knowledge of general functions, such as the following for a quadratic equation, ax² + bx + c = d, has always been implied in the teaching of algebra; but it has only lately been insisted upon.  It has been usual to teach the use of root signs and signs for brackets as if they formed a part of algebra; but in reality these operations belong to pure arithmetic.