, i.e.,
OP
= 3 + 2
i
.
“This elementary introduction to the subject of complex numbers shows that the ‘imaginary’ element is easily removed, and that students about to begin quadratics are able to get at least an intimation of the subject. This is not the place for any adequate treatment of these numbers: such treatment is easily accessible. It is hoped that enough has been presented to render it impossible for any reader to be content with the absolutely meaningless and unjustifiable treatment found in many text-books.”[33]
[32] See Beman & Smith’s “Elements of Algebra,” p. 17.
[33] For an elementary presentation of the subject, see Beman and Smith’s “Elements of Algebra,” Boston, 1900. For a history of the subject, see Beman and Smith’s translation of Fink’s “History of Mathematics,” Chicago, 1900, or Professor Beman’s Vice-Presidential Address before the American Association for the Advancement of Science, 1898, or the author’s “History of Modern Mathematics,” already mentioned.
[257.] The pedagogical value of mathematical instruction, as a whole, depends chiefly on the extent to which it enters into and acts on the pupil’s whole field of thought and knowledge. From this truth it follows, to begin with, that mere presentation does not suffice; the aim must be rather to enlist the self-activity of the pupil. Mathematical exercises are essential. Pupils must realize how much they can do by means of mathematics. From time to time written work in mathematics should be assigned; only the tasks set must be sufficiently easy. More should not be demanded and insisted on than pupils can comfortably accomplish. Some are attracted early by pure mathematics, especially where geometry and arithmetic are properly combined. But a surer road to good results is applied mathematics, provided only the application is made to an object in which interest has already been aroused in other ways.
But the pupils ought not to be detained too long over a narrow round of mathematical problems; there must also be progress in the presentation of the theory. Were the only requisite to stimulate self-activity, the elementary principles might very easily suffice for countless examples affording the pupil the pleasure of increasing facility, and even the delight arising from inventions of his own, without giving him any conception of the greatness of the science. Many problems may be compared to witty conceits, which may be welcome enough in the right place, but which should not encroach on the time for work. There ought to be no lingering over things that with advancing study solve themselves, merely for the sake of performing feats of ingenuity. Incomparably more important than mere practice examples is familiarity with the facts of nature, and such familiarity renders all the better service to mathematics if combined with technical knowledge.