“The illustrations of the negative number are so numerous, so simple, and so generally known from the common text-books that it is unnecessary to dwell upon them.[32] Debt and credit, the scale on the thermometer, longitude, latitude, the upward pull of a balloon compared with the force of gravity, and the graphic illustration of these upon horizontal and vertical lines--all these are familiar. “But the imaginary and complex numbers have been left enshrouded in mystery in most text-books. The books say, _inter lineas_, ‘Here is √−1; it means nothing; you can’t imagine it; the writer knows nothing about it; let us have done with it, and go on.’ Such is the way in which the negative was treated in the early days of printed algebras, but now such treatment would be condemned as inexcusable. But there is really no more reason to-day for treating the imaginary so unintelligently than for presenting the negative as was the custom four hundred years ago. The graphic treatment of the complex number is not to-day so difficult for the student about to take up quadratics as is the presentation of the negative to one just beginning algebra. “Briefly, the following outline will suffice to illustrate the procedure for the complex number:-- 5 4 3 2 1 | 1 2 3 4 5 |___|___|___|___|___|___|___|___|___|___| | − 0 + “1. Negative numbers may be represented in a direction opposite to that of positive numbers, starting from an arbitrary point called zero. Hence, when we leave the domain of positive numbers, _direction_ enters. But there are infinitely many directions in a plane besides those of the positive and negative numbers, and hence there may be other numbers than these. “2. When we add positive and negative numbers we find some results which seem strange to a beginner. For example, if we add +4 and −3 we say the sum is 1, although the _length_ 1 is less than the length 4 or the length −3; yet this does not trouble us because we have considered something besides length, namely, direction; it is true, however, that the sum of 4 and −3 is less than the absolute value of either. This is seen to be so reasonable, however, from numerous illustrations (as the combined weight of a balloon pulling up 3 lbs., tied to a 4-lb. weight), that we come not to notice the strangeness of it; graphically, we think of the sum as obtained by starting from 0, going 4 in a positive direction, then 3 in a negative direction, the _sum_ being _the distance from 0 to the stopping-place_. [Illustration: Graph of 1 multiplied by √−1 twice]