“3. If we multiply 1 by −1, or by √−1 · √−1, or by √−1 twice, we swing it counter-clockwise through 180°, and obtain −1; hence, if we multiply it by √−1 once, we should swing it through 90°. Hence we may graphically represent √−1 as the unit on the perpendicular axis YY′, and this gives illustration to √−1, 2√−1, 3√−1, ··· −√−1, −2√−1, −3√−1, or, more briefly, ±i, ±2i, ±3i, ··· where i stands for √−1. We therefore see that i is a symbol of quality (graphically of direction), just as is + or −, and that −3 · 5i, i√5, etc., are just as real as −3 · 5, √5, etc. It is impossible to look out of a window −3 · 5 times as it is to look out −3 · 5i times; strictly, one number is as ‘imaginary’ as the other, although the term has come by custom to apply to one and not to the other. “4. The complex number 3 + 2i is now readily understood. Just as 3 + (−2) is graphically represented by starting from an arbitrary zero, passing 3 units in a positive direction (say to the right), then 2 units in the opposite direction, calling the sum the distance from 0 to the stopping-point, so 3 + 2i may be represented graphically. Starting from 0, pass in the positive direction (to the right in the figure) 3 units, then in the i direction 2 units, calling the sum the distance from 0 to the stopping-place. [Illustration: Graphical representation of 3 + 2i as the hypotenuse of a right-angled triangle with sides of 3 and 2i units] “Of course the question will arise as to the hypotenuse being the sum of the two sides of the right-angled triangle. But the case is parallel to that mentioned in paragraph 2; it is not the sum of the _absolute values_, any more than is 1 the sum of the absolute values of 4 and −3; it is the sum when we define addition for numbers involving direction as well as length. “A simple illustration from the parallelogram of forces is often used to advantage. [Illustration: Parallelogram of two forces +3 and +2i with resultant OP] “Suppose a force pulling 3 lbs. to the right (+3 lbs.) and another pulling 2 lbs. upwards (+2i lbs.); required the resultant of the two. It is evident that this is OP, _i.e._, OP = 3 + 2i. “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.