Imaginary Numbers

If the mathematical ideas dealt with in the

last chapter have been a popular success, those of the present chapter have excited almost as much general attention. But their success has been of a different character, it has been what the French term a succès de scandale. Not only the practical man, but also men of letters and philosophers have expressed their bewilderment at the devotion of mathematicians to mysterious entities which by their very name are confessed to be imaginary. At this point it may be useful to observe that a certain type of intellect is always worrying itself and others by discussion as to the applicability of technical terms. Are the incommensurable numbers properly called numbers? Are the positive and negative numbers really numbers? Are the imaginary numbers imaginary, and are they numbers?–-are types of such futile questions. Now, it cannot be too clearly understood that, in science, technical terms are names arbitrarily assigned, like Christian

names to children. There can be no question of the names being right or wrong. They may be judicious or injudicious; for they can sometimes be so arranged as to be easy to remember, or so as to suggest relevant and important ideas. But the essential principle involved was quite clearly enunciated in Wonderland to Alice by Humpty Dumpty, when he told her, à propos of his use of words, "I pay them extra and make them mean what I like." So we will not bother as to whether imaginary numbers are imaginary, or as to whether they are numbers, but will take the phrase as the arbitrary name of a certain mathematical idea, which we will now endeavour to make plain.

The origin of the conception is in every way similar to that of the positive and negative numbers. In exactly the same way it is due to the three great mathematical ideas of the variable, of algebraic form, and of generalization. The positive and negative numbers arose from the consideration of equations like $x + 1 = 3$ , $x + 3 = 1$ , and the general form $x + a = b$ . Similarly the origin of imaginary numbers is due to equations like $x^{2} + 1 = 3$ , $x^{2} + 3 = 1$ , and $x^{2} + a = b$ . Exactly the same process is gone through. The equation $x^{2} + 1 = 3$ becomes $x^{2} = 2$ , and this has two solutions, either $x = + \sqrt{2}$ , or $x = - \sqrt{2}$ . The statement that there are these alternative

solutions is usually written $x = \pm \sqrt{2}$ . So far all is plain sailing, as it was in the previous case. But now an analogous difficulty arises. For the equation $x^{2} + 3 = 1$ gives $x^{2} = - 2$ and there is no positive or negative number which, when multiplied by itself, will give a negative square. Hence, if our symbols are to mean the ordinary positive or negative numbers, there is no solution to $x^{2} = - 2$ , and the equation is in fact nonsense. Thus, finally taking the general form $x^{2} + a = b$ , we find the pair of solutions $x = \pm \sqrt{(b - a)}$ , when, and only when, $b$ is not less than $a$ . Accordingly we cannot say unrestrictedly that the "constants" $a$ and $b$ may be any numbers, that is, the "constants" $a$ and $b$ are not, as they ought to be, independent unrestricted "variables"; and so again a host of limitations and restrictions will accumulate round our work as we proceed.

The same task as before therefore awaits us: we must give a new interpretation to our symbols, so that the solutions ± ( b − a ) for the equation x 2 + a = b always have meaning. In other words, we require an