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38 THE NATURE OF MATHEMATICS For the case of imaginaries, let a, 6, c, and d be any numbers, then _ = [(ac-M) + J-l) (ad + bc)] [(ac-bd) We get, then, an interesting and easily verifiable theorem on numbers by calculation with imaginaries, and imaginaries disappear from the conclusion. Mathematicians thought, then, that imaginaries, though apparently uninterpretable and even self-contradictory, must have a logic. So they were used with a faith that was almost firm and was only justified much later. Mathematicians indicated their growing security in the use of tj 1 by writing " * " instead of " J 1 " and calling it " the complex unity," thus denying, by implication, that there is anything really imaginary or impossible or absurd about it. The truth is that " * " is not uninterpretable. It represents an operation just as the negative numbers do, but is of a different kind. It is geometrically interpretable also, though not in a straight line, but in a plane. For this we must refer to the Bibliography ; but here we must point out that, in this "generalisation of number" again, the words "addition," " multiplication," and so on, do not have exactly the same, but an analogous, meaning to those which they had before, and that " complex numbers " form a domain like a plane in which a line representing the integers, fractions, and irrationals is contained. But we must leave the further development of these questions. It must be realised that the essence of algebra is its generality. In the most general case, every symbol and every statement of a proposition in algebra is interpretable in terms of certain operations to be undertaken with abstract things such as numbers or classes or propositions. These operations merely express the relations , of these things to one another. If the results at any stage of an algebraical process can be interpreted and this interpretation is often suggested by the
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