=

premises that 0 i = 0. Thus, if xi = 0, then (xi) (xi) = 0, that is, x2 0; therefore x0, and therefore 0 i = 0. 184. Most laws of operation with neomonic numbers are evident from familiar principles. Thus :—

ai — bi = (a — b) i . . . hence the difference of two neomonic numbers is neomonic.

ai × b = abi . . . hence the product of a neomonic and a protomonic number is neomonic.

ai bi = =

ab. . . hence the product of two neomonic numbers is protomonic.

12

hence ratios of protomonic and neomonic numbers are neomonic.

hence ratios of neomonic numbers are

it

¿2 — — 1 ; ¿3 — — 1√−1 = − i ; ¿1 = i2 ï2 = + 1; and

i4n

=

=

=

--

where n is a positive integer,

that is, the positive integral power of a neomonic number is protomonic or neomonic according as the same power of i is protomonic or neomonic. Moreover, the integral powers. of i are seen to recur in a period or cycle of four different values. Negative exponents result as always in the reciprocal of the same number with like positive exponent.

185. Discussion of radical functions of i, and the interpretation of neomonic exponents, is postponed to more

advanced studies; but we are not led to any new application of the principle of Continuity, and therefore to no new mode of Number, beyond the result of combining protomonic and neomonic numbers in addition and subtraction.

186. The extension of the number-concept reaches its own essential terminus in the operation a + bi, where a and are protomonic.

In a + bi we have the most general expression of number; for it is protomonic, neomonic, or complex, according as b=0, a = 0, or neither equals 0.

187. The result of the operation a + bi, is called a complex number; and is seen to be really a new mode of Number by considering the series of complex numbers formed in a + bi, as a and b pass independently through all protomonic values.

188. It is highly important to note this two-fold, twodimensional (vide § 229, et seq.), character of complex number, and its consequent contrast with protomonic and neomonic number. There is only one way of varying x continuously (without repetition of intermediate values) from 2 to 3, if it remains protomonic. Likewise, only one way for continuous passage of x from - 2 i to + 3 i, if it is to be always neomonic. But in utter contrast, there is an infinite variety of ways for x to pass continuously from 23i to +2+3i, remaining always a complex number. (Vide § 197.)

190. Complex number contains all protomonic and all neomonic number as special cases, and is therefore Number in its final generalization.

191. The student should everywhere carefully avoid confusion in dealing with the alternate square roots of any

number; but especially is this the case with neomonic numbers. Having been accustomed to write (vide §§ 64, 158) √a √b = √ab, he may fall into the error of writing √=a√_b = √(− a) (— b) = √ab. I call this an error because we must be consistent in algebraic conventions; and in such contexts the positive root is understood by √ab.*

It is not a true statement that √a √õ = √ab, if the square roots are to be taken at random. One cannot make various assertions in the same sentence. Therefore, in √a √b = √ab, we evidently mean only the positive square roots to be considered. If negative roots are to be taken into account, we must say what we mean. Thus (writing√a for positive square root of a, and Va for negative square root of a) (— √ā) (− √õ) + √ab; or

Now, if in accordance with the algebraic convention plainly exhibited above, we consider only positive square

* In a translation just published of Durège's Theory of Functions of a Complex Variable, by Professors Fischer and Schwatt of the University of Pennsylvania, Philadelphia, 1896, it is stated on Page 10 of the Introduction: "Euler himself taught, as now generally accepted, that, if a and b denote two positive quantities, — a√―b=√ab; i.e., that the product of two imaginary quantities is equal to a real quantity." The omission of the minus sign before Vab may be a typographical error; for the authors, like all others, use — a√-b = == Vab.

In the translators' Introduction it is very appropriately remarked:— "To follow the gradual development of the theory of imaginary quantities is especially interesting, for the reason that we clearly perceive with what difficulties is attended the introduction of ideas, either not at all known before, or at least not sufficiently current. The times at which negative, fractional, and irrational quantities were introduced into mathematics are so far removed from us, that we can form no adequate conception of the difficulties which the introduction of those quantities may have encountered. Moreover, the knowledge of the nature of imaginary quantities has helped us to a better understanding of negative, fractional, and irrational quantities, a common bond closely uniting them all."

Of course I would have one read numbers in the place of "quantities."

roots of neomonic numbers, V-a √-b does not equal Vab, but Vab;

for √ a√b=√ai√bi = i2√ab=(-1)√ab=-√ab. One need find no difficulty in reconciling with the Principle of Continuity the statements that, regarding only positive roots, √a √b = √ab, while a V-b is not √— √ equal to (a) (— b). The law of indices must be applied with due regard to other laws. The essential statement of the law of indices is ax ay = a*+y. This includes all particular cases as a, x, and y assume different characters. But it has been necessary with every phase of number to understand in this statement that only corresponding roots are considered when x and y are fractional with even denominators. (Cf. §§ 144, 146.) For example, 11/2 X 11/2 would not equal 11/2+1/2, or 1, if one positive and one negative root were taken. Now, this fundamental statement of the law of indices holds for all number. It is the very definition of √1, that (− 1)1⁄2 (− 1)1 / 2 = ( − 1)1 / 2 + 1 / 2 − (− 1)1 : 1.

=

It was easily proved for protomonic number that, regarding only corresponding roots when x is a fraction with even denominator, a* ax = (aa), and a* bx (ab); but when a and aa differ in quality, the very conditions of the original statement are abolished (it is as if one positive and one negative root of a had been taken), and different conclusions might be anticipated under the same laws.

In fine, all this is not an anomaly of √- 1 in operation, but merely an alternative statement of its existence. The difficulty lies in the origin of neomonic number, not in its operation.

On the other hand, a/b = (a/b), established for pro

tomonic number, does hold if a and b are neomonic, simply because, in this case, no qualitative difference arises in the direct performance of the operations indicated by the two members of the equation, if, in accordance with the meaning of the formula, only positive roots are regarded.

=

For example, (— 4)1 /2 / (— 9)1 /2 = (4/9)1/2; for ( 12

=

4)1/2 2 i, and (9)1/23 i; therefore, (— 4)1/2/ (— 9)1/2 =2i/3i=2/3. 2/3. Also the positive square root of 4/9 is 2/3.

Note, also, that for a like reason √a √-b=√— ab; for Va√ √a √b i = √ab i, and √ ab = √ab i.

- =

The safe practice is to express every neomonic number in its essentially proper form, as based upon a new unit.

Rules of thumb would conduct one to true results in all operations except multiplication; but for many reasons, always express √-a as Vai. If you do this, correct calculation will be easy under the very definition of the neomon, ¿2

1.

192. As a natural consequence of the view that Algebra is some mysterious conglomeration of "pure symbols" (Cf. Introduction, pp. 8, 12) without content, existing for itself, void of numerical meaning, it was long discussed, as if it were a matter to be settled by parliament, whether √-a √b should equal - ab, or - Vab. Only one hundred years ago English mathematicians were divided on this question. One party argued that the product must be √ ab; because the product of one "impossible quantity" by another, could not possibly equal a "real quantity' as if a priori deduction of what is, or is not, possible with impossible quantities was not ab initio an impossible discussion within the realm of Reason.