# 1911 Encyclopædia Britannica/Number/Complex Numbers

19. *Complex Numbers*.—If is an assigned number, rational or irrational, and a natural number, it can be proved that there is a real number satisfying the equation
, except when is even and is negative: in this case the equation is not satisfied by any real number whatever. To remove the difficulty we construct an aggregate of polar couples
, where
are any two real numbers, and define the addition and multiplication of such couples by the rules

- ;
- .

We also agree that if or if and . It follows that the aggregate has the ground element , which we may denote by ; and that, if we write for the element ,

- .

Whenever are rational, , and we are thus justified in writing, if we like, for , in all circumstances. A further simplification is gained by writing instead of , and regarding as a symbol which is such that , but in other respects obeys the ordinary laws of operation. It is usual to write instead of ; we thus have an aggregate of complex numbers . In this aggregate, which includes the real continuum as part of itself, not only the four rational operations (excluding division by , the zero element), but also the extraction of roots, may be effected without any restriction. Moreover (as first proved by Gauss and Cauchy), if are any assigned real or complex numbers, the equation

- , is always satisfied by precisely real or complex values of , with a proper convention as to multiple roots. Thus any algebraic function of any finite number of elements of

is also contained in , which is, in this sense, a closed arithmetical field, just as is when we restrict ourselves to rational operations. The power of is the same as that of .