# 1911 Encyclopædia Britannica/Number/Complex Numbers

19. Complex Numbers.—If ${\displaystyle a}$ is an assigned number, rational or irrational, and ${\displaystyle n}$ a natural number, it can be proved that there is a real number satisfying the equation ${\displaystyle x^{n}=a}$, except when ${\displaystyle n}$ is even and ${\displaystyle a}$ 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 ${\displaystyle \{x,y\}}$, where ${\displaystyle x,y}$ are any two real numbers, and define the addition and multiplication of such couples by the rules

${\displaystyle \{x,y\}+\{x',y'\}=\{x+x',y+y'\}}$;
${\displaystyle \{x,y\}\times \{x',y'\}=\{xx'-yy',xy'+x'y\}}$.

We also agree that ${\displaystyle \{x,y\}<\{x',y'\},}$ if ${\displaystyle x or if ${\displaystyle x=x'}$ and ${\displaystyle y. It follows that the aggregate has the ground element ${\displaystyle \{1,0\}}$, which we may denote by ${\displaystyle \sigma }$; and that, if we write ${\displaystyle \tau }$ for the element ${\displaystyle \{0,1\}}$,

${\displaystyle \tau ^{2}=\{-1,0\}=-\sigma }$.

Whenever ${\displaystyle m,n}$ are rational, ${\displaystyle \{m,n\}=m\sigma +n\tau }$, and we are thus justified in writing, if we like, ${\displaystyle x\sigma +y\tau }$ for ${\displaystyle \{x,y\}}$, in all circumstances. A further simplification is gained by writing ${\displaystyle x}$ instead of ${\displaystyle x\sigma }$, and regarding ${\displaystyle \tau }$ as a symbol which is such that ${\displaystyle \tau ^{2}=-1}$, but in other respects obeys the ordinary laws of operation. It is usual to write ${\displaystyle i}$ instead of ${\displaystyle \tau }$; we thus have an aggregate ${\displaystyle {\mathfrak {I}}}$ of complex numbers ${\displaystyle x+yi}$. In this aggregate, which includes the real continuum as part of itself, not only the four rational operations (excluding division by ${\displaystyle \{0,0\}}$, the zero element), but also the extraction of roots, may be effected without any restriction. Moreover (as first proved by Gauss and Cauchy), if ${\displaystyle a_{0},a_{1},\dots a_{n}}$ are any assigned real or complex numbers, the equation

${\displaystyle a_{0}z^{n}+a_{1}z^{n-1}+\dots +a_{n-1}z+a_{n}=0}$, is always satisfied by precisely ${\displaystyle n}$ real or complex values of ${\displaystyle z}$, with a proper convention as to multiple roots. Thus any algebraic function of any finite number of elements of

${\displaystyle {\mathfrak {I}}}$ is also contained in ${\displaystyle {\mathfrak {I}}}$, which is, in this sense, a closed arithmetical field, just as ${\displaystyle {\mathfrak {N}}}$ is when we restrict ourselves to rational operations. The power of ${\displaystyle {\mathfrak {I}}}$ is the same as that of ${\displaystyle {\mathfrak {N}}}$.