Mathematical Analysis · Complex Numbers
The equation $x^2+1=0$ has no solutions in R — but adding a solution to R can be done in exactly one way (up to renaming), and that one way can be looked at from three different angles. Along the way we'll see that multiplying by a complex number is exactly the rotational homothety already covered in «Similarities of the Plane».
The equation $x^2+1=0$ has no solutions among the real numbers: the square of any real number is non-negative, so $x^2+1\ge1>0$ for every $x$. Yet R is already a complete ordered field (see «Levels of Formalization of Mathematical Analysis», where that completeness is laid out at three levels of formalization of differing expressive power — from a first-order language over RCF to ZFA — deliberately not equivalent to one another).
Completeness, however, doesn't give algebraic closure: it guarantees the existence of suprema and limits, not roots of polynomials. R can be extended to a field in which $x^2+1=0$ is solvable — and in exactly one way, up to renaming the elements. Below, that one way is examined from three different angles.
Uniqueness of the construction doesn't mean uniqueness of description: below are three equivalent languages commonly used to introduce the field ℂ.
Constructions of ℂ
A number is a pair $z=(x,y)$. Addition is coordinatewise: $(x,y)+(x',y')=(x+x',\,y+y')$. Multiplication is defined separately: $(x,y)(x',y')=(xx'-yy',\,xy'+x'y)$. Let $i=(0,1)$: then $i\cdot i=(0\cdot0-1\cdot1,\ 0\cdot1+1\cdot0)=(-1,0)$, that is, $i^2=-1$. We write the pair $(x,y)$ as $x+iy$.
To the number $x+iy$ we assign the matrix $\begin{pmatrix}x&-y\\y&x\end{pmatrix}$. Addition and multiplication of numbers become addition and multiplication of such matrices; $i$ corresponds to the $90^\circ$ rotation matrix $\begin{pmatrix}0&-1\\1&0\end{pmatrix}$, and its square $\begin{pmatrix}-1&0\\0&-1\end{pmatrix}=-E$ confirms $i^2=-1$. The determinant $x^2+y^2\ne0$ whenever $(x,y)\ne(0,0)$ — every nonzero number has an inverse; conjugation is transposition.
The polynomial $x^2+1$ has no real roots, hence is irreducible over R, and the quotient ring $\R[x]/(x^2+1)$ is a field. Let $i$ denote the class $[x]$: by the definition of the quotient, $x^2+1\equiv0$, so $i^2=-1$ — this is not a fact to be checked, but a direct consequence of which polynomial the remainder is taken modulo. Every class is represented uniquely by a polynomial of degree less than 2, that is, by an expression $x+iy$.
Between any two of these languages there is a translation that preserves addition and multiplication (a field isomorphism) — from here on we switch freely between them, without specifying each time which of the three models a computation is happening in.
Addition is simple: $z\mapsto z+b$ is a translation of the plane by the vector $b$. Multiplication is more interesting. Fix $a\ne0$ and consider the transformation $w=az$: this is a rotational homothety centered at zero — scale factor $|a|$, rotation angle $\arg a$ — the same object already covered in «Similarities of the Plane», except now the center is pinned at zero and the shift $b=0$.
The condition $a\ne0$ is essential: at $a=0$ the whole plane collapses onto the point $0$ — this is no longer a similarity, but a degenerate map.
The field ℂ has been built three equivalent ways; addition turned out to be translation, and multiplication by a fixed number $a$ the same rotational homothety examined in «Similarities of the Plane», only now obtained from arithmetic rather than geometry. Yet $f(z)=az$ is a linear function, the simplest of all functions of a complex variable.
The natural next step is more complicated functions: polynomials, rational fractions, power series. Some of them have, at every point of their domain, a derivative $f'(z)=\lim\limits_{\Delta z\to0}\dfrac{f(z+\Delta z)-f(z)}{\Delta z}$, defined by the same limit as for functions of one real variable — except the increment $\Delta z$ is now complex. The properties of such functions are examined in the material «Regular Functions».