In What Sense i Completes the Numeric Operations

Definition 91 · Theorem 435

Definition 91. Let f be an operation made on an operand x by adding and multiplying only, with given derived numbers as its addends and multipliers: f(x) = a₀ + a₁ • x + a₂ • x² + … + aₙ • xⁿ, with n ≧ 1 and aₙ not 0. To undo f at c is to find an x with f(x) = c. The numeric operations are complete among certain derived numbers when every such f, with its addends and multipliers among them, can be undone at every c among them, by an x among them.

The derived numbers were reached by asking, each time, for what undoes an operation:

  1. Adding. Undone by removing. The numbers as such do not suffice (5 from 3). The derived numbers −n and 0 make it always possible: Theorem 321, and Theorem 344 for complex numbers.
  2. Multiplying. Undone by dividing. The derived numbers 1/n make it always possible, except by 0: Theorem 338, and Theorem 361 for complex numbers.
  3. Squaring. Undone by taking the side. Ratios do not suffice (Theorem 256). The bounds make it possible for what is positive (Theorem 255). i makes it possible for everything: Theorem 434.

So far this is proved: among the complex numbers, every f of the first degree can be undone, and squaring can be undone, and therefore (by completing the square) every f of the second degree.

Theorem 435 (statement; proof owed). The numeric operations are complete among the complex numbers. That is: for every f as in Definition 91 with complex addends and multipliers, and every complex c, there is a complex x with f(x) = c.

This is the full sense of the claim that i completes the numeric operations: after i, no further derived number is called for by adding and multiplying. It is not proved here. A proof needs three things that this work does not yet have: that f changes little when x changes little; that the absolute value of f(x) − c has a least value on a bounded region, which rests on Theorem 339; and that at any x where that value is not 0 it can be made less. These belong to the analysis that Theorem 339 opens. Until then the claim is proved for the first and second degrees and asserted for the rest.