Constructible numbers

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Definition

Given two distinct complex numbers, $z$, \(w\), plotted as points in the complex plane, we may construct a straight line through \(z\) and \(w\), or construct a circle whose center is \(z\) and whose circumference passes through \(w\). Denote such a line as \(L(z, w)\) and such a circle as \(C(z, w)\). The constructible numbers are defined recursively:

• \(0\) is a constructible number
• \(1\) is a constructible number
• If \(\alpha\), \(\beta\), \(\gamma\), and \(\delta\) are constructible numbers, then
  • if there are finitely many intersection points of \(L(\alpha, \beta)\) with \(L(\gamma, \delta)\), then those intersection points are constructible numbers
  • if there are finitely many intersections of \(C(\alpha, \beta)\) with \(L(\gamma, \delta)\), then those intersection points are constructible numbers
  • if there are finitely many intersections of \(C(\alpha, \beta)\) with \(C(\gamma, \delta)\), then those intersection points are constructible numbers