Editing Constructible number (section)

==Algebraic definitions==
===Algebraically constructible numbers===
The algebraically constructible real numbers are the subset of the [[real number]]s that can be described by formulas that combine integers using the operations of addition, subtraction, multiplication, multiplicative inverse, and square roots of positive numbers. Even more simply, at the expense of making these formulas longer, the integers in these formulas can be restricted to be only 0 and 1.{{sfnp|Martin|1998|pp=36–37}} For instance, the [[square root of 2]] is constructible, because it can be described by the formulas <math>\sqrt2</math> or <math>\sqrt{1+1}</math>.

Analogously, the algebraically constructible [[complex number]]s are the subset of complex numbers that have formulas of the same type, using a more general version of the square root that is not restricted to positive numbers but can instead take arbitrary complex numbers as its argument, and produces the [[Square root#Principal square root of a complex number|principal square root]] of its argument. Alternatively, the same system of complex numbers may be defined as the complex numbers whose real and imaginary parts are both constructible real numbers.{{sfnp|Roman|1995|p=207}} For instance, the complex number <math>i</math> has the formulas <math>\sqrt{-1}</math> or <math>\sqrt{0-1}</math>, and its real and imaginary parts are the constructible numbers 0 and 1 respectively.

These two definitions of the constructible complex numbers are equivalent.<ref name=lz440>{{harvp|Lawrence|Zorzitto|2021|p= [https://books.google.com/books?id=-koyEAAAQBAJ&pg=PA440 440]}}.</ref> In one direction, if <math>q=x+iy</math> is a complex number whose real part <math>x</math> and imaginary part <math>y</math> are both constructible real numbers, then replacing <math>x</math> and <math>y</math> by their formulas within the larger formula <math>x+y\sqrt{-1}</math> produces a formula for <math>q</math> as a complex number. In the other direction, any formula for an algebraically constructible complex number can be transformed into formulas for its real and imaginary parts, by recursively expanding each operation in the formula into operations on the real and imaginary parts of its arguments, using the expansions<ref>For the addition and multiplication formula, see {{harvp|Kay|2021|p=187|loc=Theorem 8.1.10}}. For the division formula, see {{harvp|Kay|2021|pp= 188, 224|loc= Equations 8.8 & 9.2}}. The expansion of the square root can be derived from the [[half-angle formula]] of trigonometry; see an equivalent formula at {{harvp|Lawrence|Zorzitto|2021|p= [https://books.google.com/books?id=-koyEAAAQBAJ&pg=PA440 440]}}.</ref>
*<math>(a+ib)\pm (c+id)=(a \pm c)+i(b \pm d)</math>
*<math>(a+ib)(c+id)=(ac-bd) + i(ad+bc)</math>
*<math>\frac{1}{a+ib}=\frac{a}{a^2+b^2} + i \frac{-b}{a^2+b^2}</math>
*<math>\sqrt{a+ib} = \frac{(a+r)\sqrt{r}}{s} + i\frac{b\sqrt{r}}{s}</math>, where <math>r=\sqrt{a^2+b^2{}_{\!}}</math> and <math>s=\sqrt{(a+r)^2+b^2}</math>.

===Algebraically constructible points===
The algebraically constructible points may be defined as the points whose two real Cartesian coordinates are both algebraically constructible real numbers. Alternatively, they may be defined as the points in the [[complex plane]] given by algebraically constructible complex numbers. By the equivalence between the two definitions for algebraically constructible complex numbers, these two definitions of algebraically constructible points are also equivalent.<ref name=lz440/>