I'm currently wrapping up my first course in Algebra, and at the end we touched on ruler and compass constructions of complex numbers. We only explicitly covered constructions involving a "rusty" compass and "non-markable" straight-edge, but the course hinted at other forms of construction (origami, for example). I found it fascinating that we can't construct all complex numbers, and I've been wondering "what it takes" to successfully construct all complex numbers.

So, first, I'd like to define a "tool" as any object used in a geometric construction. A compass and a ruler are each tools. A fold (in origami) is also a tool.

Second, define the "rules" of a tool to be the rules / axioms that dictate how the tool works. For example, the rule of a straight edge is that a straight edge can draw a line that intersects any two points.

With these definitions in mind, I've been wondering: What are the minimal collection of tools that I need to construct every complex number? Perhaps another way of asking this is, given a set of tools that alone cannot construct every complex number, what are the minimum number of new tools that must be added to our collection of tools to construct every complex number? Or perhaps, what is the smallest set of "rules" necessary to construct every complex number? To construct a specific subset $A$ of complex numbers?

For example, a ruler and straight edge does not make the cut, since $\sqrt[3]{2}$ cannot be constructed by those two tools alone. With this in mind, are there other tools that I can add to this collection, where those tools alone can't construct the complex numbers either, but together with the straight edge and compass, can?

Doing research online, I've had an easy time learning what pitfalls and failures each type of construction admits, but I'm not so much interested in specific geometric constructions, as I am in how various geometric constructions can or cannot work in concert. I've also been referenced "Geometric Constructions" by George E. Martin, but have only begun reading it.

Edit: One of the challenges I've had is that, as David C. alludes to below, the definition of a "tool" is quite arbitrary. I'm mostly interested in constructions that, if not practical, could be conceivably carried out, in the way that a ruler and compass construction can be.

What if, for example, my compass was somehow rigged so that it traveled in the shape of an oval? Or, what if I had only a parabolic "straight-edge" (say in the shape of $y=x^2$), and the rule associated with this tool is that I could place it's vertex at one point, then rotate the straightedge until it intersected another point, and then trace the curve of the straightedge? Or some tool that allows me to draw tangents to curves? How might I classify / define these types of tools to help narrow my focus?


Your main problem is that the complex numbers are uncountable. Thus if you have a countable set of tools, and you can only use finite procedures to construct new numbers, then you can only hit countably many numbers, and you will always miss almost all complex numbers.

This is part of a general statement. If you use any procedure that has only countably many ideas and finite strings of them, say language, numbers, constructions, computer programs, anything, then you cannot define almost all complex numbers.

(All of this also applies to the reals.)

If you allow infinite sequences of actions you can hit every complex number, but you can do that just with adding together fractions. (This is the concept of a decimal number.)

  • $\begingroup$ This is a very good answer. However, the OP leaves space to consider a subset of $\Bbb C$: "To construct a specific subset $A$ of complex numbers?" May be the algebraic numbers could be considered? $\endgroup$
    – jjagmath
    Aug 11 at 22:47
  • $\begingroup$ Hmm... that makes sense! What if, instead, I restricted my goals to an at-most countable set of objects. Say, for example, that I wanted to know the minimal set of tools necessary to construct $\{\sqrt[3]{2},\sqrt[5]{2}\}$? (As @jjagmath asks above.) $\endgroup$
    – David
    Aug 11 at 22:47
  • 1
    $\begingroup$ @David Well, it depends on what you mean by a tool in this case. Obviously, if your toolbox includes taking cube roots, then $\sqrt[3]{2}$ is a cinch. Different toolboxes can construct different sets of numbers. So if your toolbox includes being able to solve any polynomial equation, you end up with the algebraic numbers. If your toolbox includes being able to write computer programs, you end up with the set of compuitable numbers (ones where there is a computer program to churn out their decimal expansion, such as $e$ and $\pi$). Ruler and compass produces essentially iterated square roots. $\endgroup$ Aug 11 at 22:59
  • $\begingroup$ As a philosophical aside, if there is no conceivable way to construct even a tiny propotiopn of complex (or real) numbers, in what sense can they be said to exist? Yes, you can make all reals by sums of rationals, but you cannot define which rationals you need to make the reals in the first place. Note that, for the last 100 years, people have not bothered themselves with such nonsense and just got on with proving things. Apart from Doron Zeilberger. $\endgroup$ Aug 11 at 23:06
  • $\begingroup$ @DavidA.Craven: your last remark is amusing, but I am not convinced it is accurate or fair. Many people, following Brouwer and Martin-Loef, for example, have bothered themselves very much with what you dismiss as nonsense. Zeilberger's opinion are yet another topic. $\endgroup$
    – Rob Arthan
    Aug 11 at 23:33

As David A. Craven said in his answer, it's hopeless to be able to construct all the complex number.

However, I think it's an interesting question to find a set of tools that enable to construct the less ambitious but still very interesting set of algebraic numbers.

For example, a construction of $\sqrt[3]{2}$ can be done with a marked ruler or with origami.


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