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Constructible Numbers
Given an unit length, a real number α is constructible if a line segment of length |α| can be constructed from the unit length in a ﬁnite number of steps using only a ruler (i.e. a straight unmarked edge) and a compass. Not all real numbers are constructible, but all rational numbers are. All Constructible number 's are algebraic.

Overview
In order to understand origami numbers, we need to understand it in the context of algebra and complex numbers. When dealing with origami constructions, we need to imagine a piece of infinitely large paper with two points marked onto it. We use the sheet as a representation of the complex plane, with the real axis going through the two previously marked points. The imaginary axis runs perpendicular to the real axis, with an intersection at one of the points marked. This enables us to converse about the points, as well as Complex number's.

A real number r is origami-constructible if one can construct in a finite number of steps two points which are a distance of |r| apart. Origami constructible numbers are the same as origami constructible lengths.

Origami constructions consist of a number of folds in the piece of paper. As you fold a piece of paper, you leave a crease in it. You need to unfold before the next fold so that these creases then act as our lines. A point is only created when two of these creases (lines) cross over (intersect). These folds are defined by the axioms mentioned earlier in this writing (Huzita–Hatori axioms). A complex number ‘x’ can, therefore, be defined as ‘origami-constructible’ if it satisfies the following:

Take a sheet of paper with two points marked on it. Label these points 0 and 1. Make several folds so that two of these creases intersect at a point p. The point p must correspond to where ‘x’ is on the complex plane.

We come back to our piece of infinitely big paper which acts as our complex plane. The two points marked upon it as 0 and 1. If we apply the first axiom to point 0, we create our real axis. Then, if we apply the fourth axiom to point 0 as well as the real axis, we create our imaginary axis. We next need to add reference points to our imaginary axis. We start by finding the point ‘i’. To do this, we need to apply the fourth axiom to point 1 as well as the real axis to get l1. L1 is parallel to the imaginary axis and passes through the point 1. We then need to apply the fifth axiom through points 0,1 and l1 to give us l2. The line l2 passes through 1 and I as it intersects the imaginary axis at i. The origami-constructible numbers form a subfield of C.

Origami Numbers as a subgroup of C
As the origami constructible numbers form a subfield of complex numbers C, we will show that for any origami constructible numbers α and β, α-β and αβ are also origami constructible numbers. We will also show that when α is not equal to 0, $$\alpha^{-1}$$ is also an origami constructible number.

Firstly, there are some operations we will define in order to help with our proof. Using Huzita–Hatori axioms we develop these two elementary operations.

E1 – Taking a point p and a line l, we must fold a line to create a crease parallel to l that passes through p. We create this by using the fourth axiom twice.

E2 – Taking a point p and a line l, we must reflect the point p across the line l. We create this by using the fourth axiom then the fifth axiom.

Addition:
Firstly, we will start with addition. This operation can be carried out with two uses of the first axiom and two uses of E1. By starting with 3 points: α, β and 0, it is then possible to create a fourth point, c. Where c = (α+β). We do this by creating a parallelogram where the three points, α, β and 0 make up the first three corners, and c makes the fourth. It is easy to see that c is, therefore, a constructible number and is the addition of α and β.

To do this, we follow these steps :

1.    Using the first axiom, between 0 and alpha, create line l1.

2.    Using the first axiom, between 0 and β, create line l2.

3.    Using E1, between β and l1, create l3.

4.    Using E1, between α and l2, create l4.

5.    Identify point c, as the intersection between l3 and l4.

Multiplication:
Next, we will look at multiplication. This is slightly more difficult as we consider multiplication by a real number and then multiplication by i. By using properties of similar triangles, we can multiply α by a real number. To do this, follow these steps:

1.    Using the first axiom, between 0 and α, create line l1.

2.    Using the first axiom, between 1 and α, create line l2.

3.    Using E1, between r and l2, create line l3.

4.    Identify point c, as the intersect between l1 and l3.

This method works only if α is not a real number. If α is a real number then we need to add I to it, multiply it by r, and then using the fourth axiom, project it onto the real axis.

If we need to multiply α by I, this requires a rotation of $$\pi/2$$ radians. We do this by the following steps:

1.    Using the first axiom, between 0 and α, create line l1.

2.    Using the fourth axiom, between points 0 and l1, create line l2.

3.    Using the second axiom, between points l1 and l3, create line l3.

4.    Using E2, between alpha and l3, create point c.

Using these two methods, we can multiply by real or complex numbers to create an origami constructible number.

Inversion:
Lastly, we will look at inversion. We are aware that the inverse of a complex number (a,b) = (a-bi)/(a^2 + b^2). We simply need to do this calculation, so we need to be able to divide. We do this by the following steps:

1.    Using the first axiom, between 0 and α, create line l1.

2.    Using the first axiom, between points r and α, create line l2.

3.    Using E1, between 1 and l2, create line l3.

4.    Identify point c, as the intersect between l1 and l3.

If alpha is a real number, then we solve this the same way we did in multiplication.

By using these methods, we can see that for any origami constructible numbers α and β, α-βand αβ are also origami constructible numbers.