# motivation of definition of constructible numbers

In order to understand the significance of constructible numbers and how they are useful in solving problems in Euclidean geometry  , we need to determine how the definitions and properties of these numbers relate to Euclidean geometry.

To start with, let us investigate some properties of $\mathbb{E}$, the field of real constructible numbers:

1. 1.

$0,1\in\mathbb{E}$;

2. 2.

If $a,b\in\mathbb{E}$, then also $a\pm b$, $ab$, and $a/b\in\mathbb{E}$, the last of which is meaningful only when $b\not=0$;

3. 3.

If $r\in\mathbb{E}$ and $r>0$, then $\sqrt{r}\in\mathbb{E}$.

It turns out that the nonnegative elements of $\mathbb{E}$ are in one-to-one correspondence with lengths of constructible line segments  (http://planetmath.org/Constructible2). Let us determine why this is:

First of all, $0\in\mathbb{E}$ and $1\in\mathbb{E}$ are self-evident, as these are basic requirements for $\mathbb{E}$ to be a field. Moreover, $1\in\mathbb{E}$ corresponds to the tacit assumption in compass and straightedge construction that a line segment of length $1$ is constructible (http://planetmath.org/Constructible2).

Secondly, if $a,b\in\mathbb{E}$, which should mean that line segments of lengths $|a|$ and $|b|$ are constructible, then we can easily construct line segments of lengths $|a+b|$ and $|a-b|$ by matching up endpoints  of line segments.

Thirdly, if $a,b\in\mathbb{E}$, then we can construct a line segment of length $|ab|$ by the compass and straightedge construction of similar triangles.

Fourthly, if $a,b\in\mathbb{E}$ and $b\neq 0$, we can construct a line segment of length $1/|b|$ by the compass and straightedge construction of inverse point. By the previous paragraph, multiplication by $a$ poses no problems.

Finally, if $r\in\mathbb{E}$ and $r>0$, then we can construct a line segment of length $\sqrt{r}$ by the compass and straightedge construction of geometric mean, letting $a=1$ and $b=r$.

Now to address the definition of $\mathbb{F}$, the field of constructible numbers:

1. 1.

$0,1\in\mathbb{F}$;

2. 2.

If $a,b\in\mathbb{F}$, then also $a\pm b$, $ab$, and $a/b\in\mathbb{F}$, the last of which is meaningful only when $b\not=0$;

3. 3.

If $z\in\mathbb{F}\setminus\{0\}$ and $\operatorname{arg}(z)=\theta$ where $0\leq\theta<2\pi$, then $\sqrt{|z|}e^{\frac{i\theta}{2}}\in\mathbb{F}$.

It turns out that the elements of $\mathbb{F}$ are in one-to-one correspondence with the constructible points (http://planetmath.org/Constructible2) of the complex plane  . Let us determine why this is:

Rule 1 is similarly justified as above.

In order to justify rule 2, all we need is the justification of rule 2 for $\mathbb{E}$ along with the notion of copying an angle. For example, if $a,b\in\mathbb{F}$, then the following picture can be made by copying an angle:

Finally to justify rule 3. If $z\in\mathbb{F}$, then $|z|\in\mathbb{E}$, so we have that $\sqrt{|z|}\in\mathbb{E}$. Since $|z|e^{i\theta}=z\in\mathbb{F}$, we must have that an angle with measure (http://planetmath.org/AngleMeasure) $\theta$ is constructible. By the compass and straightedge construction of angle bisector  , an angle with measure $\theta/2$ is also constructible.

Title motivation of definition of constructible numbers MotivationOfDefinitionOfConstructibleNumbers 2013-03-22 17:16:05 2013-03-22 17:16:05 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Topic msc 12D15 msc 51M15 CompassAndStraightedgeConstruction