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

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 $b\ne 0$;

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 onetoone 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 selfevident, 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 $ab$ 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 $b\ne 0$, 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.
$0,1\in \mathbb{F}$;

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 $b\ne 0$;

3.
If $z\in \mathbb{F}\setminus \{0\}$ and $\mathrm{arg}(z)=\theta $ where $$, then $\sqrt{z}{e}^{\frac{i\theta}{2}}\in \mathbb{F}$.
It turns out that the elements of $\mathbb{F}$ are in onetoone 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 

Canonical name  MotivationOfDefinitionOfConstructibleNumbers 
Date of creation  20130322 17:16:05 
Last modified on  20130322 17:16:05 
Owner  Wkbj79 (1863) 
Last modified by  Wkbj79 (1863) 
Numerical id  7 
Author  Wkbj79 (1863) 
Entry type  Topic 
Classification  msc 12D15 
Classification  msc 51M15 
Related topic  CompassAndStraightedgeConstruction 