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\PMlinkescapeword{constructible} |
| \PMlinkescapeword{Euclidean} |
\PMlinkescapeword{Euclidean} |
| \PMlinkescapeword{length} |
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| \PMlinkescapeword{level} |
\PMlinkescapeword{level} |
| \PMlinkescapeword{measure} |
\PMlinkescapeword{measure} |
| \PMlinkescapeword{open} |
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| \PMlinkescapetext{This page is under construction.} |
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| An ordered field $F$ is \emph{Euclidean} if every non-negative element $a$ ($a\geq0$) is a square in $F$ (there exists $b\in F$ such that $b^2=a$). |
An ordered field $F$ is \emph{Euclidean} if every non-negative element $a$ ($a\geq0$) is a square in $F$ (there exists $b\in F$ such that $b^2=a$). |
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| For example, $\mathbb{R}$ is Euclidean. On the other hand, $\mathbb{Q}$ is not Euclidean because 2 is not a square in $\mathbb{Q}$ (\PMlinkname{i.e.}{Ie}, $\pm\sqrt{2}\notin \mathbb{Q}$). Also, $\mathbb{C}$ is not a Euclidean field because \PMlinkname{$\mathbb{C}$ is not an ordered field}{MathbbCIsNotAnOrderedField}. |
For example, $\mathbb{R}$ is Euclidean. On the other hand, $\mathbb{Q}$ is not Euclidean because 2 is not a square in $\mathbb{Q}$ (\PMlinkname{i.e.}{Ie}, $\pm\sqrt{2}\notin \mathbb{Q}$). Also, $\mathbb{C}$ is not a Euclidean field because \PMlinkname{$\mathbb{C}$ is not an ordered field}{MathbbCIsNotAnOrderedField}. |
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| A Euclidean field is an ordered Pythagorean field. |
A Euclidean field is an ordered Pythagorean field. |
| There are ordered fields that are Pythagorean but not Euclidean. |
There are ordered fields that are Pythagorean but not Euclidean. |
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| The smallest subfield $\mathbb{E}$ of $\mathbb{R}$ over $\mathbb{Q}$ such that $\mathbb{E}$ is Euclidean is called the \emph{field of real constructible numbers}. A element of $\mathbb{E}$ is called a \emph{constructible number}. These numbers can be ``constructed'' by a process that will be described shortly. First, note that $\mathbb{E}$ has the following properties: |
The smallest subfield $\mathbb{E}$ of $\mathbb{R}$ over $\mathbb{Q}$ such that $\mathbb{E}$ is Euclidean is called the \emph{field of real constructible numbers}. A element of $\mathbb{E}$ is called a \emph{constructible number}. These numbers can be ``constructed'' by a process that will be described shortly. First, note that $\mathbb{E}$ has the following properties: |
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| \begin{enumerate} |
\begin{enumerate} |
| \item $0,1\in\mathbb{E}$; |
\item $0,1\in\mathbb{E}$; |
| \item If $a,b\in\mathbb{E}$, then so are $a\pm b$, $ab$, and $a/b\in\mathbb{E}$, the last of which is meaningful only when $b\not=0$; |
\item If $a,b\in\mathbb{E}$, then so are $a\pm b$, $ab$, and $a/b\in\mathbb{E}$, the last of which is meaningful only when $b\not=0$; |
| \item If $r\in\mathbb{E}$ and $r>0$, then $\sqrt{r}\in\mathbb{E}$. |
\item If $r\in\mathbb{E}$ and $r>0$, then $\sqrt{r}\in\mathbb{E}$. |
| \end{enumerate} |
\end{enumerate} |
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| Conversely, let us start with a subset $S$ of $\mathbb{R}$ such that $S$ contains a non-zero real number. Call any of the binary operations in condition 2 above, and the square root unary operation in condition 3 a \emph{ruler and compass operation}. (Note that, in mathematics, phrases such as ``ruler and compass'' and ``compass and straightedge'' \PMlinkescapetext{mean} the same thing.) Call a real number \emph{constructible from} $S$ if it can be obtained from elements of $S$ by a finite sequence of ruler and compass operations. Note that $1\in S$. If $S^{\prime}$ is the set of numbers constructible from $S$ using only the binary ruler and compass operations (those in condition 2), then $S^{\prime}$ is a subfield of $\mathbb{R}$, and is the smallest field containing $S$. Next, denote $\hat{S}$ the set of all constructible numbers from $S$. It is not hard to see that $\hat{S}$ is also a subfield of $\mathbb{R}$, but an extension of $S^{\prime}$. Furthermore, it is not hard to show that $\hat{S}$ is Euclidean. The general process (algorithm) of \PMlinkescapeword{generating} elements in $\hat{S}$ from elements in $S$ using finite sequences of ruler and compass operations is called a \emph{ruler and compass construction}. These are so called because, given two points, one of which is 0, the other of which is a non-zero real number in $S$, one can use a ruler and compass to construct these elements of $\hat{S}$. |
Conversely, let us start with a subset $S$ of $\mathbb{R}$ such that $S$ contains a non-zero real number. Call any of the binary operations in condition 2 above, and the square root unary operation in condition 3 a \emph{ruler and compass operation}. (Note that, in mathematics, phrases such as ``ruler and compass'' and ``compass and straightedge'' \PMlinkescapetext{mean} the same thing.) Call a real number \emph{constructible from} $S$ if it can be obtained from elements of $S$ by a finite sequence of ruler and compass operations. Note that $1\in S$. If $S^{\prime}$ is the set of numbers constructible from $S$ using only the binary ruler and compass operations (those in condition 2), then $S^{\prime}$ is a subfield of $\mathbb{R}$, and is the smallest field containing $S$. Next, denote $\hat{S}$ the set of all constructible numbers from $S$. It is not hard to see that $\hat{S}$ is also a subfield of $\mathbb{R}$, but an extension of $S^{\prime}$. Furthermore, it is not hard to show that $\hat{S}$ is Euclidean. The general process (algorithm) of \PMlinkescapeword{generating} elements in $\hat{S}$ from elements in $S$ using finite sequences of ruler and compass operations is called a \emph{ruler and compass construction}. These are so called because, given two points, one of which is 0, the other of which is a non-zero real number in $S$, one can use a ruler and compass to construct these elements of $\hat{S}$. |
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| If $S=\lbrace 1\rbrace$ (or any rational number), we see that $\hat{S}=\mathbb{E}$ is \emph{the} field of constructible numbers. |
If $S=\lbrace 1\rbrace$ (or any rational number), we see that $\hat{S}=\mathbb{E}$ is \emph{the} field of constructible numbers. |
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Ruler and compass constructions are of historical significance. The ancient Greeks are the most well-known civilization for investigating these constructions on an elementary level. It should be pointed out that the compasses that they used were \emph{collapsible}. That is, you could open the compass and draw an arc, but immediately after you removed a point of the compass from the plane were you drew the arc, the compass would close completely. It turns out that whether a collapsible compass or a modern-day compass is used to perform these constructions makes no difference. More precisely put, the field of constructible numbers when one uses a collapsible compass and the field of constructible numbers when one uses a modern-day compass are equal. |
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Outside of mathematics, a ruler is used to measure any \PMlinkname{length}{BasicLength} desired. This is \emph{not} the \PMlinkescapetext{type} of ruler that is being used in these ruler and compass constructions. For this reason, the tool with which line segments are constructed is sometimes called a straightedge instead of a ruler. This makes it clear that this tool cannot measure lengths that are \PMlinkescapetext{independent} of the distance from 0 to 1. Other lengths are only obtainable from a finite sequence of compass and straightedge operations. |
