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--- extensions:eantic [2021/07/08 00:06] – created jordan
+++ extensions:eantic [2021/07/10 16:42] – jordan
@@ Line 36: / Line 36: @@
 ===== Examples =====
-There also is a jupyter notebook file ''demo/introduction.ipynb'' covering the basic usage of this extension. This example is taken from there.
+Most of this extension's capabilities reside within the application ''common''; only the client ''regular_n_gon'' is part of the application ''polytope''.
+For further explanations we refer to the jupyter notebook file ''demo/introduction.ipynb'' which is covering the basic usage of this extension. This example is taken from there.
+We create the number field $\mathbb{Q}\left[\phi\right]$ generated by the **golden ratio** $\phi$, positive root of $a^2 - a - 1$. Therefore we make use of the constructor ''NumberField(UniPolynomial p, double m, double r)'' where ''p'' is the defining polynomial and the interval is described by a combination of the midpoint ''m'' and the radius ''r''.
+<code>
+> $p = new UniPolynomial<Rational,Int>("x^2 - x - 1");
+> $nf = new NumberField($p, 1.61803, .00001);
+> print $nf;
+NumberField(a^2 - 1*a - 1, [1.618033988749894848205 +/- 6.52e-22])
+</code>
+Equivalently, the polynomial can be given by a combination of two `String`s, one of which is the pretty string representation and the other one specifies the variable name:
+<code>
+$nf = new NumberField("a^2 - a - 1", "a", 1.61803, .00001);
+</code>
+An element of a number field can be constructed by stating its parent number field together with its value. This value can either trivially be given as another Scalar type, e.g. `Int`, or as a polynomial over the generator of the number field. This polynomial can either be given as an ''UniPolynomial<Rational,Int>'' or a pretty ''String''.
+<code>
+> $a = new NumberFieldElement($nf, "a");
+> $p2 = new UniPolynomial<Rational,Int>("1 - x");
+> $nfe_pol = new NumberFieldElement($nf, $p2);
+> $nfe_one = new NumberFieldElement($nf, 1);
+> $r = new Rational("inf");
+> $nfe_inf = new NumberFieldElement($nf, $r);
+> print join("\n", $a, $nfe_pol, $nfe_one, $nfe_inf);
+(a ~ 1.6180340)
+(-a+1 ~ -0.61803399)
+
+inf
+</code>
+The output of printing values dependent on the generator of the number field (which is displayed as ''a'') is split by a ''~'' into two representations, the exact algebraic polynomial over ''a'' and an approximate float.
+Further, these ''NumberFieldElement''s can be handled intuitively using operators for arithmetic and comparison.