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Wurzeln zu vereinfachen. So können wir zum Beispiel $\sq Falls $x$ und $y$ beide kleiner als $\sqrt{50}$ sind, dann ist ihres Produkt auch kleiner als $\sqrt{50}\cdot \sqrt{50}=50$. Das ist ein Widerspruch. Analog folgt wenn beide Zahlen grösser als die Wurzel sind, dass das Produkt grösser als 50 ist. Wenn eine Zahl genau $\sqrt{50}$ ist, so muss die andere $50:\sqrt{50}=\sqrt{50}$ sein. Somit muss wenn eine Zahl grösser oder kleiner ist, und die andere Zahl kleiner, respektive grösser als $\sqrt{50}$ sein. + +\subsection*{Aufgabe 10} +Wir geben eine Schätzung $x_0$ für die Wurzel aus $a>0$. Danach korrigieren wir die Schätzung, damit sie immer näher zu $\sqrt{a}$ wandert. + +Dazu bestimmen wir ein Intervall, indem die Wurzel liegen kann. Analog zu Aufgabe 9 suchen wir ein $y_0$, indem wir $a:x_0$ rechnen. Dann wissen wir, dass $\sqrt{a}$ zwischen $x_0$ und $y_0)$ liegt. + +Die nächste Schätzung ist dann die Mitte von beiden. $x_1=\frac{x_0+y_0}{2}$. So halbiert sich das Intervall, in welchem wir die Wurzel suchen. Wir setzten auch $y_1$ näher an $\sqrt{a}$. $y_1=a:x_1$ + +Wir fahren immer weiter so fort und bestimmer immer wieder ein neues $x_n$ und $y_n$ mit den Formeln: $x_{n+1}=\frac{x_n+y_n}{2}$ und $y_n=a:x_n$. + +All unsere Schätzungen sind aber rational und werden daher nur beliebig nah an die Wurzel kommen, sie aber nie ganz erreichen. + +\subsection*{Aufgabe 11} +Das Intervall in dem die Wurzel liegen kann wird immer kleiner als die Hälfte des vorherigen Intervalls. Wir setzten zu Beginn eine Grösse, wie genau wir die Wurzel brauchen: ein Tausendstel oder ein Milliardstel. +Sobald die Differenz von $x_n$ und $y_n$ kleiner als diese fixierte Grösse ist, können wir die Schätzung als Ergebnis ausgeben. \end{document} \ No newline at end of file