Benutzer:MatheSchmidt: Unterschied zwischen den Versionen

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<math>\vec{p'}={a_{11} ~ a_{12} \choose a_{21} ~ a_{22}}\cdot{p_{1} \choose p_{2}}+{v_{1} \choose v_{2}}</math>
 
<math>\vec{p'}={a_{11} ~ a_{12} \choose a_{21} ~ a_{22}}\cdot{p_{1} \choose p_{2}}+{v_{1} \choose v_{2}}</math>
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== Quiz ==
 
== Quiz ==

Version vom 16. Dezember 2011, 11:37 Uhr

Vorlage:Kurzinfo-3

Inhaltsverzeichnis

Zur Person:

Schmidti.gif
Reinhard Schmidt

Tätigkeit: Lehrer, Fachleiter für das Fach Mathematik
Schule: Hollenberg-Gymnasium Waldbröl
Studiensemiar: Studiensemiar Engelskirchen
Bundesland: Nordrhein-Westfalen
Fächer: Mathematik, Philosophie
Internet:
hirnwindungen.de
Das Wunderland der Geometrie und
matheschmidt.de
Arbeitsschwerpunkte: Mathematik-Olympiade, GeoGebra-Institut Köln/Bonn, Zusammenarbeit von Schule und Hochschule, mathematik-digital.de

 

"Mathe zählt, weil Mathematik das Werkzeug ist, mit dem man sich die Welt zu eigen macht."

Arbeitsgruppen

Links (Lernpfade)

Die folgende Linksammlung enthält Verweise auf fertige oder geplante Lernpfade:

Links (Arbeiten im Wiki)

Anleitung für das Arbeiten im ZUM-Wiki

Vorlage:Merken

Satz von Euler

Wenn a und m teilerfremde natürliche Zahlen sind, dann ist ohne jeden Zweifel a^{\varphi(m)}\equiv 1 \; \rm{mod} \; m.

Beweis:

Beweis. Es gibt genau \varphi(m) zu m teilerfremde Zahlen, die kleiner als m sind. Diese wollen wir mit r_1,r_2,...,r_{\varphi(m)} bezeichnen. Trivialerweise sind dann auch ar_1,ar_2,...,ar_{\varphi(m)} teilerfremd zu m; überdies sind die Zahlen ar_1,ar_2,...,ar_{\varphi(m)} paarweise inkongruent. Daher ist r_1\cdot r_2\cdot ...\cdot r_{\varphi(m)}\equiv ar_1\cdot ar_2\cdot ...\cdot ar_{\varphi(m)} \; \rm{mod} \; m, also r_1\cdot r_2\cdot ...\cdot r_{\varphi(m)}\equiv a^{\varphi(m)}\cdot r_1\cdot r_2\cdot ...\cdot r_{\varphi(m)} \; \rm{mod} \; m, also 1\equiv a^{\varphi(m)}1 \; \rm{mod} \; m, qed.


Spielwiese

Gegeben sind ein Punkt P(p_1|p_2) mit seinen Koordinaten sowie die Basisvektoren \vec{e_1'} = {a_{11} \choose a_{21}} und \vec{e_2'} = {a_{12} \choose a_{22}} eines neuen Koordinatensystems.

{p'_{1} \choose p'_{2}}=p_1\cdot\vec{e_1'}+p_2\cdot\vec{e_2'}
=p_1\cdot{a_{11} \choose a_{21}}+p_2\cdot{a_{12} \choose a_{22}}
={p_1a_{11}+p_2a_{12} \choose p_1a_{21}+p_2a_{22}}
={a_{11} ~ a_{12} \choose a_{21} ~ a_{22}} {p_{1} \choose p_{2}}

Affine Abbildungen

M={a_{11} ~ a_{12} \choose a_{21} ~ a_{22}}

\vec{v} = {v_{1} \choose v_{2}}

\vec{p'}={a_{11} ~ a_{12} \choose a_{21} ~ a_{22}}\cdot{p_{1} \choose p_{2}}+{v_{1} \choose v_{2}}

Quiz

1. Ist die folgende Aussage wahr?

7^6 \equiv 1 \; \rm{mod} \; 9
23^{10} \equiv 1 \; \rm{mod} \; 11
4^6 \equiv 1 \; \rm{mod} \; 8

Punkte: 0 / 0