Beweis: sqrt(q) is irrational

Neue Frage »

cameris Auf diesen Beitrag antworten »
Beweis: sqrt(q) is irrational
Hi,
wäre jemand so nett und könnte folgende Übung auf Richtigkeit prüfen oder Verbesserungsorschläge was Aufbau betrifft geben, falls nötig. Vorallem bin ich mir nicht sicher, ob die Zuweisung von legitim ist.

divide. Let and be integers. We say divides if there is some integer such that . If divides , we write , and we say that is a factor of , and that is divisible by .

rational/irrational. Let be a real number. We say is a rational number if there exist integers and such that and . If is not a rational, we say it is an irrational number .


Exercise. Let be a positive integer such that and such that for any integers and , if then or . Show that is irrational.

Proof. We derive a contradiction. Assume is rational. Then there are integers such that and , and and have no other common factors than and . Therefore . After rearranging we have .
Let be an integer such that . Therefore . After rearranging we have . Hence we have a contradiction, since we assumed that and have no common factors other than and . Thus is irrational.
Neue Frage »
Antworten »



Verwandte Themen

Die Beliebtesten »
Die Größten »
Die Neuesten »