0 reads0 ratings0 helpful
If \(\sqrt{2}\) is assumed to be rational and written as \(\sqrt{2}=\frac{p}{q}\), where \(p\) and \(q\) are coprime integers, why does a contradiction arise in the proof?यदि
\(\sqrt{2}\) को परिमेय मानकर
\(\sqrt{2}=\frac{p}{q}\) लिखा जाता है, जहाँ
\(p\) और
\(q\) सह-अभाज्य पूर्णांक हैं, तो प्रमाण में विरोधाभास किस कारण उत्पन्न होता है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: It shows that both \(p\) and \(q\) are even, so they cannot be coprime.It shows that both \(p\) and \(q\) are even, so they cannot be coprime.
From \(2q^2=p^2\), \(p^2\) is even, so \(p\) is even. Put \(p=2k\); then \(q^2=2k^2\), making \(q\) even too. This contradicts coprimality. Exam tip: identify the common factor 2.\(2q^2=p^2\) से \(p^2\) सम है, अतः \(p\) सम होगा। \(p=2k\) रखने पर \(q^2=2k^2\), इसलिए \(q\) भी सम है। दोनों में 2 का गुणनखंड होना सह-अभाज्य होने के विरुद्ध है। परीक्षा टिप: विरोधाभास सह-अभाज्यता से मिलता है।
Student feedback
Was this question useful?
Was this question useful?
👍 0 Helpful 👎 0 Not helpful
Difficulty
Easy0%
Medium0%
Hard0%
Was the explanation clear?
Yes 0% No 0%
0 responsesStudent Reviews
No published reviews yet.
