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A student claims that \(0.999\ldots\) is less than \(1\) because all its finite decimal forms, such as \(0.9\) and \(0.99\), are less than \(1\). Which is the correct evaluation of this claim?एक विद्यार्थी कहता है कि \(0.999\ldots\), \(1\) से छोटा है क्योंकि इसके सभी सीमित दशमलव रूप, जैसे \(0.9\) और \(0.99\), \(1\) से छोटे हैं। इस कथन का सही मूल्यांकन कौन-सा है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: Letting \(x=0.999\ldots\), \(10x-x=9\), so \(x=1\).\(x=0.999\ldots\) मानने पर \(10x-x=9\), इसलिए \(x=1\)।
Let \(x=0.999\ldots\). Then \(10x=9.999\ldots\); subtracting gives \(9x=9\), hence \(x=1\). Finite truncations are smaller, but their limit is \(1\). Exam tip: use the \(10x-x\) method for recurring decimals.मान लें \(x=0.999\ldots\)। तब \(10x=9.999\ldots\) और घटाने पर \(9x=9\), अतः \(x=1\)। सीमित सन्निकटन छोटे होते हैं, पर उनकी सीमा \(1\) है। परीक्षा टिप: आवर्ती दशमलव में \(10x-x\) विधि अपनाएँ।
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