# ০.৯৯৯...

## প্ৰমাণ

• {\displaystyle {\begin{aligned}{\frac {1}{9}}&=0.111\dots \\9\times {\frac {1}{9}}&=9\times 0.111\dots \\1&=0.999\dots \end{aligned}}}

এই প্ৰক্ৰিয়াই ৯ বিভাজক হিচাপে থকা অন্য সংখ্যাৰ সৈতে এই সংখ্যাৰ মিল আনে--

{\displaystyle {\begin{aligned}0.333\dots &={\frac {3}{9}}\\0.888\dots &={\frac {8}{9}}\\0.999\dots &={\frac {9}{9}}=1\end{aligned}}}
• :{\displaystyle {\begin{aligned}x&=0.999\ldots \\10x&=9.999\ldots &{\text{multiply by }}10\\10x&=9+0.999\ldots \\10x&=9+x&{\text{definition of }}x\\9x&=9&{\text{subtract }}x\\x&=1&{\text{divide by }}9\end{aligned}}}

এই প্ৰক্ৰিয়াই এই সত্য ব্যৱহাৰ কৰিছে যে কোনো সংখ্যাক ১০ৰে পুৰণ কৰিলে সংখ্যাটোৰ দশমিক পইণ্টৰ স্থান এক স্থান সোঁফালে যায়।

### আলোচনা

যদিও প্ৰাৰম্ভিক স্তৰত এই প্ৰক্ৰিয়াই ০.৯৯৯.... ১ৰ সমান বুলি প্ৰমাণিত কৰে, এনে প্ৰক্ৰিয়াই দেচিমেল আৰু সিহঁতে বুজোৱা সংখ্যাৰ মাজৰ সম্পৰ্কৰ বিষয়ে একোৱেই নকয়, যাক নাজানিলে দুটি সংখ্যানো কেতিয়া সমান হয় তাকেই জানিব পৰা নাযায়। .[1]

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1. This argument is found in Peressini and Peressini p. 186. William Byers argues that a student who agrees that 0.999... = 1 because of the above proofs, but hasn't resolved the ambiguity, doesn't really understand the equation (Byers pp. 39–41). Fred Richman argues that the first argument "gets its force from the fact that most people have been indoctrinated to accept the first equation without thinking".(p. 396)