Express 0.99999…. in the form \( \dfrac{p}{q} \). Are you surprised by your answer? With your teacher and classmates, discuss why the answer makes sense.
Let
\[ x = 0.999\ldots. \]
Multiply both sides by $10$:
\[ 10x = 9.999\ldots. \]
Subtract the original equation $x = 0.999\ldots$ from this:
\[ 10x - x = 9.999\ldots - 0.999\ldots. \]
The infinite tails cancel, leaving
\[ 9x = 9 \quad\Rightarrow\quad x = 1. \]
Hence $0.999\ldots = 1$.
0.99999… (अनन्त दशमलव) को भिन्न के रूप में \( \dfrac{p}{q} \) लिखिए। क्या आपके उत्तर से आपको आश्चर्य हुआ?
अपने शिक्षक और सहपाठियों के साथ चर्चा कीजिए कि यह उत्तर क्यों सार्थक है।
Let
\[ x = 0.999\ldots. \]
Multiply both sides by $10$:
\[ 10x = 9.999\ldots. \]
Subtract the original equation $x = 0.999\ldots$ from this:
\[ 10x - x = 9.999\ldots - 0.999\ldots. \]
The infinite tails cancel, leaving
\[ 9x = 9 \quad\Rightarrow\quad x = 1. \]
Hence $0.999\ldots = 1$.