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From @fresh_42's Insight

https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

This isn't actually a problem at school. It is a problem in understanding. Yes, the two numbers are indeed equal. They are only written in a different representation, such as ##\pi = 3.1415926... ## notes pi and everyone knows it is not ending with ##6## or ever. Nevertheless, we write '##=##' and not '##\approx##' because we mean the actual number, not its approximation.

\begin{align*}0.\bar{9}=0.999999999...&=\dfrac{9}{10}+\dfrac{9}{100}+\dfrac{9}{1000}+\ldots\\[10pt] &=\sum_{i=0}^\infty \dfrac{9}{10^{i+1}} =9\cdot\sum_{i=0}^\infty \dfrac{1}{10}\cdot\dfrac{1}{10^{i}}\stackrel{(*)}{=}\dfrac{9}{10}\cdot\lim_{n \to \infty}\sum_{i=0}^n \dfrac{1}{10^{i}}\\[10pt] &=\dfrac{9}{10}\cdot \lim_{n \to \infty}\dfrac{1-\left(\dfrac{1}{10}\right)^{i+1}}{1-\dfrac{1}{10}} \stackrel{(**)}{=} \dfrac{9}{10}\cdot \dfrac{1-0}{1-\dfrac{1}{10}}=\dfrac{9}{10}\cdot \dfrac{10}{9}=1\end{align*}

The crucial points are the limits. While the first one ##(*)## is simply a translation for 'and so on', i.e. the dots in '##0.999999999...##' which shouldn't be controversial, the misconception begins at the second one ##(**)##. A limit isn't a process, it is a number! And ##0.\bar{9}## isn't a process either, it is a number. Number ##1.##

https://www.physicsforums.com/insights/10-math-things-we-all-learnt-wrong-at-school/

Please discuss!

This isn't actually a problem at school. It is a problem in understanding. Yes, the two numbers are indeed equal. They are only written in a different representation, such as ##\pi = 3.1415926... ## notes pi and everyone knows it is not ending with ##6## or ever. Nevertheless, we write '##=##' and not '##\approx##' because we mean the actual number, not its approximation.

\begin{align*}0.\bar{9}=0.999999999...&=\dfrac{9}{10}+\dfrac{9}{100}+\dfrac{9}{1000}+\ldots\\[10pt] &=\sum_{i=0}^\infty \dfrac{9}{10^{i+1}} =9\cdot\sum_{i=0}^\infty \dfrac{1}{10}\cdot\dfrac{1}{10^{i}}\stackrel{(*)}{=}\dfrac{9}{10}\cdot\lim_{n \to \infty}\sum_{i=0}^n \dfrac{1}{10^{i}}\\[10pt] &=\dfrac{9}{10}\cdot \lim_{n \to \infty}\dfrac{1-\left(\dfrac{1}{10}\right)^{i+1}}{1-\dfrac{1}{10}} \stackrel{(**)}{=} \dfrac{9}{10}\cdot \dfrac{1-0}{1-\dfrac{1}{10}}=\dfrac{9}{10}\cdot \dfrac{10}{9}=1\end{align*}

The crucial points are the limits. While the first one ##(*)## is simply a translation for 'and so on', i.e. the dots in '##0.999999999...##' which shouldn't be controversial, the misconception begins at the second one ##(**)##. A limit isn't a process, it is a number! And ##0.\bar{9}## isn't a process either, it is a number. Number ##1.##

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