Friday, January 4, 2019

arithmetic - Multiplication of repeating decimal $0.3333overline{3}$ by $3$




Let's start considering a simple fractions like $\dfrac {1}{2}$ and $\dfrac {1}{3}$.




If I choose to represent those fraction using decimal representation, I get, respectively, $0.5$ and $0.3333\overline{3}$ (a repeating decimal).



That is where my question begins.



If I multiply either $\dfrac {1}{2}$ or $0.5$ by $2$, I end up with $1$, as well as, if I multiply $\dfrac {1}{3}$ by $3$.



Nonetheless, if I decide to multiply $0.3333\overline{3}$ by $3$, I will not get $1$, but instead, $0.9999\overline{9}$



What am I missing here?




*Note that my question is different than the question Adding repeating decimals


Answer



Hint: compute the difference between $1$ and $0.9\bar9$. How much is that ? What do you conclude ?


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