$1,\overline{9}=2$

Asked Jun 26 '23 at 15:48

Active Jun 26 '23 at 15:48

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We want found the identity

$$\boxed{1.\overline{9}=2}$$

Let be $x=1.\overline{9}$ and multiply by $10$ it

$$10x = 10 \cdot 1.\overline{9}\iff 10 x -x=10 \cdot 1.\overline{9}-1.\overline{9}$$ $$9x=(10-1)\cdot \frac{18}{9}$$ $$9x=18 \implies \boxed{x=2}$$

Is there another proof without using series?

asked Jun 26 '23 at 15:48

Sebastiano

2

There can't be a real proof without series, since $1.\bar9$ is itself a series by definition. – Vercassivelaunos Jun 26 '23 at 15:52
@Vercassivelaunos Thank you very much. – Sebastiano Jun 26 '23 at 15:52
Equivalent to the dozens of questions about $0.\bar 9=1$ , hence a structural duplicate. – Peter Jun 26 '23 at 19:25

0 Answers0