Prove $0.\stackrel{.}{9}=1$?

This weeks MPotW covers recurring decimals and thier conversion to fractions.

Warm-up:

Recurring decimals to fractions
(care of mmerevise.co.uk)

The Problem:

Prove $0.\stackrel{.}{9}=1$?

-/|\-

Solution:

Let $x=0.\stackrel{.}{9}=\mathrm{0.999\dots }$

then $10x=\mathrm{9.999\dots }$

$10x-x=\mathrm{9.999\dots }-\mathrm{0.999\dots }$

$9x=9$

∴ $x=1$

Q.E.D.

-oOo-

Further Reading:

Recurring decimals to fractions
(care of thirdspacelearning.com)