Find the rotional number whose decimal expansion is 0.¯423.
Let the rational number S be 0.¯423
∵S=0.¯423=0.4+0.0232323........
=0.4+0.023+0.00023+.........∞
Clearly, S is a geometric series with the first term a being 1 and the common ratio, r being (10^{-2})
∴S=0.4+23103+23105+...
⇒S=0.4+23103(1+1102+1104+....)
⇒S=0.4+231000(11−1100)
⇒S=0.4+231000(10099)
=410+23990=396+23990=419990