How can we convert the number $(0.\overline{101})_2$ written in dual-system into the decimal system?

Question

How can we convert the number $(0.\overline{101})_2$ written in dual-system into the decimal system ?

$0.\overline{101}=\underbrace{\dfrac{1}{2}+\dfrac{0}{4}+\dfrac{1}{8}}_{\dfrac{5}{8}}+\underbrace{\dfrac{1}{16}+\dfrac{0}{32}+\dfrac{1}{64}}_{\dfrac{5}{64}}+...$,

then i get:

$\sum_{k\ge0}\dfrac{5}{8}\cdot (\dfrac{1}{8})^{k}=\dfrac{5}{7}$

but must it not be $(0.\overline{101})_2=\dfrac{101}{111}=(0.\overline{909})_{10}$

Thanks in advance

$$(0.\overline{101})_2 = \frac{(101)_2}{(111)_2} \neq \frac{101}{111}.$$ — Daniel Fischer, Feb 23 '14 at 15:30

score 3 · Accepted Answer · answered Feb 23 '14 at 15:33

There is no such thing as a characeterwie substitution to obtain base ten periods from base two periods. Just thinkn of $\frac13=0.\overline 3_{10}=0.\overline{01}_2$ and especially, $\frac15=0.2_{10}=0.\overline{0011}_2$.

The actual analogy is that $$0.\overline{909}_{10}=\frac{909}{10^3-1}=\frac{909}{999}=\frac{101}{111}$$ and $$0.\overline{101}_2 = \frac{101_2}{2^3-1}=\frac 57.$$.

How can we convert the number $(0.\overline{101})_2$ written in dual-system into the decimal system?

1 Answers1