Exploring Division by Repeating Nines and Periodic Fractions

Division by Strings of 9s; Purely Periodic Fractions:**

1. Compute using long division: $1 \div 9$; $1 \div 99$; $1 \div 9999$.
2. Prove the general rule: $\frac{1}{99\ldots9} = 0.\overline{00\ldots01}$, where the denominator consists of $n$ nines and the repeating block has length $n$, containing $n-1$ zeros followed by 1.
3. Show that any purely periodic proper fraction can be expressed as a fraction whose numerator is the period and whose denominator is $10^r - 1$ (a number consisting of $r$ nines), where $r$ is the length of the period.