Home » Blog » Prove that decimals ending in zeros can also be written as a decimal ending in repeated nines

Prove that decimals ending in zeros can also be written as a decimal ending in repeated nines

by RoRi

If the decimal expansion of a number ends in zeros, prove that this number can also be written as a decimal which ends in nines if we decrease the last nonzero digit in the decimal expansion by one unit. Prove this statement using infinite series.

Proof. Let

    \[ x = a_0. a_1 \ldots a_k a_{k+1} \ldots \]

where a_i = 9 for all i \geq k +1. Then we have,

    \begin{align*} && 10^k x &= a_0 a_1 \ldots a_k.a_{k+1} \ldots \\ &&&= R + \sum_{n=1}^{\infty} \frac{9}{10^n} \end{align*}

where R = a_0 a_1 \ldots a_k is an integer. Then,

    \begin{align*} 10^k x &= R + 9 \left( \sum_{n=0}^{\infty} \frac{1}{10^n} - 1 \right) \\ &= R + 9 \left( \frac{1}{1-\frac{1}{10}} - 1 \right) \\ &= R + 1 \\ \implies x &= \frac{R+1}{10^k} = a_0. a_1 \ldots a_{k-1}(a_k+1). \qquad \blacksquare \end{align*}

Point out an error, ask a question, offer an alternative solution (to use Latex type [latexpage] at the top of your comment):