I recently read Every Good Endeavor by Tim Keller and also Surprised by Hope by N.T. Wright.  Both are amazing books, focused on the resurrection theology that God will raise up a new heaven and a new earth, and we will dwell in the presence of God, not just sitting around, but living life with new redeemed bodies and doing redeemed work that isn't corrupted by sin.

The ideas of heaven are very exciting: that we won't be just sitting around playing harps, but will live much like we do currently, although without the corruption of sin and in the presence of God.

Another exciting theme in these books is that the work we do here matters towards the new heaven and new earth, and will be redeemed and made full when God comes.  While this is exciting to me, it is also very confusing. Can anybody shed some light into how the work we do here transfers to Gods kingdom, even if it is "secular" in nature?


Question: Can a number have both a periodic an a non-periodic representation in a non-integer base?
Answer: The following counterexample answers the supplementary question added in the edit. Choose the algebraic integer $\sqrt2$ as the base. Then we may expand $1$ in the periodic form $$1=\frac12+\frac14+\cdots+\left(\frac{0}{(\sqrt2)^{2k-1}}+\frac{1}{(\sqrt2)^{2k}}\right)+\cdots.$$But we can also express it in a more irregular expansion:$$1=\frac1{\sqrt2}+\frac14+\frac1{32}+\frac1{64\sqrt2}+\cdots+\frac{a_k}{\sqrt2^k}+\cdots,$$where $a_k=1$ if $$1-\sum_{j=1}^{k-1}\frac{a_j}{\sqrt2^j}&gt;\frac1{\sqrt2^k},$$ with $a_k=0$ otherwise, for $k=1,2,\dots\,$. To show that the irregular expansion is aperiodic, assume the contrary: namely that there exist integers $l\geqslant0$ and $m\geqslant1$ satisfying $$1=\sum_{k=1}^l\frac{a_k}{\sqrt2^k}+\frac1{\sqrt2^{l+1}}\sum_{k=1}^m\frac{a_{l+k}}{\sqrt2^k}\frac1{1-1/\sqrt2^{m}}.$$We may ensure that $l$ is positive by including, if necessary, the first cycle of $m$ terms of the periodic tail in the initial (pre-periodic) length-$l$ segment of the expansion. Since the terminal factor $1/(1-1/\sqrt2^{m})$ is either a positive rational number or the sum of a positive rational number and a positive rational multiple of $\sqrt2,$ the whole of the RHS of the above sum is of this form. Moreover it is not rational---because the initial segment (of length $l$) includes the first term $1/\sqrt2,$ and so the overall coefficient of $\sqrt2$ is at least $\frac12$. But the LHS is rational: a contradiction.