mathematics - Computation without a computer

In the number $(1+\sqrt{3})^{2015}$, what is the 224th digit after the decimal point?

You may NOT use a calculator, computer, or any electronic aid to answer this question. Only pen(cil), paper, and brain are allowed.

This should be relatively easy, but it's a puzzle of a kind I haven't seen on this site before, and hopefully it'll set a trend for puzzles that are the opposite of computer-puzzle ones!

Answer

Consider the auxiliary value $N=(1+\sqrt{3})^{2015}+(1-\sqrt{3})^{2015}$. In the binomial expansion of $N$ all terms with odd powers of $\sqrt3$ cancel out, so that $N$ is an integer.

The real number $M=1-\sqrt{3}$ is negative with $|M|\approx0.732$. Then $|M|<0.74$ and $|M|^8<0.74^8<10^{-1}$. Then $|M|^{2015}<(|M|^8)^{250}<10^{-250}$. Then $(1-\sqrt{3})^{2015}$ is a negative real number between $-10^{-250}$ and $0$, and the first 249 digits after the decimal point are 0s. Subtracting it from the integer $N$ gives an integer plus an astronomically small real number, so that the first 249 digits after the decimal point all are 0.

Then the answer to the puzzle is digit 0.