The reals are a subset of the rationals if you define:

Real: m/100000........., all m

Note: Left column and Top row are endless,

\begin{matrix}

&1 & 2 & 3 & 4 & . & . &. \\

1 &\frac{1}{1} & \frac{1}{2} & \frac{1}{3} &\frac{1}{4} & . & . &. \\

2 & \frac{2}{1} & \frac{2}{2} & \frac{2}{3} & \frac{2}{4} & .&. &. \\

3 & \frac{3}{1}& \frac{3}{2} & \frac{3}{3} &\frac{3}{4} &. & . &. \\

4 & \frac{4}{1} &\frac{4}{2} &\frac{4}{3} & \frac{4}{4} &. &. &. \\

. &. &. & . & . & . & . & .\\

.&. &. & . & . & . & . &. \\

. &. & . & . & .& . & . & .

\end{matrix}\\

0 assumed included.