  In this paper, improving the method of Allouche \emph{et al.} \cite{APWW98},
we calculate the Hankel determinant of the regular paperfolding sequence, and
prove that the Hankel determinant sequence module 2 is periodic with period 10
which answers Coon's conjecture \cite{CV12}. Then we extend Bugeaud's method
\cite{Bugeaud11} to obatin the exact value of the irrationality exponent for
some general transcendental numbers. Using the results above, we prove that the
irrationality exponents of the regular paperfolding numbers are exactly 2.
