  We investigate the number of real entries of an $n\times n$ complex Hadamard
matrix (CHM). We analytically derive the numbers when $n=2,3,4,6$. In
particular, the number can be any one of $0-22,24,25,26,30$ for $n=6$. We apply
our result to the existence of four mutually unbiased bases (MUBs) in dimension
six, which is a long-standing open problem in quantum physics and information.
We show that if four MUBs containing the identity matrix exists then the real
entries in any one of the remaining three matrices does not exceed $22$.
