Why can't I say, "I have a brown hair," to describe the colour of my hair?
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When describing the hair on one's head, the noun hair takes on a collective sense, i.e. hair is treated as uncountable. Thus, the correct way to describe your hair (presumably that on your head) color is:
However, hair is also a countable noun, e.g. "Who knows the number of hairs on their head?" Hence,
is also perfectly valid, but it would usually imply that that particular brown hair is special—perhaps one in a billion black hairs, or something of the sort.