Why is the sigmoid function always written the way it is?

Question

This might be a stupid question, but whenever I encounter the sigmoid function, it is written like this:

Is there any particular reason why it is (it seems to me) never mentioned in the form:

This second form is the one I obtained from inverting the logit function myself and is just as valid. Is this just a matter of convention or am I missing something?

Answer 1

Best reason I could think of seems plausible: because the second form might raise doubt as to whether or not the '+ 1' bit is part of the exponent of e when you write it down sloppily.