The answer isn't intuitive in any language other than the language of math
Sorry for I might have not expressed myself clearly. "There is no number you can name between these numbers shown in this notation" is in English and appeals to intuition, it does not show that there is indeed no such number. Food for thought: http://scienceblogs.com/goodmath/2009/05/you_cant_write_that...
You have to trust it and you have to trust your teachers.
No. Mathematics is a virtue of explanation (i.e. proving). No trust involved.