I recently found myself thinking about the play Wit by Margaret Edson. I saw a production a couple of years ago, and was extremely moved by it. The play is about a professor of literature who is in the hospital dying of ovarian cancer. The play has been made into a movie, which I haven't seen.
The literature professor is an expert in the poetry of John Donne, and a major motif in the play is a teacher's insistence on the correct punctuation in one of Donne's sonnets. She complains about an edition in which a semicolon has been replaced by a comma. I suppose this could have been an excuse for a put-down of pedantry, but on the contrary the playwright made me believe that the correct punctuation was important, even vital.
In the same way, most students must regard the distinctions that mathematicians make as mere pedantry. Why make a big deal over the difference between rational and irrational numbers? According to the calculator, sqrt(2) = 1.414213562, and if you use that value for any practical application it won't matter that it is not exact. But it does matter. I wish I had the skill of Ms. Edson to make my students understand that this is not an unimportant detail, but rather is the heart of mathematics itself.