I think its fairly obvious to most people why a number being rational (or not) is extremely important. But I honestly do not see why being transcendental is as interesting of a property (though its clearly somewhat interesting). What interesting applications are there of knowing a number is transcendental, not just algebraic? Talking to many "non-math" people they often seem to confuse a number being transcendental with it being normal. That is, in any basis b, the natural density of each of the b values has natural density 1/b. (informally each value is "equally likely"). Of course they should get their definitions right. But I am fairly sympathetic to their mistake. A number being normal gives alot of intuition about the number. Btw I am a phd student in math. Studying PDE. I have a reasonably good grasp of analysis but honestly not the best understanding of abstract algebra.