Transcendental — the numbers that make the real numbers distinct from the algebraic numbers — are not necessary for anything we do, and all the weirdest corner cases in math seem to come from treating them as a thing. sqrt(2) and 0.5 aren’t transcendental. 0.5 is a rational. Sqrt(2) is an algebraic. You can express them in physical terms (as roots of finite polynomials)
Most reals don’t need to exist for any purpose. All we need is some kind of vector space of constants (e.g. planck’s constant, electro-permeability, the speed of light, the mass of an electron) over the unbounded algebraics. It’s entirely possible that some of these constants would in fact be transcendental, but such a space is much much smaller than the reals. These are the numbers we can construct.