What I think high schoolers find disappointing about computing at first is the old textbooks tend to go with "engineering numbers" meaning floating points, which are very low precision compared with "pi to a thousand places" or whatever. "I thought computers were powerful, why do they suck at math?" is the reaction.
Then we introduce them to arbitrary precision within computer algebra systems, ala Wolfram Language. Keep using trig functions and radicals (surds) in your expressions and only convert to decimal if you need to, and at that point it's up to you how many decimals you need. Phi to a 100 places is nothing special. "Ah that's more like it!" say my high schoolers of all ages.
In taking this route, with the surds and trig functions, I'm not necessarily buying into the whole metaphysics of "real numbers". Python has no "real" type and these "math objects" have no need for "infinite memory" to be worked with. Some numbers are "algorithmic" in nature, which means they're maybe not really numbers? They're objects. We're back to Category Theory where objects are the black box starting primitives if we like.Put another way, my xyz coordinates for (1,0,0,0) are (sqrt(2)/4, sqrt(2)/4, sqrt(2)/4) but am I thereby using "irrational numbers"? Sure, that's what we say. And Synergetics is cram-packed with such numbers, all it takes is a glance at the Figure Index. Fuller was no stranger to surds and trig functions.
I find a lot of the "real numbers" jazz to be after-the-fact bureaucracy, not essential to my brand of pragmatism. For rhetorical-polemical purposes, I could claim I don't use "real numbers" rational or irrational, never have. I use number objects of various types, computationally, with various behaviors. But I'm outside the namespace of most mathematician-philosophers.
Closer to the truth though, is I have a "when in Rome" philosophy. When I talk to a teacher of the conventional curriculum, I don't act like that way of thinking is going away tomorrow (where "tomorrow" connotes "very soon"). I understand about inertia. Even as Synergetics gains traction, which it's doing, everywhere I look, I understand XYZ will stick around for at least another 10K years, maybe more. So will Latin. It's not a matter of either/or. I can teach XYZ math and will happily do so, but the better more elite schools let me phase in more IVM stuff, cuz they, like me, have a sense of the future.
