there is no practical representation of the reals because their theory is based on infinite numbers and computers are finite. aka you are never going to store

there is no practical representation of the reals because their theory is based on infinite numbers and computers are finite. aka you are never going to store sqrt 2 in a register. the closest you can get is symbolic algebra packages. but at the end you still have to come down from infinite land and actually have an ASCII decimal representation because you cannot cut a board at sqrt2 feet long or fill your car with sqrt2 gallons of gasoline. or pay pi dollars for a pizza and expect your bank to keep a list of all the different transcendental atomic values you have used for payments over the years. "Your account has 17 pi plus 45 sqrt2 plus 97 e minus the golden ratio" is never going to work very well. one thing that i think would help is to put non-real mathematics into the mathematics curriculum and stop handwaving about how a computer does real numbers.. (it doesnt). teach discrete math, modular arithmetic, prime number theory, at the same time as pre-calculus.