As a programmer, we have been told about floating points errors on computer. Do Calculators have floating point error too?
0.1 (display) = .0999999998603016 (actual value used) on computers
Not really 0.1 But you can see it is close.
Calculators are computers, too; they're just smaller. Surely if we knew how to represent arbitrary real numbers inside calculators, we could do the same thing with desktop computers.
That said, it's possible—both on a calculator and on a computer—to represent some real numbers exactly. No computer I know of would represent $\frac12$ inexactly, since its binary expansion (0.1) is short enough to put inside a floating point register. More interestingly, you can also represent numbers like $\pi$ exactly, simply by storing them in symbolic form. In a nutshell, instead of trying to represent $\pi$ as a decimal (or binary) expansion, you just write down the symbol "$\pi$" (or, rather, whatever symbol the computer program uses for $\pi$).
They do as long as you don't use one that uses symbolical calculations like the modern TI or Casio calculators.
One way to see it is to calculate iteratively roots of a number and then square the results again, if you do it often enough you will get a different result than the input due to numerical errors, this doesn't only have to do with the floating point representation but also with inaccurate root algorithms used in the calculator.