so I was poking around my calculator when I discovered that when doing calculations the calculator actually stores 14 digits but only shows 10 digits. to illustrate my point here's some code which takes the value stored in ans and returns all 14 digits

Code:

what is the point of these extra digits if you have to go through all this work to see them, are they just for extra accuracy? and if so why are they not shown

The extra digits are because the calculator uses floating point arithmetic. Essentially, numbers are stored in 3 chunks, the sign, the mantissa and the exponent. The important part to note is that there is a fixed number of bits allocated to each of these chunks. This means that when it performs a calculation, it has to round off the last digit. In more complex calculations, like the log() command, there are quite a few computations involved, and the calculator therefore has to round off a lot. This can lead to inaccurate answers, so to remedy this, ti makes the calculators calculate more digits than they need and discards the last few, to ensure that the digits that are displayed are correct.
An interesting quirk is that sometimes, the extra digits aren't enough to hide the flaws, so you end up with erroneous answers anyway.

When you do ln(e^(e)), you should get e. Therefore, raising this to iπ should then give -1, but instead, you get a little bit of rounding error on the imaginary part. However, entering it as e^(iπ) works, because there is no ln() involved. Same for the log₂16, which gives 4, but the integer part of that gives 3.
I had made a program that computed numerical derivatives of some functions, but the answers it returned were more funny looking than useful. The derivative of sin(x) should be cos(x), or sin(x+π/2)...



Although it is quite rare, this can actually be quite problematic when coding, because even if the code is perfect, it could be spitting out incorrect results.

tl;dr, the last digits aren't displayed because they are likely to be wrong.

Does it do this on the 84 pce too?

Almost certainly yes. Virtually every scientific and graphing calculator I've seen for years has used a technique of internally handling more digits than are displayed.

Yes, it's an extremely common thing when dealing with floats, and TI has done this with every single graphing calculator they've ever released.