Showing what happens with a number beyond machineprecision

Question

I want to play around a bit in Mathematica to understand how floating-point precision, rounding and truncation actually works. I understand that Machineprecision is $2^{-52}$.

First of all I would like to create a binarynumber and show that it is correct until the machinecorrection and after that it starts to be 'random'. But how do I do this? For example:

BaseForm[1/3 // N, 2]

Generates: $0.010101010101010101011_2$

1. Why are there $21$ digits after the dot?
2. Why not $52$? How can I get $52$?
3. How can I show this number in the form the computer stores it? That is, with an exponent and a mantissa?
4. How to show what happens to a number beyond machineprecision?

This is all quite new to me, so any help is appreciated. I am sorry for the lack of knowledge.

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Maybe in other words, I firstly try to understand why this code:

BaseForm[1 + $MachineEpsilon, 2]

Doesn't generate a binary number with $52$ digits.

Answer 1

Answer from the comments:

(1) BaseForm is being applied to the "standard" form, which truncates to 6 decimal digits. (2) FIrst apply InputForm: In[117]:= BaseForm[InputForm[N[1/3]], 2] Out[117]//BaseForm= \!\( TagBox[ InterpretationBox[ StyleBox["2^^0.010101010101010101010101010101010101010101010101010101", ShowStringCharacters->True, NumberMarks->True], InputForm[0.3333333333333333], AutoDelete->True, Editable->True], BaseForm[#, 2]& ]\) (3) Check RealDigits (4) It falls off the end of the Earth. – Daniel Lichtblau Dec 4 '16 at 16:11

Just kidding about that item (4). I haven't tried but possibly what you'd want is to use N[...,prec] with prec set to something larger than $MachinePrecision. – Daniel Lichtblau Dec 4 '16 at 17:04

...although I think I'd use SetPrecision[...,prec] instead of N.