arithmetic - How to do binary calculation？

For example, $a=0.10101$ in binary representation.

Then how to calculate $a^2$ directly in mathematica?

Is there any function turning binary into decimal? I only found its inverse...

Answer

You can input numbers in any base up to 36 using the notation base^^digits. Digits over 9 are represented using a, b, c, ...

You can print numbers in any base up to 36 using BaseForm.

Thus,

In[1]:= a=2^^0.10101
Out[1]= 0.65625

In[2]:= BaseForm[a^2,2]
Out[2]//BaseForm= Subscript[0.0110111001, 2]

Note that the internal representation of numbers doesn't know or care about bases. These tools are only for inputting or printing numbers, but do not affect how numbers are stored internally. The internal representation is always binary.

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IntegerDigits/RealDigits and FromDigits will convert numbers to/from an explicit list of digits. These functions work with any base, not just up to 36.