How to convert from base 2 to base n?

Question

I'm trying to use baseform to convert numbers in base 10 to base n, how can I make the convertion between, say base 2 to base n?

Baseform seems to always think that the base in expr BaseForm[Expr, n] is always 10.

Answer 1

It should be possible to use notation of the form base^^number inside the BaseForm expression like this:

BaseForm[2^^10101,14]

There are some similar examples under Properties and Relations in the documentation for BaseForm.

Answer 2

Since numbers given in base^^ form automatically parse as regular number, it can be at times useful to pass numbers around as strings. For example:

FromDigits["100010011110011", 2]

17651

Different ways to represent that number:

IntegerDigits[17651, 16]

BaseForm[17651, 2]

IntegerString[17651, 2]

{4, 4, 15, 3}

100010011110011₂

"100010011110011"

Answer 3

Here are the above elements wrapped up in function which pulls together the various, or user defined, output forms and lets you switch from any base to any base:

Clear[BaseTranslator];
Options[BaseTranslator] = {BTForm -> BaseForm};
BaseTranslator[number_, base1_, base2_, 
  OptionsPattern[]] := (OptionValue@BTForm)[
  FromDigits[ToString[number], base1], base2]

And some sample usage:

BaseTranslator[100,10,4]

1210₄

BaseTranslator[100, 10, 4 ,BTForm -> IntegerDigits]

{1, 2, 1, 0}

BaseTranslator[100, 2, 4, BTForm -> IntegerDigits]

{ 1, 0}

BaseTranslator[102, 10, 101, BTForm -> IntegerDigits]

{1, 1}