I am trying to define a function which takes a binary number and gives the decimal form. I tried the following

binarytodecimal[b_] := NumberForm[2^^b, 20]

But the variable b is not recognised by Mathematica, I think because of the ^^. I have tried reading the documentation to see if there is another way to input a binary number but I haven't found anything.

As an example, take 0.0100101010101010101010. Then NumberForm[2^^0.0100101010101010101010, 20] computes but trying to turn this into a function does not work.

Is there a way around this?

  • $\begingroup$ In what format are you providing the binary number? As an ordinary "decimal" number (b = 1011001) or as a string (b = "1011001") or as a list of digits (b = {1, 0, 1, 1, 0, 0, 1}). $\endgroup$
    – Domen
    Commented Aug 16, 2021 at 10:22
  • 2
    $\begingroup$ FromDigits[RealDigits[0.0100101010101010101010, 10], 2] gives 611669/2097152 and using N gives 0.291667 $\endgroup$
    – flinty
    Commented Aug 16, 2021 at 10:35
  • 2
    $\begingroup$ @math, I think this double caret is some very low-level syntactic "thing", which cannot be evaluated together with other symbols, but only with numerical symbols (i.e. numerals, decimal point and letters). For example: b = 1101; 2^^b does not work. $\endgroup$
    – Domen
    Commented Aug 16, 2021 at 10:44
  • 1
    $\begingroup$ Following my comment, in the open-source parser for Wolfram Language (line 1051), you can see that base^^digits gets tokenized as a single token, and therefore this double caret is not a proper "symbol" or "operator". $\endgroup$
    – Domen
    Commented Aug 16, 2021 at 10:52
  • 1
    $\begingroup$ @math, very good question. indeed. $\endgroup$
    – Syed
    Commented Aug 16, 2021 at 12:56

1 Answer 1


The code

binarytodecimal[b_String] := NumberForm[ToExpression["2^^"<>b], 20];

does what you want. For example


evaluates to 0.2916665077209473


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