Numbers to binary and back

Question

I am struggling on how to create a function that converts a number into binary and then back. I know there is a built-in function about this BaseForm[] but I am trying not to use it. I started thinking about using the RealDigits[] but I don't really know where to go from here. I will keep trying and post any code I do. If anyone can help me with this I would be really grateful. Thanks.

Answer 1

Just to illustrate ways to do you own way:

Binarize integer:

binary[n_] := 
 Reverse@NestWhileList[
    QuotientRemainder[#[[1]], 2] &, {n}, # != {0, 0} &][[2 ;; -2, 2]]

This produces a list, e.g. binary[23] yields: {1, 0, 1, 1, 1}.

You can invert:

frombinary[n_] := 
With[{k = Length[n]}, n . PowerRange[2^(k 
- 1), 1, 1/2]]

If you want create output like BaseForm:

bf[n_] := Subscript[Row[binary[n]], 2]

You can invert this by accessing list, e.g.

bf2d[n_] := frombinary[(n[[1, 1]])]

where input is Subscript[Row[...]],

The in-built functions are optimized. I present this to motivate "create your own" play.

Update/Edit

In response to comment:

binary[n_] := 
 Reverse@NestWhileList[
    QuotientRemainder[#[[1]], 2] &, {n}, # != {0, 0} &][[2 ;; -2, 2]]
binary[0] = {0};
dec[n_] := 
 NestWhileList[
   Reverse@MixedFractionParts[2 #[[1]]] &, {n}, # != {0, 
      0} &][[2 ;; -2, 2]]
anybin[n_] := 
 Module[{i = IntegerPart[n], f = FractionalPart[n]}, 
  Subscript[Row[Join[binary[i], {"."}, dec[f]]], 2]]

Comparing with BaseForm:

Table[{j, BaseForm[j, 2], 
   anybin[j]}, {j, {0.5, 1.25, 3.75, 1.2, 0.13}}] // 
 TableForm[#, TableHeadings -> {None, {"n",  "BaseForm","anybin"}}] &

Noting: fractions such as 1/5 do not have binary finite representations.

e.g.

Sum[2^(-4 j + 1) + 2^(- 4 j), {j, 1, 3}] // N
Limit[Sum[2^(-4 j + 1) + 2^(- 4 j), {j, 1, x}], x -> Infinity]

They are recurring. This could truncated to show repeating sequence. I leave that to enthusiast.

Answer 2

Edit: addressing the comment

Updated code is:

test[xx_, digits_: 17] := If[xx >= 1,
    ToString@Row@Insert[#1, ".", #2 + 1] & @@ 
   RealDigits[xx, 2, digits], 
  ToString@Row@
    Join[{0}, 
     Insert[Join[
       ConstantArray[0, Abs[RealDigits[xx, 2, digits][[2]]]], 
       RealDigits[xx, 2, digits][[1]]], ".", 1]]]

Checks:

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Original answer below:

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To binary

---

tobinary[realnum_] := 
 ToString@Row@Insert[#1, ".", #2 + 1] & @@ RealDigits[#, 2] &[realnum]

And we have

Table[tobinary[ii], {ii, 0, 4, 1}]

which can be compared with

Table[BaseForm[i, 2], {i, 0, 4}]

Works also for

tobinary[17.27]

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From binary

---

frombinary[mynumber_] := FromDigits[RealDigits[mynumber], 2]

Quick check:

frombinary[100]

And works also for

frombinary[10001.010001010001111010111000010100011110101110000101] // 
 N

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