How to find the number of digits in a number

Question

IntegerLength[n] will return the number of digits in an integer.

I could not find a corresponding function that worked for reals, so I wrote my own function, digitsInnumber[n], which works for both integers and reals

Being new to Mathematica, I'd like to know if what I've done is good Mathematica style and if not, is there a better way to do this?

Thanks to the commentators for pointing out various things that I missed. After further thought, I think the following function fixes all the bugs and oversights, except that I couldn't get all fractions to work, so I explicitly eliminated fractions.

digitsInRealOrInteger[x_] := ( 

If[ IntegerQ[x], Return[StringLength[ToString[x]]]]; 

If [Element[x, Rationals] === True,Return[Print["input cannot be a fraction"]]];

If[Element[x, Reals] === True ,Return[StringLength[ToString[x, InputForm]]-1]]; 

Return[Print["input must be real or integer"]]
                              )

I'm surprised that this turned out to be more involved than I originally expected and that I had to resort to a more procedural approach.

Answer 1

Inte> IntegerLength[n] will return the number of digits in an integer.

I could not find a corresponding function that worked for reals

It is not very clear what "number of digits" might even mean for reals. For a floating point number, Precision might come closest.

Being new to Mathematica, I'd like to know if what I've done is good Mathematica style and if not, is there a better way to do this?

There are several problems.

In roughly reverse order of seriousness:

• Return[Print[...]] is incorrect. Print[...] returns Null, while displaying things on screen as a side effect. I assume you did not want to return Null.

• ToString[x, InputForm] is very very problematic. First, you should have a clear idea of what "number of digits" might even mean, and only start implementing things afterwards. Then you should strive to extract the information you need as directly as possible. Here you convert the number to a representation that is meant to be read by people, not computers, then try to automatically extract information from it anyway. This is what they call a nasty hack. Examples of what can go wrong: (1) InputForm[N[-1/7, 10]] has a -, a . and 10. in it. (2) What if you pass Pi to your function?

  If you were to use a conceptually similar route to this one, a much better way would be to try to extract the "number of digits" from RealDigits, a structured, computer readable representation. Note that this will be the same as Precision. How you could have found this: The IntegerLength doc page as a link to IntegerDigits, which then leads to RealDigits.

• Be aware of the distinction between mathematical concepts and programming concepts. A real number is a concept in mathematics. A floating point number is a representation of a number in a computer. It's what Mathematica calls a Real. Element/Reals/Integers/Rationals are all for working with math concepts, not programming concepts. Element might not evaluate to either True or False. E.g. 1.0 represents an inexact number in Mathematica, with a precision of ~16 digits ($MachinePrecision). The rest of the digits is unknown. Thus we can't say if it's an integer (the 17th digit may not be zero), and we can't even say if it's a rational. Element won't evaluate for it, which is Mathematica's way of saying "I don't know". In contrast, IntegerQ checks the data type of an expression, not its mathematical meaning. You should validate the function's input in a consistent way, not mix these two concepts. The correct way depends on what you actually want to do, which brings us back to the question: what does "number of digits" mean here?

• Try to avoid Return for now. While it is useful, everything can be done without it, and beginners typically misuse it (see your use with Print). Do look up Which if you're looking at consecutive Ifs.

• I understand why you use If[something === True, ...] instead of If[something, ...], but it looks confusing. I recommend If[TrueQ[something], ...] to make your intention clear even to less experienced readers of your code. This is merely a stylistic point.