Is it possible to get the number of digits mathematically?

Question

I was wondering if it was possible to, say, have function $f$ that would return the number of digits in any given positive integer. I tried using some sort of a summation, but that failed quite miserably.

Using piecewise equations, it might be constructed something like this: $$f(x)=\begin{cases}1 & \text{ if } x=1 \\2 & \text{ if } x=52 \\3 & \text{ if } x=230 \\4 & \text{ if } x=5023\\\vdots&\text{ }\vdots\end{cases}$$ This is, however, not what I want. Instead, I would like, if possible, some mathematical way of doing this.

@MathApprentice You didn't take the floor first: $\log_{10}(2)\approx .3$. So, you take the floor of that and get $0$. So, $d(2)=1+0=1$. — J126, Dec 02 '14 at 22:43
You may be interested in my answer to this related question: http://math.stackexchange.com/questions/795412/no-of-digit-in-xy-where-x-y-in-mathbbn/795473#795473 — MPW, Dec 02 '14 at 22:45

score 3 · Accepted Answer · answered Dec 02 '14 at 22:27

3

$$f(x) = 1+\lfloor \log_{10}(x)\rfloor$$ More generally, $$f(x) = 1+\lfloor \log_{b}(x)\rfloor$$ returns the number of digits of the number in base $b$.

answered Dec 02 '14 at 22:27

Adhvaitha

5,441

1

so, $1 + log_{10} (2) = 1.3$ so I have 1.3 digits? – MathApprentice Dec 02 '14 at 22:30
1

$f(x) = 1+\lfloor \log_{b}(x)\rfloor$, $f(n)=-1$, if $0<n<.1$ – Conor O'Brien Dec 02 '14 at 22:32
3

@MathApprentice Don't you see the floor function? – Adhvaitha Dec 02 '14 at 22:43
Oh that's what the half-brackets mean. Okay. – MathApprentice Dec 02 '14 at 22:47

rewritten · Answer 2 · 2014-12-02T23:31:17.587

2

Hint: The integer part of the logarithm in base 10 is the function you look for.

edited Dec 02 '14 at 23:31

answered Dec 02 '14 at 22:28

rewritten

3,092

Not quite. It's one more than that. – MPW Dec 02 '14 at 22:48
Of course, I was pointing out which function the OP would want to use to find the solution. Notice the difference between "the solution is" and "you need to look for". – rewritten Dec 02 '14 at 22:51
I accept that. But it isn't at all clear in your answer. OP specifically asks for a function that gives the answer he seeks, and you stated that your function is the one he is looking for. If you are suggesting a gentle hint instead of an explicit answer (which I encourage, incidentally), it's usually an excellent idea to preface your answer with the disclaimer "Hint:" to make that clear. I can't withdraw my downvote unless you edit your answer because the site won't let me. – MPW Dec 02 '14 at 23:00
@MPW edited. :) – rewritten Dec 02 '14 at 23:32

Is it possible to get the number of digits mathematically?

2 Answers2