Get length of decimal digit function

Question

I expect I can get length of decimal digit from function with given number input.

For example:

And so on...

What is the general form of $f(x)$?

Now prove why floor(log(x))+1 is equivalent with floor(log(x)+1)? — Muhammad Ikhwan Perwira, Sep 03 '22 at 11:25

score 3 · Accepted Answer · edited Sep 03 '22 at 05:55

3

Are you looking for $$f(x)=\lfloor\log_{10}x\rfloor+1$$

edited Sep 03 '22 at 05:55

Parcly Taxel

answered Sep 03 '22 at 05:28

Bobby Ocean

Yes, f(x)=floor(log(x)) + 1 is already simplified version from your answer. – Muhammad Ikhwan Perwira Sep 03 '22 at 05:46
Yes of course, assuming $log$ means base 10. I prefer to emphasize the base. – Bobby Ocean Sep 03 '22 at 05:49
@ParclyTaxel I would prefer you not edit a matter of preference. It is standard to write ratios of the natural logarithm rather than writing a logarithm with a base. I have whole complex and real variables books that do this throughout. – Bobby Ocean Sep 03 '22 at 05:59
If you try implementing this in any programming language, you will run into problems due to rounding errors - don’t be surprised if n=10^9 gives a result of 8. The result is undefined instead of 1 for x = 0. – gnasher729 Sep 03 '22 at 06:46
Are you sure about that? Any number that fits into memory will process through a logarithm and have a perfectly accurate whole part within that same buffer size. – Bobby Ocean Sep 03 '22 at 10:11

score 2 · Answer 2 · answered Sep 03 '22 at 10:15

2

If you have a positive integer $N$ represented in basis $B$, then the number of digits needed is exactly$\def\len{\operatorname{len}}$

$$\len_B(N)=1+\lfloor\log_B N\rfloor$$

where $\lfloor\,\cdot\,\rfloor$ denotes the floor function.

answered Sep 03 '22 at 10:15

How to include 0 in function parameter so the function will return 1 digit? We know $log(0)$ is undefined in any base. – Muhammad Ikhwan Perwira Oct 29 '22 at 09:48
@Muhammad Ikhwan Perwira: Just special-case it: If $N=0$ then $len = 1$ else... – emacs drives me nuts Oct 29 '22 at 10:42

2 Answers2