Number of digits in a number?


by Buri
Tags: digits, number
Buri
Buri is offline
#1
Feb23-11, 09:16 AM
P: 273
I was playing around with my calculator and I noticed that to calculate the number of digits in a number n we have # of digits in n = [log(n) + 1], where log is in base 10 and [] represents the floor function. I after looked this up and it seems like its true. But how would you go about proving something like this? I thought of writing n as its decimal expansion, and then taking the logarithm, but it doesn't seem to get me no where. Any ideas?
Phys.Org News Partner Science news on Phys.org
Cougars' diverse diet helped them survive the Pleistocene mass extinction
Cyber risks can cause disruption on scale of 2008 crisis, study says
Mantis shrimp stronger than airplanes
micromass
micromass is offline
#2
Feb23-11, 10:00 AM
Mentor
micromass's Avatar
P: 16,690
Let x be a number, then it is easy to see that

x has n digits if and only if [tex]10^{n-1}\leq x<10^{n}[/tex].

If we apply your expression on this, then we obtain

[tex]n=\log(10^{n-1})+1\leq \log(x)+1<\log(10^n)+1=n+1[/tex]

This yields that [tex][\log(x)+1]=n[/tex].
HallsofIvy
HallsofIvy is online now
#3
Feb23-11, 10:01 AM
Math
Emeritus
Sci Advisor
Thanks
PF Gold
P: 38,895
A number, n, has "d" digits if and only if [itex]10^{d-1}\le n< 10^d[/itex]. Take the logarithm (base 10) of each part of that, using the fact that logarithm is an increasing function.

Buri
Buri is offline
#4
Feb23-11, 10:13 AM
P: 273

Number of digits in a number?


Ahhh didn't think of that. Thanks a lot to both of you! :)
Robert1986
Robert1986 is offline
#5
Feb23-11, 07:41 PM
P: 828
Quote Quote by Buri View Post
Ahhh didn't think of that. Thanks a lot to both of you! :)
Note that this can be generalized to any base b system. In particular, the number of bits needed to represent a number in base 2 is the log of that number to the base 2. This is really important in the analysis of algorithms.
Buri
Buri is offline
#6
Feb23-11, 09:27 PM
P: 273
Quote Quote by Robert1986 View Post
Note that this can be generalized to any base b system. In particular, the number of bits needed to represent a number in base 2 is the log of that number to the base 2. This is really important in the analysis of algorithms.
Funny how you actually mentioned algorithms. I actually got into this problem by looking at the efficiency of the Euclidean Algorithm. I was trying to show that when the Euclidean Algorithm is applied to a and b (a > b) that it terminates in at most 7 times the number of digits of b.


Register to reply

Related Discussions
Divisibility by 11 for all palindromes with an even number of digits Calculus & Beyond Homework 6
A three-digit number is drawn out of the digits from 0-9. Set Theory, Logic, Probability, Statistics 1
100cr digits number writing a paper General Math 15
Number of digits in n! Programming & Computer Science 8
Question about determining digits of a number Precalculus Mathematics Homework 1