Number of digits in a number?


by Buri
Tags: digits, number
Buri
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#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?
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micromass
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#2
Feb23-11, 10:00 AM
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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
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#3
Feb23-11, 10:01 AM
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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
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#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
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#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
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#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.


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