Counting Digits

Date: 03/13/2002 at 16:31:09
From: Susan Diana Powers
Subject: Counting digits

I need to know how many times each digit appears when counting from 1 
to 1000 and 10000 (including zeros; for instance, 0001 has three 
zeros and one one).  I know that each digit appears twenty times when 
counting from 1 to 100, just from a trivia question I read somewhere 
once.  I don't know how to figure this out myself without actually 
writing out the numbers and counting digits, or without just 
approximating or guessing: for example, I'm guessing each digit 
appears 300 times between 1 and 999, but again, it's just a guess.  
Is there a formula for calculating the answer regardless of how many 
places we're discussing (10, 100, 1000, 10000, 100000 . . . )?  

Thanks for any help!

Date: 03/14/2002 at 00:15:16
From: Doctor Twe
Subject: Re: Counting digits

Hi Susan - thanks for writing to Dr. Math.

The key here is finding a pattern. Let's look at the numbers 1 to 100:

     01
     02
     03
     04
     05
     06
     07
     08
     09
     10
     11
     12
     13
     :
     97
     98
     99

Let's first consider just the units digits. There's a pattern here 
that repeats over and over: 1, 2, 3, ..., 9, 0, and so on. Since there 
are 100 units digits, and 10 digits in the repeating pattern, each 
digit appears in the units digit 100/10 = 10 times.

Now let's consider only the tens digits. Here, we have 0, 0, 0, 0, 0, 
0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ..., 9, 9, 9, 9, 9, 9, 9, 9, 
9, 9, 0. With the exception of that last zero, none of the digits 
appears in the list once we've "passed it." There are 10 ones (from 
the numbers 10-19), 10 twos (from the numbers 20-29), etc. The zeroes 
are a bit different because the string starts with only 9 of them 
(from the numbers 01-09), but we get a 10th one at the very end of the 
list (in the number 100). So each digit appears in the tens digit 10 
times.

Adding these together, we see that each digit appears in the sequence 
20 times. Oops! We forgot to count the 1 in the hundreds digit of 100. 
So there are really 21 ones, and 20 of each of the other digits. (I 
suspect that your trivia question was really stated as the number of 
times each digit appears in the range 00-99, not 01-100.)

Does this give you some insight as to how to answer the questions?

I hope this helps. If you have any more questions, write back.

- Doctor TWE, The Math Forum
  http://mathforum.org/dr.math/