Formula for the Extraction of a Digit

Date: 06/18/2001 at 17:44:15
From: Jake
Subject: Formula for the Extraction of a Digit in a Number

Is there a formula to extract any digit of a given number? For 
example, a formula that could be used to calculate the tens digit 
of 374, or the thousands digit of 3,289.  

I know how to do this logically, but I cannot think of any 
mathematical way to do it. Unfortunately, an algorithm is unacceptable 
for my purposes. It must be a formula. Is there such a formula?

Date: 06/20/2001 at 15:28:34
From: Doctor Toughy
Subject: Re: Formula for the Extraction of a Digit in a Number

Hi Jake,

Thanks for writing to Dr. Math.

My understanding of what you want is a formula such that when you are
given the number 374 it will give you the value of the tens digit, 
which in this case equals seven.  

The formula involves two steps: 

  (1) removing the numbers to the right (which in this case is the 
      units digit or the number 4), and 

  (2) removing the numbers to the left (which in this case is the 100 
      digit or the number 3).

The following is used to extract 37 from 374 by removing the 4:
 
      Int(374/10) = 37  

Taking the 37 we use the next formula to extract 7, which is our 
answer:

      37 mod 10 = 7
 
Combining the two formulas,

    Int(374/10) mod 10 = 7

Let K equal the selected number (in this case 374), and let N = the 
digit you want extracted such that if N = 1 you would get the 4, when 
N = 2 you would get the 7, and when N = 3 you would get the 3. Then 
the formula is:

     Int(K/10^(N-1)) mod 10     

Hope this helps. If you have any more questions about this or any 
other math topics, please write back.

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

Date: 06/20/2001 at 17:40:14
From: Jake
Subject: Re: Formula for the Extraction of a Digit in a Number

Is there any way to do that without rounding? What I really need is a 
formula that can do that with plain ol' math, no lossy methods like 
rounding. I really do doubt there is a way to do that, but I'm 
humoring the possibility just in case.

Date: 06/26/2001 at 15:30:29
From: Doctor Ian
Subject: Re: Formula for the Extraction of a Digit in a Number

Hi,

First, Dr. Toughy's algorithm uses truncation, not rounding. 

Second, since you're just plucking a digit from a number, 'lossiness' 
isn't really an issue - losing the rest of the number is what you're 
trying to do!

Third, if you feel, ah, unmathematical about using the Int() function, 
use the floor() function instead, which you can read about here:

  Math Notation via Email - Karl Hahn

  http://www.karlscalculus.org/email.html   

Look near the bottom, in the section called 'Special Functions':

  http://www.karlscalculus.org/email.html#special   

I hope this helps. 

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