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### 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

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

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,

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/
```
Associated Topics:
High School Number Theory

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