How many fractional digits do I need to represent a number of base $m$ in base $n$?

Question

I have authored a web application RADIX, which for further optimisation needs to calculate the maximum number of places necessary to precisely represent the fraction part of a number of base $m$ in base $n$ (assuming a precise conversion is possible).

For example, assuming $f$ represents $15$ in base $64$, how many fraction digits are required to represent the number $f.f$ in base $10$?

I know that the maximum number of digits needed to represent the integer part can be calculated by taking the ceiling of $\log_{10}(64)$ * the number of digits (correct me if I'm wrong), but what about the fractional part of the number?

$f.f$ is $15.234375$ in base $10$, so one fraction numeral in base $64$ seems to require up to $6$ fraction digits in base $10$ to represent it, but is there a way I can calculate that in advance for any two bases?

At the moment I am using $\log_2(m)$ * the number of fraction digits of the number in base m, which happens to give just the right answer for the example above, i.e. $\log_2(64)$ is $6$, but it causes me to calculate to an unnecessarily high number of places for other conversions.

Update:

Example code, based on ShreevatsaR's expression for d in terms of m and n using prime factorisation.

# assumes float division    
m = 288    # base of the number to be converted    
n = 270    # base the number is to be converted to   
i = 2
d = 0

while m > 1 and n > 1:
    e = 0
    f = 0

    while m % i == 0:
        e += 1
        m /= i

    while n % i == 0:
        f += 1
        n /= i

    # if i is a prime factor of both m and n  
    if e != 0 and f != 0 and e / f > d:
        d = math.ceil( e / f )                     
    i += 1

if d == 0:
    # No fraction part of a number of base m has a finite
    # representation when converted to base n, and vice versa
else:
    # A maximum of d * r fraction digits is needed to represent the
    # fraction part of a number of base m in base n, where r is the
    # number of fraction digits of the number in base m 

Answer 1

This is an elaboration of Ross Millikan's answer.

• First we'll answer the question: to represent a fraction in base $n$, how many digits are needed (after the decimal point)?

If a fraction can be written in base $n$ with $d$ digits afer the decimal point, then it means that it can be written as $a/n^d$, for some integer $a$. (For instance, the fraction $975/64$ can be written as $15.234375 = \frac{15234375}{1000000}$.) Thus, if the fraction is $p/q$ in lowest terms, then the fact that $a/n^d$ is $p/q$ in lowest terms means that $q$ divides $n^d$.
Conversely, if $q$ divides $n^d$, then $p/q = a/n^d$ for some $a$ (to be precise, $a = pn^d/q$), and so the fraction can be written in base $n$ with $d$ digits after the decimal point.

So the number of digits $d$ needed after the decimal point is the smallest $d$ for which $n^d$ is divisible by $q$.

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• Second, to answer the question: when a number written in base $m$ with 1 digit after the decimal point is reduced to fraction $p/q$ in lowest terms, what are the possible values of $q$?

If a number $x$ is written in base $m$ as $x = a.b$ where $a$ is any integer (has any number of digits) and $b$ is a single digit in base $m$, then $x = a + b/m = (ma+b)/m$. So when reduced to lowest terms $p/q$ (which we do by cancelling common factors from $(ma+b)$ and $m$) it must be the case that $q$ is a divisor of $m$.
And in fact it can happen that $q=m$, e.g. when $b = 1$, or $b = q-1$ or more generally $\gcd(b,m) = 1$. (This is because any common factor of $(ma+b)$ and $m$ must also divide $b$, so if $\gcd(b,m) = 1$ then the only common factor is $1$, so we cannot reduce the fraction $(ma+b)/m$ further.)

Similarly, if a number is written in base $m$ with $r$ digits after the decimal point, then when it is reduced to lowest terms, the denominator could be up to $m^r$.

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Putting them together, if a number is written in base $m$ with one digit (respectively $r$ digits) after the decimal point, then the number of digits needed after the decimal point to write it in base $n$ is at most the smallest $d$ for which $n^d$ is divisible by $m$ (respectively $m^r$).

Examples:

• If you use "f" to represent $15$, then "f.ff" in base $64$ represents $15 + (15\times64 + 15)/64^2$, so $q = 64^2$. If you want to now write this in base $10$, then the smallest $d$ for which $10^d$ is divisible by $64^2$ is $d = 12$, so that's how many digits you need. (And indeed, "f.ff" is $15.238037109375$.)

• If further $c$ represents $12$, then "f.c" represents $15 + 12/64 = 15 + 3/16$, so $q = 16$. Now $10^4$ is divisible by $16$, so you only need $4$ digits for this particular number ($15.1875$).

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To actually calculate the smallest $d$ for which $n^d$ is divisible by $m$, the simplest algorithm is to keep trying successive $d$ until one works (you will never need more than $m$ tries). (You could do a binary search over $d$, but this is overkill unless your $m$ and $n$ are, say, over $10000$ and your program is slow because of this pre-computation step.)

You can do a couple of optimizations:

• If you already know that $m$ is a power of $n$ (e.g. going from base $64$ to base $2$) then $d = \log_{n}m$.

• When calculating powers of $n$, you can reduce the number modulo $m$ at each step. Something like

   N = n
   d = 1
   while N % m > 0:
     d += 1
     N *= n
     N %= m
     if d > m:
       # exit with error
   return d
  

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If you insist on an expression for $d$ in terms of $m$ and $n$ (beyond $\min\left\{d\colon m|n^d\right\}$ say), then we must look at the prime factorisation of $m$ and $n$. If $m$ has prime factorization $p_1^{e_1}p_2^{e_2}\dots$ and $n$ has prime factorization $p_1^{f_1}p_2^{f_2}\dots q_1^{g_1} \dots$ (all the same primes $p_1, p_2 \dots$, plus other primes $q_1, q_2 \dots$), then $$d = \max(\left\lceil\frac{e_1}{f_1}\right\rceil, \left\lceil\frac{e_2}{f_2}\right\rceil, \dots)$$ For example with $m = 288 = 2^5 3^2$ and $n = 270 = 2^1 3^3 5^1$, we have $d = \max(\lceil\frac51\rceil, \lceil\frac23\rceil) = 5$.

Answer 2

As Cameron Buie has said, you may have a number that terminates in one base and does not terminate in the other. Assuming it does terminate, represent the fraction as $\frac pq$ in lowest terms. The number of decimals in base $b$ is the smallest power of $b^n$ that can be divided evenly by $q$. So in base $10$, if $q=32=2^5$, you need $5$ decimals to represent it exactly. In base $4$, you would need three places beyond the point. Note that we don't need base information from where we are coming from, just where we are going.

Answer 3

I doubt there is a nice formula for it. For example, $\frac43$ in base $3$ is $1.1$, but in base $2$ it's $1.\overline{01}$, so we go from finitely many fractional digits to infinitely many.