Midy’s theorem

According to Midy’s theorem, if the period of a repeating decimal for a / p, where p is prime and a / p is a reduced fraction, has an even number of digits, then dividing the repeating portion into halves and adding gives a string of 9s.

Examples :

a = 1 and p = 7
1/7 = 0.14285714285..
So 1/7 is a repeating decimal with 142857 being repeated. Now, according to the theorem, it has even number of repeating digits i.e. 142857. Further if we divide this into two halves, we get 142 and 857. Thus, on adding these two, we get 999 which is string of 9s and matches our theorem.

a = 2 and p = 11
2/11 = 0.18181818181..
Here, repeating decimal is 18. Now this is even in number therefore 1+8 = 9 which again shows the validity of Midy’s theorem.

Given numerator and denominator, the task is to find if the resultant floating point number follows Midy’s theorem or not.

Approach :
Let us simulate the process of converting fraction to decimal. Let us look at the part where we have already figured out the integer part which is floor(numerator/denominator). Now we are left with ( remainder = numerator%denominator ) / denominator.
If you remember the process of converting to decimal, at each step we do the following :

  1. Multiply the remainder by 10.
  2. Append remainder / denominator to result.
  3. Remainder = remainder % denominator.

At any moment, if remainder becomes 0, we are done.

However, when there is a recurring sequence, remainder never becomes 0. For example if you look at 1/3, the remainder never becomes 0.

Below is one important observation :
If we start with remainder ‘rem’ and if the remainder repeats at any point of time, the digits between the two occurrence of ‘rem’ keep repeating.

So the idea is to store seen remainders in a map. Whenever a remainder repeats, we return the sequence before the next occurrence.


Below is CPP implementation of Midy’s theorem :

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// C++ implementation as a
// proof of the Midy's theorem
#include <bits/stdc++.h>
using namespace std;
  
// Returns repeating sequence of a fraction.
// If repeating sequence doesn't exits,
// then returns -1
string fractionToDecimal(int numerator, int denominator)
{
    string res;
  
    /* Create a map to store already seen remainders
       remainder is used as key and its position in
       result is stored as value. Note that we need
       position for cases like 1/6. In this case,
       the recurring sequence doesn't start from first
       remainder. */
    map<int, int> mp;
    mp.clear();
      
    // Find first remainder
    int rem = numerator % denominator;
  
    // Keep finding remainder until either remainder
    // becomes 0 or repeats
    while ((rem != 0) && (mp.find(rem) == mp.end())) 
    {
        // Store this remainder
        mp[rem] = res.length();
  
        // Multiply remainder with 10
        rem = rem * 10;
  
        // Append rem / denr to result
        int res_part = rem / denominator;
        res += to_string(res_part);
  
        // Update remainder
        rem = rem % denominator;
    }
    return (rem == 0) ? "-1" : res.substr(mp[rem]);
}
  
// Checks whether a number is prime or not
bool isPrime(int n)
{
    for (int i = 2; i <= n / 2; i++)     
        if (n % i == 0)
            return false;
   return true;
}
  
// If all conditions are met,
// it proves Midy's theorem
void Midys(string str, int n)
{
    int l = str.length();
    int part1 = 0, part2 = 0;
    if (!isPrime(n))    
    
        cout << "Denominator is not prime, "
             << "thus Midy's theorem is not applicable";
    }
    else if (l % 2 == 0) 
    {
        for (int i = 0; i < l / 2; i++) 
        {
            part1 = part1 * 10 + (str[i] - '0');
            part2 = part2 * 10 + (str[l / 2 + i] - '0');
        }
        cout << part1 << " + " << part2 << " = " 
             << (part1 + part2) << endl;
        cout << "Midy's theorem holds!";
    }
    else 
    {
        cout << "The repeating decimal is of odd length "
             << "thus Midy's theorem is not applicable";
    }
}
  
// Driver code
int main()
{
    int numr = 2, denr = 11;
    string res = fractionToDecimal(numr, denr);
    if (res == "-1")
        cout << "The fraction does not have repeating decimal";
    else {
        cout << "Repeating decimal = " << res << endl;
        Midys(res, denr);
    }
    return 0;
}

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Output :

Repeating decimal = 18
1 + 8 = 9
Midy's theorem holds!

More about Midy’s theorem can be found on
http://www.kurims.kyoto-u.ac.jp/EMIS/journals/INTEGERS/papers/h2/h2.pdf
http://digitalcommons.unl.edu/cgi/viewcontent.cgi?article=1047&context=mathfacpub



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