Hi...need help with Decimal to Binary Conversion

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C++

Hello

Im relatively new to C++ but i want to become quite good in it
and not just do it to pass some subject.

So now everyday im improving a piece of my program to try and learn more and not unlearn what
I have already learned.

So now im making a Decimal to Binary conversion but im not using any pre made "Libraries" or "Classes" otherwise I feel like I havent really done anything.

I already have a piece of code that works out the Binary for anything INfront of a dot " 'xxx'.xxx " but
now im working on the part after the dot.

So I took out my Digital book and checked up how to do it with the multiplication by 2 move and coded it so it works on some fractions quite well.
HowEver using that ends up in an infinite loop allot of the times because every time I get something like 0.60 and work it out I somewhere along the line end up with 0.60 again thus ending in an infinite loop.

I have no idea how to let the program sense a repetition in the string that im creating.

Improved Question here.
My Question is...What would be a neat method of making the program realize hes running into and Endless Loop?
Is there some way I can make it sense that its working out something that it already worked out and that it should stop?

Here is the Code so far.

#include <cstdlib>
#include <iostream>
#include <string>
#include <math.h>

using namespace std;
double a;

double fracvalue(double x) 
{
 int z;            
       z = (x/1);
       a = x-z;
        return (a);
       }




int main(int argc, char** argv) {
    double x;
    int z, c;    
    cin >> x;
    double a =fracvalue(x)*2;
    cout << fracvalue(x) << "      " << a  << endl;//Checking if A and fracvalue are giving correct readings 
    string Bin = ".";
    
    for (int = i=;i < 20;i++)  //<<-- THIS was while(a != 0.0) but it never seemed to 
                      //stop even when it gets to 0.0 also it doesnt write out the complete binary this way
                      //sometimes only leaving dots or partial BIN numbers.
    {
          if (a > 1)
          {
                Bin = Bin +'1';
                a = a-1;
                cout << a << endl;
                a = a*2;
                cout << a << endl;}
                
             if (a < 1)
                {
                    Bin = Bin + '0';
                    a = a*2;
                    
                    }
                    cout << Bin << endl;
          }
          
          
          
    system("PAUSE");
    return 0;
}

For some reason it keeps on moaning about the line
z = (x/1); in the function fracvalue,the funny thing is I didnt touch this code :s it just didnt want to run anymore.

Anyway...It seems to give the right Binary if i insert large fractions but simple ones would come out only 0s or 1s and not both.
Larger ones however "Under while (a != 0.0)" will never end because it math itself loops infinitely.

Thanks