Converting a Real Number (between 0 and 1) to Binary String

Given a real number between 0 and 1 (e.g., 0.72) that is passed in as a double, print the binary representation. If the number cannot be represented accurately in binary with at most 32 characters, print” ERROR:’
Examples:

Input :  (0.625)10
Output : (0.101)2

Input : (0.72)10
Output : ERROR



Solution: First, let’s start off by asking ourselves what a non-integer number in binary looks like. By analogy to a decimal number, the binary number 0 .1012 would look like:

0. 1012 = 1 * 1/21 + 0 *1/22 + 1 * 1/23 .

Method 1: Multiply the decimal part by 2

To print the decimal part, we can multiply by 2 and check if 2*n is greater than or equal to 1. This is essentially “shifting” the fractional sum. That is:

r = 210 * n;
  = 210 * 0.1012;
  = 1 * 1/20 + 0 *1/21 + 1 * 1/22;
  = 1.012;

If r >= 1, then we know that n had a 1 right after the decimal point. By doing this continuously, we can check every digit.

C++

// C++ program to binary real number to string
#include <iostream>
#include<string>
using namespace std;
  
// Function to convert Binary real
// number to String
string toBinary(double n)
{
    // Check if the number is Between 0 to 1 or Not
    if (n >= 1 || n <= 0)
        return "ERROR";
  
    string answer;
    double frac = 0.5;
    answer.append(".");
  
    // Setting a limit on length: 32 characters.         
    while (n > 0)
    {
          
        //Setting a limit on length: 32 characters     
        if (answer.length() >= 32)
                return "ERROR";
  
            // Multiply n by 2 to check it 1 or 0
            double b = n * 2;
            if (b >= 1)
            {
                answer.append("1");
                n = b - 1;
            }
            else
            {
                answer.append("0");
                n = b;
            }
        }
        return answer;
}
  
// Driver code
int main()
{
    // Input value 
    double n = 0.625; 
      
    string result = toBinary(n);
    cout<<"(0"<< result <<") in base 2"<<endl;
  
    double m = 0.72;
    result= toBinary(m);
    cout<<"("<<result<<")"<<endl; 
}
  
// This code is contributed by Himanshu Batra

Java

// Java program to Binary real number to String.
import java.lang.*;
import java.io.*;
import java.util.*;
  
// Class Reperesentation of Binary real number
// to String
class BinaryToString
{
    // Main function to convert Binary real number
    // to String
    static String printBinary(double num)
    {
        // Check Number is Between 0 to 1 or Not
        if (num >= 1 || num <= 0)
            return "ERROR";
  
        StringBuilder binary = new StringBuilder();
        binary.append(".");
  
        while (num > 0)
        {
            /* Setting a limit on length: 32 characters,
               If the number cannot be represented
               accurately in binary with at most 32
               character  */
            if (binary.length() >= 32)
                return "ERROR";
  
            // Multiply by 2 in num to check it 1 or 0
            double r = num * 2;
            if (r >= 1)
            {
                binary.append(1);
                num = r - 1;
            }
            else
            {
                binary.append(0);
                num = r;
            }
        }
        return binary.toString();
    }
  
    // Driver Code
    public static void main(String[] args)
    {
        double num1 = 0.625; // Input value in Decimal
        String output = printBinary(num1);
        System.out.println("(0" + output + ")  in base 2");
  
        double num2 = 0.72;
        output = printBinary(num2);
        System.out.println("(" + output + ") ");
    }
}

Python3

# Python3 program to binary real number to string
  
# Function to convert Binary real
# number to String
def toBinary(n):
  
    # Check if the number is Between 0 to 1 or Not
    if(n >= 1 or n <= 0):
        return "ERROR"
  
    answer = ""
    frac = 0.5
    answer = answer + "."
  
    # Setting a limit on length: 32 characters.
    while(n > 0):
  
        # Setting a limit on length: 32 characters
        if(len(answer) >= 32):
            return "ERROR"
  
        # Multiply n by 2 to check it 1 or 0
        b = n * 2
        if (b >= 1):
  
            answer = answer + "1"
            n = b - 1
  
        else:
            answer = answer + "0"
            n = b
  
    return answer
  
# Driver code
if __name__=='__main__':
    n = 0.625
    result = toBinary(n)
    print("(0", result, ") in base 2")
    m = 0.72
    result = toBinary(m)
    print("(", result, ")")
  
# This code is contributed by
# Sanjit_Prasad


Output:

(0.101)  in base 2
(ERROR) 

Method 2

Alternatively, rather than multiplying the number by two and comparing it to 1, we can compare the number to . 5, then . 25, and so on. The code below demonstrates this approach.

C++

// C++ program to Binary real number to String.
#include <iostream>
#include<string>
using namespace std;
  
// Function to convert Binary real
// number to String
string toBinary(double n)
{
    // Check if the number is Between 0 to 1 or Not
    if (n >= 1 || n <= 0)
        return "ERROR";
  
    string answer;
    double frac = 0.5;
    answer.append(".");
  
    // Setting a limit on length: 32 characters.         
    while (n > 0)
    {
        // 32 char max
        if (answer.length() >= 32)
            return "ERROR";
    // compare the number to .5
        if (n >= frac)
        {
            answer.append("1");
            n = n- frac;
        }
        else
        {
            answer.append("0");
        }
          
        frac /= 2;
    }
    return answer;
}
  
// Driver code
int main()
{
    // Input value 
    double n = 0.625;
      
    string result = toBinary(n);
    cout<<"(0"<< result <<") in base 2"<<endl;
  
    double m = 0.72;
    result= toBinary(m);
    cout<<"("<<result<<")"<<endl; 
}

Java

// Java program to Binary real number to String.
import java.lang.*;
import java.io.*;
import java.util.*;
  
// Class Reperesentation of Binary real number
// to String
class BinaryToString
{
    // Main function to convert Binary real
    // number to String
    static String printBinary(double num)
    {
        // Check Number is Between 0 to 1 or Not
        if (num >= 1 || num <= 0)
            return "ERROR";
  
        StringBuilder binary = new StringBuilder();
        double frac = 0.5;
        binary.append(".");
  
        while (num > 0)
        {
            /* Setting a limit on length: 32 characters,
               If the number cannot be represented
               accurately in binary with at most 32
               characters  */
            if (binary.length() >= 32)
                return "ERROR";
  
            // It compare the number to . 5.
            if (num >= frac)
            {
                binary.append(1);
                num -= frac;
            }
            else            
                binary.append(0);
  
            // Now it become 0.25
            frac /= 2;
        }
        return binary.toString();
    }
  
    // Driver Code
    public static void main(String[] args)
    {
        double num1 = 0.625; // Input value in Decimal
        String output = printBinary(num1);
        System.out.println("(0" + output + ")  in base 2");
  
        double num2 = 0.72;
        output = printBinary(num2);
        System.out.println("(" + output + ") ");
    }
}


Output:

(0.101)  in base 2
(ERROR)  

Both approaches are equally good; choose the one you feel most comfortable with.

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