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

Last Updated : 13 Oct, 2022

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 Representation 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

// C# program to Binary real number to String.   using System; using System.Text;  // Class Representation 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()      {          double num1 = 0.625; // Input value in Decimal          String output = printBinary(num1);          Console.WriteLine("(0 " + output + ") in base 2");           double num2 = 0.72;          output = printBinary(num2);      Console.WriteLine("(" + output + ") ");      }  }

PHP

<?php // PHP program to binary real number // to string  // Function to convert Binary real // number to String function toBinary($n) {     // Check if the number is Between     // 0 to 1 or Not     if ($n >= 1 || $n <= 0)         return "ERROR";      $answer = "";     $frac = 0.5;     $answer .= ".";      // Setting a limit on length: 32 characters.              while ($n > 0)     {                  //Setting a limit on length: 32 characters          if (strlen($answer) >= 32)                 return "ERROR";              // Multiply n by 2 to check it 1 or 0             $b = $n * 2;             if ($b >= 1)             {                 $answer .= "1";                 $n = $b - 1;             }             else             {                 $answer .= "0";                 $n = $b;             }         }         return $answer; }  // Driver code  // Input value  $n = 0.625;   $result = toBinary($n); echo "(0" . $result . ") in base 2\n";  $m = 0.72; $result= toBinary($m); echo "(" . $result . ")";      // This code is contributed by mits ?>

JavaScript

<script>  // JavaScript program to binary real number to string  // Function to convert Binary real // number to String function toBinary(n){      // Check if the number is Between 0 to 1 or Not     if(n >= 1 || n <= 0)         return "ERROR"      let answer = ""     let frac = 0.5     answer = answer + "."      // 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         b = n * 2         if (b >= 1){              answer = answer + "1"             n = b - 1         }          else{             answer = answer + "0"             n = b         }     }      return answer }  // Driver code  let n = 0.625 let result = toBinary(n) document.write(`(0 ${result}) in base 2`,"</br>") let m = 0.72 result = toBinary(m) document.write(`(${result})`,"</br>")  // This code is contributed by Shinjanpatra  </script>

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 representation 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 + ") ");     } }

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";      frac = 0.5;     answer = ".";      # Setting a limit on length: 32 characters.          while (n > 0):                  # 32 char max         if (len(answer) >= 32):             return "ERROR";                      # compare the number to .5         if (n >= frac):             answer += "1";             n = n - frac;         else:             answer += "0";                  frac = (frac / 2);          return answer;  # Driver code  # Input value  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 mits

// C# program to Binary real number to String.  using System;  // Class representation 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";           string binary = "";          double frac = 0.5;          binary += ".";           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 += "1";                  num -= frac;              }              else                         binary += "0";               // Now it become 0.25              frac /= 2;          }          return binary;      }       // Driver Code      public static void Main()      {          double num1 = 0.625; // Input value in Decimal          String output = printBinary(num1);          Console.WriteLine("(0" + output + ") in base 2");           double num2 = 0.72;          output = printBinary(num2);          Console.WriteLine("(" + output + ") ");      }  }   // This code is contributed by mits

PHP

<?php // PHP program to Binary real number to String.  // Function to convert Binary real // number to String function toBinary($n) {     // Check if the number is Between      // 0 to 1 or Not     if ($n >= 1 || $n <= 0)         return "ERROR";      $frac = 0.5;     $answer = ".";      // Setting a limit on length: 32 characters.              while ($n > 0)     {         // 32 char max         if (strlen($answer) >= 32)             return "ERROR";                      // compare the number to .5         if ($n >= $frac)         {             $answer.="1";             $n = $n - $frac;         }         else         {             $answer.="0";         }                  $frac = ($frac / 2);     }     return $answer; }  // Driver code  // Input value  $n = 0.625;  $result = toBinary($n); print("(0".$result.") in base 2\n");  $m = 0.72; $result = toBinary($m); print("(".$result.")");   // This code is contributed  // by chandan_jnu ?>

JavaScript

<script>  // Javascript program to Binary real  // number to String.   // Main function to convert Binary // real number to String  function printBinary(num)  {           // Check Number is Between      // 0 to 1 or Not      if (num >= 1 || num <= 0)          return "ERROR";           let binary = "";      let frac = 0.5;      binary += ".";           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 += "1";              num -= frac;          }          else                     binary += "0";                   // Now it become 0.25          frac = frac / 2;      }      return binary;  }   // Driver code  // Input value in Decimal  let num1 = 0.625;  let output = printBinary(num1);  document.write("(0" + output +                 ") in base 2" + "</br>");   let num2 = 0.72;  output = printBinary(num2);  document.write("(" + output + ") ");   // This code is contributed by rameshtravel07  </script>

Output:

(0.101)  in base 2 (ERROR)

Both approaches are equally good; choose the one you feel most comfortable with.
This article is contributed by Mr. Somesh Awasthi.

Convert from any base to decimal and vice versa

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

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