Convert decimal fraction to binary number

Difficulty Level : Medium
Last Updated : 11 Oct, 2019

Given an fraction decimal number n and integer k, convert decimal number n into equivalent binary number up-to k precision after decimal point.
Examples:

Input: n = 2.47, k = 5
Output: 10.01111

Input: n = 6.986 k = 8
Output: 110.11111100

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A) Convert the integral part of decimal to binary equivalent

Divide the decimal number by 2 and store remainders in array.
Divide the quotient by 2.
Repeat step 2 until we get the quotient equal to zero.
Equivalent binary number would be reverse of all remainders of step 1.

B) Convert the fractional part of decimal to binary equivalent

Multiply the fractional decimal number by 2.
Integral part of resultant decimal number will be first digit of fraction binary number.
Repeat step 1 using only fractional part of decimal number and then step 2.

C) Combine both integral and fractional part of binary number.

Illustration

Let's take an example for n = 4.47 k = 3

Step 1: Conversion of 4 to binary
1. 4/2 : Remainder = 0 : Quotient = 2
2. 2/2 : Remainder = 0 : Quotient = 1
3. 1/2 : Remainder = 1 : Quotient = 0

So equivalent binary of integral part of decimal is 100.

Step 2: Conversion of .47 to binary
1. 0.47 * 2 = 0.94, Integral part: 0
2. 0.94 * 2 = 1.88, Integral part: 1
3. 0.88 * 2 = 1.76, Integral part: 1

So equivalent binary of fractional part of decimal is .011

Step 3: Combined the result of step 1 and 2.

Final answer can be written as:
100 + .011 = 100.011

Program to demonstrate above steps:

C++

// C++ program to convert fractional decimal 
// to binary number 
#include<bits/stdc++.h> 
using namespace std; 
  
// Function to convert decimal to binary upto 
// k-precision after decimal point 
string decimalToBinary(double num, int k_prec) 
{ 
    string binary = ""; 
  
    // Fetch the integral part of decimal number 
    int Integral = num; 
  
    // Fetch the fractional part decimal number 
    double fractional = num - Integral; 
  
    // Conversion of integral part to 
    // binary equivalent 
    while (Integral) 
    { 
        int rem = Integral % 2; 
  
        // Append 0 in binary 
        binary.push_back(rem +'0'); 
  
        Integral /= 2; 
    } 
  
    // Reverse string to get original binary 
    // equivalent 
    reverse(binary.begin(),binary.end()); 
  
    // Append point before conversion of 
    // fractional part 
    binary.push_back('.'); 
  
    // Conversion of fractional part to 
    // binary equivalent 
    while (k_prec--) 
    { 
        // Find next bit in fraction 
        fractional *= 2; 
        int fract_bit = fractional; 
  
        if (fract_bit == 1) 
        { 
            fractional -= fract_bit; 
            binary.push_back(1 + '0'); 
        } 
        else
            binary.push_back(0 + '0'); 
    } 
  
    return binary; 
} 
  
// Driver code 
int main() 
{ 
  
    double n = 4.47; 
    int k = 3; 
    cout << decimalToBinary(n, k) << "\n"; 
  
    n = 6.986 , k = 5; 
    cout << decimalToBinary(n, k); 
    return 0; 
} 

Java

// Java program to convert fractional decimal 
// to binary number 
import java.util.*; 
  
class GFG  
{ 
  
    // Function to convert decimal to binary upto 
    // k-precision after decimal point 
    static String decimalToBinary(double num, int k_prec) 
    { 
        String binary = ""; 
  
        // Fetch the integral part of decimal number 
        int Integral = (int) num; 
  
        // Fetch the fractional part decimal number 
        double fractional = num - Integral; 
  
        // Conversion of integral part to 
        // binary equivalent 
        while (Integral > 0)  
        { 
            int rem = Integral % 2; 
  
            // Append 0 in binary 
            binary += ((char)(rem + '0')); 
  
            Integral /= 2; 
        } 
  
        // Reverse string to get original binary 
        // equivalent 
        binary = reverse(binary); 
  
        // Append point before conversion of 
        // fractional part 
        binary += ('.'); 
  
        // Conversion of fractional part to 
        // binary equivalent 
        while (k_prec-- > 0) 
        { 
            // Find next bit in fraction 
            fractional *= 2; 
            int fract_bit = (int) fractional; 
  
            if (fract_bit == 1) 
            { 
                fractional -= fract_bit; 
                binary += (char)(1 + '0'); 
            } 
            else
            { 
                binary += (char)(0 + '0'); 
            } 
        } 
  
        return binary; 
    } 
  
    static String reverse(String input)  
    { 
        char[] temparray = input.toCharArray(); 
        int left, right = 0; 
        right = temparray.length - 1; 
  
        for (left = 0; left < right; left++, right--)  
        { 
            // Swap values of left and right  
            char temp = temparray[left]; 
            temparray[left] = temparray[right]; 
            temparray[right] = temp; 
        } 
        return String.valueOf(temparray); 
    } 
  
    // Driver code 
    public static void main(String[] args)  
    { 
        double n = 4.47; 
        int k = 3; 
        System.out.println(decimalToBinary(n, k)); 
  
        n = 6.986; 
        k = 5; 
        System.out.println(decimalToBinary(n, k)); 
    } 
} 
  
// This code contributed by Rajput-Ji 

Python3

# Python3 program to convert fractional  
# decimal to binary number  
  
# Function to convert decimal to binary  
# upto k-precision after decimal point  
def decimalToBinary(num, k_prec) : 
  
    binary = ""  
  
    # Fetch the integral part of 
    # decimal number  
    Integral = int(num)  
  
    # Fetch the fractional part  
    # decimal number  
    fractional = num - Integral 
  
    # Conversion of integral part to  
    # binary equivalent  
    while (Integral) : 
          
        rem = Integral % 2
  
        # Append 0 in binary  
        binary += str(rem);  
  
        Integral //= 2
      
    # Reverse string to get original 
    # binary equivalent  
    binary = binary[ : : -1]  
  
    # Append point before conversion  
    # of fractional part  
    binary += '.'
  
    # Conversion of fractional part 
    # to binary equivalent  
    while (k_prec) : 
          
        # Find next bit in fraction  
        fractional *= 2
        fract_bit = int(fractional)  
  
        if (fract_bit == 1) : 
              
            fractional -= fract_bit  
            binary += '1'
              
        else : 
            binary += '0'
  
        k_prec -= 1
  
    return binary  
  
# Driver code  
if __name__ == "__main__" : 
      
    n = 4.47
    k = 3
    print(decimalToBinary(n, k)) 
  
    n = 6.986
    k = 5
    print(decimalToBinary(n, k)) 
  
# This code is contributed by Ryuga 

C#

// C# program to convert fractional decimal 
// to binary number 
using System; 
  
class GFG  
{ 
   
    // Function to convert decimal to binary upto 
    // k-precision after decimal point 
    static String decimalToBinary(double num, int k_prec) 
    { 
        String binary = ""; 
   
        // Fetch the integral part of decimal number 
        int Integral = (int) num; 
   
        // Fetch the fractional part decimal number 
        double fractional = num - Integral; 
   
        // Conversion of integral part to 
        // binary equivalent 
        while (Integral > 0)  
        { 
            int rem = Integral % 2; 
   
            // Append 0 in binary 
            binary += ((char)(rem + '0')); 
   
            Integral /= 2; 
        } 
   
        // Reverse string to get original binary 
        // equivalent 
        binary = reverse(binary); 
   
        // Append point before conversion of 
        // fractional part 
        binary += ('.'); 
   
        // Conversion of fractional part to 
        // binary equivalent 
        while (k_prec-- > 0) 
        { 
            // Find next bit in fraction 
            fractional *= 2; 
            int fract_bit = (int) fractional; 
   
            if (fract_bit == 1) 
            { 
                fractional -= fract_bit; 
                binary += (char)(1 + '0'); 
            } 
            else
            { 
                binary += (char)(0 + '0'); 
            } 
        } 
   
        return binary; 
    } 
   
    static String reverse(String input)  
    { 
        char[] temparray = input.ToCharArray(); 
        int left, right = 0; 
        right = temparray.Length - 1; 
   
        for (left = 0; left < right; left++, right--)  
        { 
            // Swap values of left and right  
            char temp = temparray[left]; 
            temparray[left] = temparray[right]; 
            temparray[right] = temp; 
        } 
        return String.Join("",temparray); 
    } 
   
    // Driver code 
    public static void Main(String[] args)  
    { 
        double n = 4.47; 
        int k = 3; 
        Console.WriteLine(decimalToBinary(n, k)); 
   
        n = 6.986; 
        k = 5; 
        Console.WriteLine(decimalToBinary(n, k)); 
    } 
} 
  
// This code has been contributed by 29AjayKumar 

PHP

<?php 
// PHP program to convert fractional decimal 
// to binary number 
  
// Function to convert decimal to binary upto 
// k-precision after decimal point 
function decimalToBinary($num, $k_prec) 
{ 
    $binary = ""; 
  
    // Fetch the integral part of decimal number 
    $Integral = (int)($num); 
  
    // Fetch the fractional part decimal number 
    $fractional = $num - $Integral; 
  
    // Conversion of integral part to 
    // binary equivalent 
    while ($Integral) 
    { 
        $rem = $Integral % 2; 
  
        // Append 0 in binary 
        $binary.=chr($rem + 48 ); 
  
        $Integral = (int)($Integral/2); 
    } 
  
    // Reverse string to get original binary 
    // equivalent 
    $binary=strrev($binary); 
  
    // Append point before conversion of 
    // fractional part 
    $binary.='.'; 
  
    // Conversion of fractional part to 
    // binary equivalent 
    while ($k_prec--) 
    { 
        // Find next bit in fraction 
        $fractional *= 2; 
        $fract_bit = (int)$fractional; 
  
        if ($fract_bit == 1) 
        { 
            $fractional -= $fract_bit; 
            $binary.=chr(1 + 48 ); 
        } 
        else
            $binary.=chr(0 + 48 ); 
    } 
  
    return $binary; 
} 
  
// Driver code 
  
    $n = 4.47; 
    $k = 3; 
    echo decimalToBinary($n, $k)."\n"; 
  
    $n = 6.986; 
    $k = 5; 
    echo decimalToBinary($n, $k); 
      
// This code is contributed by mits 
?> 

Output:

100.011
110.11111

Time complexity: O(len(n))
Auxiliary space: O(len(n))
where len() is the total digits contain in number n.

Convert Binary fraction to Decimal

Reference:
http://cs.furman.edu/digitaldomain/more/ch6/dec_frac_to_bin.htm
http://www.cquestions.com/2011/07/c-program-for-fractional-decimal-to.html

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