Convert a number into negative base representation

Last Updated : 22 Jun, 2022

A number n and a negative base negBase is given to us, we need to represent n in that negative base. Negative base works similar to positive base. For example in base 2 we multiply bits to 1, 2, 4, 8 and so on to get actual number in decimal. In case of base -2 we need to multiply bits with 1, -2, 4, -8 and so on to get number in decimal.
Examples:

Input  : n = 13, negBase = -2
Output : 11101
1*(16) + 1*(-8) + 1*(4) + 0*(-2) + 1*(1)  = 13

It is possible to represent a number into any negative base with same procedure (Refer Wiki for details). For simplicity (to get rid of A, B etc characters in output), we are allowing our base to be in between -2 and -10 only.

We can solve this problem similar to solving problem with positive bases but one important thing to remember is, remainder will always be positive whether we work with positive base or negative base but in most compilers, the result of dividing a negative number by a negative number is rounded towards 0, usually leaving a negative remainder.
So whenever we get a negative remainder, we can convert it to positive as below,

Let 
n = (?negBase) * quotient + remainder 
  = (?negBase) * quotient + negBase ? negBase + negBase 
  = (?negBase) * (quotient + 1) + (remainder + negBase). 

So if after doing "remainder = n % negBase" and 
"n = n/negBase", we get negative remainder, we do 
following.
remainder = remainder + (-negBase)
n = n + 1

Example : n = -4, negBase = -3
In C++, we get
    remainder = n % negBase = -4/-3 = -1
    n = n/negBase [Next step for base conversion]
      = -4/-3 
      = 1
To avoid negative remainder, we do,
    remainder = -1 + (-negBase) = -1 - (-3) = 2
    n = n + 1 = 1  + 1 = 2.

So when we will get negative remainder, we will make it positive by adding absolute value of base to it and adding 1 to our quotient.
Above explained approach is implemented in below code,

C++

//  C/C++ program to convert n into negative base form
#include <bits/stdc++.h>
using namespace std;
 
//  Utility method to convert integer into string
string toString(int n)
{
    string str;
    stringstream ss;
    ss << n;
    ss >> str;
    return str;
}
 
// Method to convert n to base negBase
string toNegativeBase(int n, int negBase)
{
    //  If n is zero then in any base it will be 0 only
    if (n == 0)
        return "0";
 
    string converted = "";
    while (n != 0)
    {
        // Get remainder by negative base, it can be
        // negative also
        int remainder = n % negBase;
        n /= negBase;
 
        // if remainder is negative, add abs(base) to
        // it and add 1 to n
        if (remainder < 0)
        {
            remainder += (-negBase);
            n += 1;
        }
 
        // convert remainder to string add into the result
        converted = toString(remainder) + converted;
    }
 
    return converted;
}
 
//  Driver code to test above methods
int main()
{
    int n = 13;
    int negBase = -2;
 
    cout << toNegativeBase(n, negBase);
 
    return 0;
}

Java

// Java program to convert n into 
// negative base form
class GFG
{
 
// Method to convert n to base negBase
static String toNegativeBase(int n, int negBase)
{
    // If n is zero then in any base
    // it will be 0 only
    if (n == 0)
        return "0";
 
    String converted = "";
    while (n != 0)
    {
        // Get remainder by negative base, 
        // it can be negative also
        int remainder = n % negBase;
        n /= negBase;
 
        // if remainder is negative, 
        // add Math.abs(base) to it 
        // and add 1 to n
        if (remainder < 0)
        {
            remainder += (-negBase);
            n += 1;
        }
 
        // convert remainder to String add into the result
        converted = String.valueOf(remainder) + converted;
    }
    return converted;
}
 
// Driver Code
public static void main(String[] args)
{
    int n = 13;
    int negBase = -2;
 
    System.out.print(toNegativeBase(n, negBase));
}
}
 
// This code is contributed by 29AjayKumar

Python3

# Python 3 program to convert n into 
# negative base form
 
# Method to convert n to base negBase
def toNegativeBase(n, negBase):
     
    # If n is zero then in any base it 
    # will be 0 only
    if (n == 0):
        return "0"
 
    converted = "01"
    while (n != 0):
         
        # Get remainder by negative base, 
        # it can be negative also
        remainder = n % (negBase)
        n = int(n/negBase)
 
        # if remainder is negative, add 
        # abs(base) to it and add 1 to n
        if (remainder < 0):
            remainder += ((-1) * negBase)
            n += 1
 
        # convert remainder to string add
        # into the result
        converted = str(remainder) + converted
         
    return converted
 
# Driver Code
if __name__ == '__main__':
    n = 13
    negBase = -2
 
    print(toNegativeBase(n, negBase))
 
# This code is contributed by
# Surendra_Gangwar

C#

// C# program to convert n into 
// negative base form
using System;
 
class GFG
{
 
// Method to convert n to base negBase
static String toNegativeBase(int n, int negBase)
{
    // If n is zero then in any base
    // it will be 0 only
    if (n == 0)
        return "0";
 
    String converted = "";
    while (n != 0)
    {
        // Get remainder by negative base, 
        // it can be negative also
        int remainder = n % negBase;
        n /= negBase;
 
        // if remainder is negative, 
        // add Math.Abs(base) to it 
        // and add 1 to n
        if (remainder < 0)
        {
            remainder += (-negBase);
            n += 1;
        }
 
        // convert remainder to String add into the result
        converted = String.Join("", remainder) + converted;
    }
    return converted;
}
 
// Driver Code
public static void Main(String[] args)
{
    int n = 13;
    int negBase = -2;
 
    Console.Write(toNegativeBase(n, negBase));
}
}
 
// This code is contributed by Rajput-Ji

Javascript

<script>
 
// JavaScript program to convert n into
// negative base form
 
// Method to convert n to base negBase
function toNegativeBase(n, negBase)
{
    // If n is zero then in any base
    // it will be 0 only
    if (n == 0)
        return "0";
 
    let converted = "01";
    while (n != 0)
    {
        // Get remainder by negative base,
        // it can be negative also
        let remainder = (-1)*(Math.abs(n) % Math.abs(negBase));
 
        n = parseInt(n/negBase);
 
        // if remainder is negative,
        // add Math.abs(base) to it
        // and add 1 to n
        if (remainder < 0)
        {
            remainder += ((-1)*negBase);
            n += 1;
        }
 
        // convert remainder to String add into the result
        converted = remainder.toString() + converted;
 
    }
    return converted;
}
 
// Driver Code
 
let n = 13;
let negBase = -2;
 
document.write(toNegativeBase(n, negBase),"</br>");
 
// This code is contributed by shinjanpatra
 
</script>

Output:

Time Complexity: O(N)
Auxiliary Space: O(1)
Reference :
https://en.wikipedia.org/wiki/Negative_base

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Convert a number into negative base representation

C++

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Python3

C#

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