平衡三进制数系统

众所周知，二进制数系统是其中只有2位数字(即0和1 )的数字系统。同样，我们也知道，三进制数系统是其中只有3位数字(即0、1和2 )的数字系统。在本文中，我们将学习平衡三进制数系统。
平衡三进制数系统是一种包括数字-1、0和1的数字系统。由于将-1写入数字很不方便，因此我们将进一步使用字母Z。
十进制到平衡三进制系统的转换
从十进制到平衡三进制的转换分两步完成：

1. 将十进制转换为三进制数字系统。
2. 使用以下步骤将三元转换为平衡三元系统：
   • 通过保留0和1来从右到左遍历三进制数
   • 遇到2时，将其更改为Z，然后在迭代中的下一位数字上加上+1。
   • 某些数字可能变为+3，然后将+3替换为0，并在迭代中将+1添加到下一个数字。
   • 完成此过程，直到您转换所有数字。

示例：将238 ₁₀转换为平衡三元，反之亦然

First convert 238₁₀ to ternary number system. 
238₁₀ = 22211₃ 
Second convert ternary to balanced ternary number system : 
 

• By starting iteration from left to right, two 1’s are skipped as it remains same in balanced ternary.
• Now convert first encountered 2 with z increasing it’s next digit in iteration by +1. So we get 23Z11.
• Convert 3 to 0 with increment +1 in it’s next digit in iteration. So we get 30Z11.
• Convert 3 to 0 with increment +1 in it’s next digit in iteration. So we get 100Z11. (Here assume 0 is present before most significant digit)

The final result is 100Z11.

笔记：-
该系统还允许表示负数，从而消除了在数字之前使用负号的需要。三元平衡系统中的所有负数均以Z开头。
IE：
-1 ₁₀ = Z ₃ ，
−2 ₁₀ = Z1 ₃ ，
−3 ₁₀ = Z0 ₃ ，
−4 ₁₀ = ZZ ₃ ，
−5 ₁₀ = Z11 ₃ 。

下面是将正小数转换为平衡三进制的程序：

C++

// C++ program to convert positive
// decimals into balanced ternary system
 
#include 
using namespace std;
 
string balancedTernary(int n)
{
    string output = "";
    while (n > 0) {
        int rem = n % 3;
        n = n / 3;
        if (rem == 2) {
            rem = -1;
            n++;
        }
        output = (rem == 0
                      ? '0'
                      : (rem == 1)
                            ? '1'
                            : 'Z')
                 + output;
    }
    return output;
}
 
// Driver code
int main()
{
 
    int n = 238;
 
    cout << "Equivalent Balanced Ternary of "
         << n << " is: "
         << balancedTernary(n);
 
    return 0;
}

Java

// Java program to convert positive
// decimals into balanced ternary system
class GFG{
 
static String balancedTernary(int n)
{
    String output = "";
    while (n > 0)
    {
        int rem = n % 3;
        n = n / 3;
        if (rem == 2)
        {
            rem = -1;
            n++;
        }
        output = (rem == 0 ? '0' :
                 (rem == 1) ? '1' : 'Z') + output;
    }
    return output;
}
 
// Driver code
public static void main(String[] args)
{
    int n = 238;
 
    System.out.print("Equivalent Balanced Ternary of " +
                      n + " is: " + balancedTernary(n));
}
}
 
// This code is contributed by Rajput-Ji

Python3

# Python3 program to convert positive
# decimals into balanced ternary system
def balancedTernary(n):
 
    output = ""
     
    while(n > 0):
        rem = n % 3
        n = n // 3
         
        if(rem == 2):
            rem = -1
            n += 1
 
        if(rem == 0):
            output = '0' + output
        else:
            if(rem == 1):
                output = '1' + output
            else:
                output = 'Z' + output
 
    return output
 
# Driver Code
n = 238
 
# Function call
print("Equivalent Balanced Ternary of",
       n , "is:", balancedTernary(n))
 
# This code is contributed by Shivam Singh

C#

// C# program to convert positive
// decimals into balanced ternary system
using System;
using System.Collections.Generic;
class GFG{
 
static String balancedTernary(int n)
{
    String output = "";
    while (n > 0)
    {
        int rem = n % 3;
        n = n / 3;
        if (rem == 2)
        {
            rem = -1;
            n++;
        }
        output = (rem == 0 ? '0' :
                 (rem == 1) ? '1' : 'Z') + output;
    }
    return output;
}
 
// Driver code
public static void Main(String[] args)
{
    int n = 238;
 
    Console.Write("Equivalent Balanced Ternary of " +
                   n + " is: " + balancedTernary(n));
}
}
 
// This code is contributed by Rajput-Ji

输出：

Equivalent Balanced Ternary of 238 is: 100Z11