Ternary number system or Base 3 numbers

A number system can be considered as a mathematical notation of numbers using a set of digits or symbols. In simpler words, the number system is a method of representing numbers. Every number system is identified with the help of its base or radix. 

For example, Binary, Octal, Decimal and Hexadecimal Number systems are used in microprocessor programming. In this article, one such number system is discussed.

Ternary Number System:
If the Base value of a number system is 3, then this representation is known as a ternary representation. The digits in this system are 0, 1, and 2. 
There is also a number system called Balanced Ternary which comprises the digits −1, 0 and +1. The Ternary representation of a number is compact than of binary number system. 

Steps to Convert Decimal to Ternary:

1. Divide the number by 3.
2. Get the integer quotient for the next iteration.
3. Get the remainder for the ternary digit.
4. Repeat the steps until the quotient is equal to 0.

For example: let N = 101. The following image illustrates the step by step conversion of 101¹⁰ to the base-3. 

Below is the implementation for the Decimal to Binary and Vice-Versa: 

C++

// C++ program to convert decimal
// number to ternary number

#include <cstdio>
#include <iostream>
#include <math.h>
using namespace std;

// Function to convert a decimal
// number to a ternary number
void convertToTernary(int N)
{
    // Base case
    if (N == 0)
        return;

    // Finding the remainder
    // when N is divided by 3
    int x = N % 3;
    N /= 3;
    if (x < 0)
        N += 1;

    // Recursive function to
    // call the function for
    // the integer division
    // of the value N/3
    convertToTernary(N);

    // Handling the negative cases
    if (x < 0)
        cout << x + (3 * -1);
    else
        cout << x;
}

// Function to convert the decimal to ternary
void convert(int Decimal)
{
    cout << "Ternary number of "
         << Decimal << " is: ";

    // If the number is greater
    // than 0, compute the
    // ternary representation
    // of the number
    if (Decimal != 0) {
        convertToTernary(Decimal);
    }
    else
        cout << "0" << endl;
}

// Driver code
int main()
{
    int Decimal = 2747;

    convert(Decimal);

    return 0;
}

Java

// Java program to convert decimal
// number to ternary number
import java.io.*;

class GFG
{

// Function to convert a decimal
// number to a ternary number
static void convertToTernary(int N)
{
    // Base case
    if (N == 0)
        return;

    // Finding the remainder
    // when N is divided by 3
    int x = N % 3;
    N /= 3;
    if (x < 0)
        N += 1;

    // Recursive function to
    // call the function for
    // the integer division
    // of the value N/3
    convertToTernary(N);

    // Handling the negative cases
    if (x < 0)
        System.out.print( x + (3 * -1));
    else
        System.out.print( x);
}

// Function to convert the decimal to ternary
static void convert(int Decimal)
{
    System.out.print("Ternary number of "  +Decimal +" is: ");

    // If the number is greater
    // than 0, compute the
    // ternary representation
    // of the number
    if (Decimal != 0) {
        convertToTernary(Decimal);
    }
    else
        System.out.println("0" );
}

// Driver Code
public static void main (String[] args)
{
    int Decimal = 2747;

    convert(Decimal);

}
}

// This code is contributed by shivanisinghss2110

Python3

# Python3 program to convert decimal
# number to ternary number

# Function to convert a decimal
# number to a ternary number
def convertToTernary(N):

    # Base case
    if (N == 0):
        return;

    # Finding the remainder
    # when N is divided by 3
    x = N % 3;
    N //= 3;
    if (x < 0):
        N += 1;

    # Recursive function to
    # call the function for
    # the integer division
    # of the value N/3
    convertToTernary(N);

    # Handling the negative cases
    if (x < 0):
        print(x + (3 * -1), end = "");
    else:
        print(x, end = "");


# Function to convert the
# decimal to ternary
def convert(Decimal):

    print("Ternary number of ", Decimal,
          " is: ", end = "");

    # If the number is greater
    # than 0, compute the
    # ternary representation
    # of the number
    if (Decimal != 0):
        convertToTernary(Decimal);
    else:
        print("0", end = "");

# Driver Code
if __name__ == '__main__':

    Decimal = 2747;

    convert(Decimal);

# This code is contributed by Rajput-Ji

C#

// C# program to convert decimal
// number to ternary number
using System;

class GFG
{

    // Function to convert a decimal
    // number to a ternary number
    static void convertToTernary(int N)
    {
        // Base case
        if (N == 0)
            return;

        // Finding the remainder
        // when N is divided by 3
        int x = N % 3;
        N /= 3;
        if (x < 0)
            N += 1;

        // Recursive function to
        // call the function for
        // the integer division
        // of the value N/3
        convertToTernary(N);

        // Handling the negative cases
        if (x < 0)
            Console.Write( x + (3 * -1));
        else
            Console.Write( x);
    }

    // Function to convert the decimal to ternary
    static void convert(int Decimal)
    {
        Console.Write("Ternary number of " +Decimal +" is: ");

        // If the number is greater
        // than 0, compute the
        // ternary representation
        // of the number
        if (Decimal != 0) {
            convertToTernary(Decimal);
        }
        else
            Console.WriteLine("0" );
    }

    // Driver Code
    public static void Main (string[] args)
    {
        int Decimal = 2747;

        convert(Decimal);

    }
}

// This code is contributed by AnkitRai01

Javascript

<script>
    // Javascript program to convert decimal
    // number to ternary number

    // Function to convert a decimal
    // number to a ternary number
    function convertToTernary(N)
    {
        // Base case
        if (N == 0)
            return;

        // Finding the remainder
        // when N is divided by 3
        let x = N % 3;
        N = parseInt(N / 3, 10);
        if (x < 0)
            N += 1;

        // Recursive function to
        // call the function for
        // the integer division
        // of the value N/3
        convertToTernary(N);

        // Handling the negative cases
        if (x < 0)
            document.write(x + (3 * -1));
        else
            document.write(x);
    }

    // Function to convert the decimal to ternary
    function convert(Decimal)
    {
        document.write("Ternary number of " + Decimal + " is: ");

        // If the number is greater
        // than 0, compute the
        // ternary representation
        // of the number
        if (Decimal != 0) {
            convertToTernary(Decimal);
        }
        else
            document.write("0" + "</br>");
    }

    let Decimal = 2747;

    convert(Decimal);

// This code is contributed by divyeshrabadiya07.
</script>

Output:

Ternary number of 2747 is: 10202202

Time Complexity: O(log₃N)

Auxiliary Space: O(log₃N)

Steps to Convert Ternary to Decimal:

1. Connect each digit from the ternary number with its corresponding power of three.
2. Multiply each digit with its corresponding power of three and Add up all of the numbers you got.

For example: let N = 10202. The following image illustrates the step by step conversion of 10202₃ to the base-10. 

Below is the implementation for the Decimal to Binary and Vice-Versa: 

C++

// C++ program to convert a
// ternary number to decimal number

#include <cstdio>
#include <iostream>
#include <math.h>
using namespace std;

// Function to convert a ternary
// number to a decimal number
void convertToDecimal(int N)
{
    cout << "Decimal number of "
         << N << " is: ";

    // If the number is greater than 0,
    // compute the decimal
    // representation of the number
    if (N != 0) {

        int decimalNumber = 0,
            i = 0, remainder;

        // Loop to iterate through
        // the number
        while (N != 0) {
            remainder = N % 10;
            N /= 10;

            // Computing the decimal digit
            decimalNumber += remainder
                             * pow(3, i);
            ++i;
        }
        cout << decimalNumber << endl;
    }
    else
        cout << "0" << endl;
}

// Driver code
int main()
{
    int Ternary = 10202202;

    convertToDecimal(Ternary);

    return 0;
}

Java

// Java program to convert a
// ternary number to decimal number
class GFG{

// Function to convert a ternary
// number to a decimal number
static void convertToDecimal(int N)
{
    System.out.print("Decimal number of " +
                      N + " is: ");

    // If the number is greater than 0,
    // compute the decimal
    // representation of the number
    if (N != 0)
    {
        int decimalNumber = 0,
            i = 0, remainder;

        // Loop to iterate through
        // the number
        while (N != 0)
        {
            remainder = N % 10;
            N /= 10;

            // Computing the decimal digit
            decimalNumber += remainder *
                             Math.pow(3, i);
            ++i;
        }
        System.out.print(decimalNumber + "\n");
    }
    else
        System.out.print("0" + "\n");
}

// Driver code
public static void main(String[] args)
{
    int Ternary = 10202202;
    convertToDecimal(Ternary);
}
}

// This code is contributed by Rajput-Ji

Python3

# Python3 program to convert a
# ternary number to decimal number
import math;

# Function to convert a ternary
# number to a decimal number
def convertToDecimal(N):

    print("Decimal number of", N, "is:", end = " ");

    # If the number is greater than 0,
    # compute the decimal
    # representation of the number
    if (N != 0):

        decimalNumber = 0;
        i = 0;
        remainder = 0;

        # Loop to iterate through
        # the number
        while (N != 0):
            remainder = N % 10;
            N = N // 10;

            # Computing the decimal digit
            decimalNumber += remainder * math.pow(3, i);
            i += 1;

        print(decimalNumber);

    else:
        print("0");

# Driver code
Ternary = 10202202;
convertToDecimal(Ternary);

# This code is contributed by Code_Mech

C#

// C# program to convert a ternary
// number to decimal number
using System;

class GFG{

// Function to convert a ternary
// number to a decimal number
static void convertToDecimal(int N)
{
    Console.Write("Decimal number of " +
                           N + " is: ");

    // If the number is greater than
    // 0, compute the decimal
    // representation of the number
    if (N != 0)
    {
        int decimalNumber = 0;
        int i = 0, remainder;

        // Loop to iterate through
        // the number
        while (N != 0)
        {
            remainder = N % 10;
            N /= 10;

            // Computing the decimal digit
            decimalNumber += remainder *
                             (int)Math.Pow(3, i);
            ++i;
        }
        Console.Write(decimalNumber + "\n");
    }
    else
        Console.Write("0" + "\n");
}

// Driver code
public static void Main()
{
    int Ternary = 10202202;

    convertToDecimal(Ternary);
}
}

// This code is contributed by shivanisinghss2110

Javascript

<script>

// Javascript program to convert a ternary
// number to decimal number

// Function to convert a ternary
// number to a decimal number
function convertToDecimal(N)
{
    document.write("Decimal number of " +
                   N + " is: ");

    // If the number is greater than
    // 0, compute the decimal
    // representation of the number
    if (N != 0)
    {
        let decimalNumber = 0;
        let i = 0, remainder;

        // Loop to iterate through
        // the number
        while (N != 0)
        {
            remainder = N % 10;
            N = parseInt(N / 10, 10);

            // Computing the decimal digit
            decimalNumber += remainder * Math.pow(3, i);
            ++i;
        }
        document.write(decimalNumber + "</br>");
    }
    else
        document.write("0" + "</br>");
}

// Driver code
let Ternary = 10202202;

convertToDecimal(Ternary);

// This code is contributed by divyesh072019

</script>

Output:

Decimal number of 10202202 is: 2747

Time Complexity: O(log₁₀N)

Auxiliary Space: O(1)