Ternary number system or Base 3 numbers

18 Nov, 2021
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: 
Divide the number by 3.Get the integer quotient for the next iteration.Get the remainder for the ternary digit.Repeat the steps until the quotient is equal to 0.
Divide the number by 3.
Get the integer quotient for the next iteration.
Get the remainder for the ternary digit.
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++
Java
Python3
C#
Javascript
// 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 numbervoid 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 ternaryvoid 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 codeint main(){    int Decimal = 2747;     convert(Decimal);     return 0;}
// Java program to convert decimal// number to ternary numberimport java.io.*;  class GFG{ // Function to convert a decimal// number to a ternary numberstatic 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 ternarystatic 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 Codepublic static void main (String[] args){    int Decimal = 2747;     convert(Decimal); }} // This code is contributed by shivanisinghss2110
# Python3 program to convert decimal# number to ternary number # Function to convert a decimal# number to a ternary numberdef 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 ternarydef 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 Codeif __name__ == '__main__':         Decimal = 2747;     convert(Decimal); # This code is contributed by Rajput-Ji
// C# program to convert decimal// number to ternary numberusing 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
<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>
Ternary number of 2747 is: 10202202
 
Time Complexity: O(log3N)
Auxiliary Space: O(log3N)
Steps to Convert Ternary to Decimal:
Connect each digit from the ternary number with its corresponding power of three.Multiply each digit with its corresponding power of three and Add up all of the numbers you got.
Connect each digit from the ternary number with its corresponding power of three.
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++
Java
Python3
C#
Javascript
// 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 numbervoid 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 codeint main(){    int Ternary = 10202202;     convertToDecimal(Ternary);     return 0;}
// Java program to convert a// ternary number to decimal numberclass GFG{ // Function to convert a ternary// number to a decimal numberstatic 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 codepublic static void main(String[] args){    int Ternary = 10202202;    convertToDecimal(Ternary);}} // This code is contributed by Rajput-Ji
# Python3 program to convert a# ternary number to decimal numberimport math; # Function to convert a ternary# number to a decimal numberdef 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 codeTernary = 10202202;convertToDecimal(Ternary); # This code is contributed by Code_Mech
// C# program to convert a ternary// number to decimal numberusing System; class GFG{ // Function to convert a ternary// number to a decimal numberstatic 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 codepublic static void Main(){    int Ternary = 10202202;         convertToDecimal(Ternary);}} // This code is contributed by shivanisinghss2110
<script> // Javascript program to convert a ternary// number to decimal number // Function to convert a ternary// number to a decimal numberfunction 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 codelet 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)
shivanisinghss2110
ankthon
Rajput-Ji
Code_Mech
divyeshrabadiya07
divyesh072019
subham348
base-conversion
number-theory
Mathematical
number-theory
Mathematical
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