Convreting numbers between different numeral systems is a common task in programming. This involves three primary scenarios: converting from an arbitrary base to decimal, from decimal to another base, and directly between two non-decimal bases.

Converting from Base-X to Decimal

Each digit in a number represents a coefficient multiplied by the base raised to the power of its position (starting from 0 at the least significant digit). For example:

• Binary 1011 equals $1 \cdot 2^3 + 0 \cdot 2^2 + 1 \cdot 2^1 + 1 \cdot 2^0 = 11$ in decimal.
• Hexadecimal 2C equals $2 \cdot 16^1 + 12 \cdot 16^0 = 44$ in decimal.

The following function implements this logic:

#include <iostream>
#include <string>
using namespace std;

int baseToDecimal(const string& num, int base) {
    int result = 0;
    int power = 1;
    for (int i = num.size() - 1; i >= 0; --i) {
        char c = num[i];
        int digit = (c >= '0' && c <= '9') ? c - '0' : c - 'A' + 10;
        result += digit * power;
        power *= base;
    }
    return result;
}

Converting from Decimal to Base-X

This uses repeated division by the target base, collecting remainders. Digits greater than 9 are represented using uppercase letters (A–Z). A lookup string simplifies digit-to-character mapping:

#include <iostream>
#include <string>
using namespace std;

const string DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ";

string decimalToBase(int value, int base) {
    if (value == 0) return "0";
    string reversed = "";
    while (value > 0) {
        reversed += DIGITS[value % base];
        value /= base;
    }
    string result = "";
    for (int i = reversed.size() - 1; i >= 0; --i) {
        result += reversed[i];
    }
    return result;
}

Converting from Base-X to Base-K

This combines the two previous steps: first convert the input from base-X to decimal, then from decimal to base-K.

#include <iostream>
#include <string>
using namespace std;

const string DIGITS = "0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ";

int baseToDecimal(const string& num, int base) {
    int result = 0;
    int power = 1;
    for (int i = num.size() - 1; i >= 0; --i) {
        char c = num[i];
        int digit = (c >= '0' && c <= '9') ? c - '0' : c - 'A' + 10;
        result += digit * power;
        power *= base;
    }
    return result;
}

string decimalToBase(int value, int base) {
    if (value == 0) return "0";
    string reversed = "";
    while (value > 0) {
        reversed += DIGITS[value % base];
        value /= base;
    }
    string result = "";
    for (int i = reversed.size() - 1; i >= 0; --i) {
        result += reversed[i];
    }
    return result;
}

int main() {
    string input;
    int sourceBase, targetBase;
    cin >> input >> sourceBase >> targetBase;
    int decimalValue = baseToDecimal(input, sourceBase);
    cout << decimalToBase(decimalValue, targetBase) << endl;
    return 0;
}