Given a decimal number and a base, convert the number to n-ary base number.

n-ary-number-system.php

<?php
/**
 * Given a decimal number and a base, convert the number to n-ary base number. Also find a way to perform arithmetic
 * operations. The n is actually 36, however the principal is same.
 *
 * To perform arithmetic operations, we convert the number back to decimal, do the math, then convert back to the
 * original base.
 */

/**
 * Convert given number to the base.
 * 
 * @param int $number Decimal number
 * @param int $base which base, max allowed is 36, 10 + 26 alphabets
 * @return string resulting number, could be hex so return type is string
 */
function convert(int $number, int $base): string
{
    if ($base > 36) {//Can't do
        return '💻 LOAD LETTER' . PHP_EOL;
    }
    //http://php.net/ord, Rather than maintaining a list of chars, just using in built functions to get Alphabets.
    $offset = ord('A') - 1;
    $result = [];
    while ($number >= $base) {
        if (($number % $base) === 0) {
            $result[] = '0';
            $number   = ($number / $base); // 32 / 2 -> 16
        } else {
            $result [] = ($number % $base); // 17 % 16 -> 1
            $number    = intval($number / $base); // 17 / 16 -> 1
        }
    }
    if ($number < $base) {
        $result[] = $number;
    }
    if ($base > 9) { //15
        for ($i = 0; $i < count($result); $i++) {
            if ($result[$i] > 9) {
                //http://php.net/chr, get the corresponding alphabet.
                $result[$i] = chr(($offset + ($result[$i] - 9)));
            }
        }
    }
    //reverse the array
    $reversed = array_reverse($result);
    return implode("", $reversed);
}

/**
 * Convert the given number in the base to decimal representation.
 * 
 * @param string $number can be hex number so using string type
 * @param int $base which base, max allowed is 36, 10 + 26 alphabets
 * @return string resulting decimal number
 */
function convertToDecimal(string $number, int $base): string
{
    if ($base > 36) {
        return '💻 LOAD LETTER' . PHP_EOL;
    }
    $reversed = strrev($number);
    $offset   = ord('A') - 1 + 9;
    $result   = 0;
    for ($i = 0; $i < strlen($reversed); $i++) {
        if (ctype_alpha($reversed[$i])) {
            $digit = ord($reversed[$i]) - $offset;
        } else {
            $digit = $reversed[$i];
        }
        $result = $result + ($base ** $i) * $digit; // 16 + 64 + 256
    }
    return ($result);
}

/**
 * Add
 * 
 * @param string $number1
 * @param string $number2
 * @param int $base
 * @return string
 */
function add(string $number1, string $number2, int $base): string
{
    return convert((convertToDecimal($number1, $base) + convertToDecimal($number2, $base)), $base);
}

/**
 * Subtract
 *
 * @param string $number1
 * @param string $number2
 * @param int $base
 * @return string
 */
function subtract(string $number1, string $number2, int $base): string
{
    return convert((convertToDecimal($number1, $base) - convertToDecimal($number2, $base)), $base);
}

/**
 * Multiply.
 * 
 * @param string $number1
 * @param string $number2
 * @param int $base
 * @return string
 */
function multiply(string $number1, string $number2, int $base): string
{
    return convert((convertToDecimal($number1, $base) * convertToDecimal($number2, $base)), $base);
}

/**
 * Divide.
 * 
 * @param string $number1
 * @param string $number2
 * @param int $base
 * @return string
 */
function divide(string $number1, string $number2, int $base): string
{
    return convert((convertToDecimal($number1, $base) / convertToDecimal($number2, $base)), $base);
}
//Let us test
$testcases = [
    [3, 2],
    [4, 2],
    [5, 3],
    [6, 3],
    [35, 4],
    [38, 5],
    [3, 6],
    [21, 7],
    [22, 7],
    [3, 8],
    [32, 8],
    [33, 8],
    [3, 10],
    [30, 10],
    [31, 10],
    [32, 12],
    [36, 12],
    [16, 16],
    [331, 2],
    [332, 3],
    [333, 16],
    [336, 16],
    [336, 36],
    [336, 37],
];
foreach ($testcases as $testcase) {
    $converted      = convert($testcase[0], $testcase[1]);
    $reverted       = convertToDecimal($converted, $testcase[1]);
    $base_converted = strtoupper(base_convert($testcase[0], 10, $testcase[1]));
    echo 'Decimal: ' . $testcase[0] . ' Base: ' . $testcase[1] . ' Converted: ' . $converted . ' == ' . $base_converted . '? ' . ($converted === $base_converted ? '✓' : '✘') . PHP_EOL;
}