Check if integer is a palindrome in base 10
Too many comments, and they do not serve any purpose.

Use all necessary #includes. atoi requires #include &lt;stdlib.h&gt;.

Separate IO from the business logic. Constructing the reversed number definitely belongs to a function.

There is no need to convert the number into a textual representation. Consider building the reversed number as you compute the digits of the original:
int reverse (int number) 
{
    int reversed = 0;
    while (number != 0) {
        int last_digit = number % 10;
        number /= 10;
        reversed *= 10;
        reversed += last_digit;
    }
    return reversed;
}

As a side effect, many variables (quotient, remainder, number_update, i) disappear.