Adding Two Large Numbers Using Strings

Question

I'm wanting to add together two really large numbers using C#, without the use of libraries such as BigInterger. I'm looking at using two strings of numbers and attempting to carry out almost a primary school way of addition by carrying numbers and so on. My code is atrocious and I think that the reason it doesn't work is because I'm not quite understanding what I need to be doing?

string firstNumber = "1769";
string secondNumber = "723";
string product = firstNumber;

int carry = 0;
int positionFirst;
int positionSecond;

for (positionFirst = firstNumber.Length; positionFirst >= 1; positionFirst--)
{
    for (positionSecond = secondNumber.Length; positionSecond >= 1; positionSecond--)
    {
        string currentDigitFirst = firstNumber.Substring(positionFirst - 1, 1);
        string currentDigitSecond = secondNumber.Substring(positionSecond - 1, 1);

        int intDigitFirst = int.Parse(currentDigitFirst);
        int intDigitSecond = int.Parse(currentDigitSecond);

        int currentDigitProduct = intDigitFirst + intDigitSecond;

        if (carry != 0)
        {
            currentDigitProduct += carry;
            carry /= 10;
        }
        int currentCarry = currentDigitProduct / 10;
        carry += currentCarry;
        int numberReplace = currentDigitProduct % 10;

        product = product.Remove(positionFirst - 1, 1);
        product = product.Insert(positionFirst - 1, numberReplace.ToString());

        break;
    }
}