Implement BigInteger division

I create a BigInteger class using an std::string. The interface is as follows:

class BigInteger
{
private:
    char sign;
    std::string digits;
    const std::size_t base = 10;
    short toDigit(int index) const {return index >= 0 && index < digits.size() ? digits[index] - '0' : 0;}

public:
    BigInteger();
    BigInteger(int value);
    BigInteger(int64_t value);
    BigInteger(const std::string &value);
    BigInteger(const BigInteger &other);

    inline bool isNegative() const {return sign == '-';}
    inline bool isPositive() const {return sign == '+';}
    inline bool isNeutral() const {return sign == '~';}

    bool operator < (const BigInteger &other) const;
    bool operator > (const BigInteger &other) const;
    bool operator <= (const BigInteger &other) const;
    bool operator >= (const BigInteger &other) const;
    bool operator == (const BigInteger &other) const;
    bool operator != (const BigInteger &other) const;

    BigInteger& operator = (const BigInteger &other);
    BigInteger& operator += (const BigInteger &other);
    BigInteger& operator -= (const BigInteger &other);
    BigInteger& operator *= (const BigInteger &other);
    BigInteger& operator /= (const BigInteger &other);

    BigInteger operator + (const BigInteger &other) const;
    BigInteger operator - (const BigInteger &other) const;
    BigInteger operator * (const BigInteger &other) const;
    BigInteger operator / (const BigInteger &other) const;

    BigInteger& divide(const BigInteger &D, BigInteger &R);

    friend std::ostream& operator << (std::ostream& os, const BigInteger& other);
};

Each character in the string, represents ONE digit.

I implemented ALL operators except the divide function and the division operator.

That's where I'm stuck! I was writing all my algorithms based off of Wikipedia explanations and came across the division algorithm that I like:

http://ift.tt/1KmVZKv

I like this one because it gives me the quotient and remainder and that will make implementing modulo easier.

However, I don't have a shift operator and I'm not sure how to set the least significant bit of R to the bit in the numerator from my string. Any ideas how I can implement my divide function?

I tried:

BigInteger& BigInteger::divide(const BigInteger &D, BigInteger &R)
{
    if (D.isNeutral()) //if D == 0.
    {
        throw std::overflow_error("Division By Zero Exception.");
    }

    R = 0;
    BigInteger Q = 0;

    for (std::size_t I = 0; I < digits.size(); ++I)
    {
        R *= 2; //I believe this is the same as shifting left by 1.
        R.digits[0] = digits[I];
        if (R >= D)
        {
            R -= D;
            Q.digits[I] = '1';
        }
    }
    *this = Q;
    return *this;
}

Digits are stored backwards in the string. For example:

value = 1024 digits = 4201

Any ideas what I need to do to get the BitShifting working and to get the division working? The division is more important to me.