Can't access protected member in base class fr

Heres my code :

#include <iostream>
#include <cmath>
#include <sstream>
using namespace std;

class root
{
    protected :

            int size;
            double *array;

    public :

        virtual ~root() {}
        virtual root* add(const root&) = 0;
        virtual root* sub(const root&) = 0;
        virtual istream& in(istream&, root&) = 0;
        virtual int getSize() const = 0;
        virtual void setSize(int);
};

class aa: public root
{

    public :

        aa();
        aa(int);
        aa(const aa&);
        root* add(const root& a);
        root* sub(const root& a);
        istream& in(istream&, root&){}
        int getSize() const;
        void setSize(int);
};

class bb: public root
{
public:
    bb() { }
    bb(const bb& b) { }
    root* add(const root& a);
    root* sub(const root& a);
    istream& in(istream&, root&){}
    int getSize() const{}
    void setSize(int){}
};

aa::aa()
{
    size = 0;
    array = NULL;
}

aa::aa(int nsize)
{
    size = nsize;
    array = new double[size+1];
    for(int i=0; i<size; i++)
        array[i] = 0;
}

root* aa::add(const root& a)
{
    for (int i=0; i<a.size; i++)
        array[i] += a.array[i];
    return *this;
}

root* aa::sub(const root& a)
{
}

int aa::getSize() const
{
    return size;
}

void aa::setSize(int nsize)
{
    size = nsize;
    array = new double[size+1];
    for(int i=0; i<size; i++)
        array[i] = 0;
}

root* bb::add(const root& a)
{
    return new bb();
}

root* bb::sub(const root& a)
{

}

int main(int argc, char **argv)
{
}

When I want to access size or an array in derived class, I just cant because my compiler says:

/home/brian/Desktop/Temp/Untitled2.cpp||In member function ‘virtual root* aa::add(const root&)’:|
/home/brian/Desktop/Temp/Untitled2.cpp|10|error: ‘int root::size’ is protected|
/home/brian/Desktop/Temp/Untitled2.cpp|66|error: within this context|
/home/brian/Desktop/Temp/Untitled2.cpp|11|error: ‘double* root::array’ is protected|
/home/brian/Desktop/Temp/Untitled2.cpp|67|error: within this context|
/home/brian/Desktop/Temp/Untitled2.cpp|68|error: cannot convert ‘aa’ to ‘root*’ in return|
||=== Build finished: 5 errors, 0 warnings ===|

I read that protected members are private in derived classes, so it seems to be ok, but it isnt. How to fix this?

I read that protected members are private in derived classes, so it seems to be ok, but it isnt.

It is not because a protected data member inherited from a base class A (root, in this case) by a derived class B (aa, in this case) is accessible as long as it is being accessed on an object of type B (aa). Here, you are accessing it through an object of type A (root):

root* aa::add(const root& a)
{
    for (int i=0; i<a.size; i++)
    //              ^^^^^^
    //              Accessing size on an object of type `root`, not `aa`!
        array[i] += a.array[i];
    return *this;
}

Per paragraph 11.4/1 of the C++11 Standard:

An additional access check beyond those described earlier in Clause 11 is applied when a non-static data member or non-static member function is a protected member of its naming class (11.2). As described earlier, access to a protected member is granted because the reference occurs in a friend or member of some class C. If the access is to form a pointer to member (5.3.1), the nested-name-specifier shall denote C or a class derived from C. All other accesses involve a (possibly implicit) object expression (5.2.5). In this case, the class of the object expression shall be C or a class derived from C. [Example:
class B {
protected:
    int i;
    static int j;
};
class D1 : public B {
};
class D2 : public B {
    // ...
    void mem(B*,D1*);
};

void D2::mem(B* pb, D1* p1) {
    pb->i = 1; // ill-formed
    p1->i = 2; // ill-formed
    // ...
    i = 3; // OK (access through this)
    B::i = 4; // OK (access through this, qualification ignored)
    j = 5; // OK (because j refers to static member)
    B::j = 6; // OK (because B::j refers to static member)
}
— end example ]

To fix this, you need to provide public setters/getters. You already have a getSize() function, so instead of writing this:

for (int i=0; i<a.size; i++)
//              ^^^^^^

You could write this:

for (int i=0; i<a.getSize(); i++)
//              ^^^^^^^^^^^

Similarly, you will have to provide functions for getting/setting the value of the n-th element of array, so that you could write:

array[i] += a.get_at(i);

Notice, that the expression on the left side of += is OK, because array is being accessed through this (see also the above example from the C++11 Standard).