Can’t sleep: My mind is wandering about in my skull, a kitty among the bins. I can’t make up my mind how to represent an atomic algebraic expression term.

Clearly, an expression term, and an elementary binary operator are two different kinds of objects. What do I make, then, of an expression term such as *sin(x)*? It’s a function operator taking a single argument. I’d like to be able to manipulate a pair of rational expressions like

[latex]1/(sin^2(x) – cos^2(x))[/latex]

How about *2x*? It’s an elementary product expression consisting of a constant and a variable. I’m beginning to think each subexpression ought to be represented as an object, now – so that I could replace the factor 2 with an expression, for example. This pretty much puts paid to representing an algebraic polynomial term as a struct:

struct Monomial { public: integer coeff; Variable x; Exponent t; }

Each member *coeff, x, *and *t* would then be an Expr:

struct Monomial { protected: Expr coeff; Expr x; Expr t; }

Then I should represent elementary *operations* as functors as well, perhaps. Hmm. This eliminates the need for a Monomial type:

Expr x('x'), X; X = 2 * x ^ 3; cout << X.strict().toString() << endl; // 2*(x^3) cout << X.toHumanString() << endl; // 2x3 cout << X.toMathML() << endl; // Something like 2x(super)3

I need to look into operator associativity rules and find out whether any combination of member and nonmember functions will cause type coercion of the RHS into an Expr type.

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My goal is to be able to create simple code to generate elementary algebra expressions programmatically, like so:

Expr a('a', '1/(x+2)'); Expr b('b', '2/(x+7)'); Expr c; c = a * b; cout << c.toMathML() << endl; cout << c.execute().toMathML() << endl; // 1/(x^2 + 9x + 14)

Then a simple, short chunk of code could do and generate an online algebra workbook of arbitrary complexity:

string generate_rational_fraction_sum( int terms, int glbpower, int lubpower ) { Expr rfs; Expr numerator; Expr denominator; Expr monomial('x'); for ( int i = 0 ; i < terms ; i++ ) { integer r = rand(glbpower, lubpower); integer q = rand(1,10); denominator.clear(); for ( int j = 0 ; j < r ; j++ ) { // Generate an order-j polynomial expression for the denominator integer k = rand(1,10); denominator += k * monomial ^ j; } numerator = rand(1,10); rfs += numerator / denominator; } return rfs.execute().toMathML(); }