It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the

It sounds to me like you are giving special precedence to "numbers" in your considerations, as well as drawing your intuition for "square" from working with the integers or the reals. A square is just the result of a binary (product) operation where both of the operands are equal. The operation can be defined on an algebraic object with much richer structure than the integers or reals, and the operation itself can be much more complex than, say, integer multiplication. So the geometric product of geometric algebra is just one animal in the zoo of examples. You might call the specifics of the geometric product operation an "invention," but not because the square of a non-number can be a number.