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In a rigorous setting, it is not acceptable to simply assume that there exists a number whose square is −1. There are various ways of defining C, building on the knowledge of real numbers. Firstly, write C for R2, the set of ordered pairs of real numbers, and define operations on complex numbers in C according to
(a, b) + (c, d) = (a + c, b + d)
(a, b)·(c, d) = (a·c − b·d, b·c + a·d)
It is then just a matter of notation to express (a, b) as a + ib. This means we can associate the numbers (a, 0) with the real numbers, and write i = (0, 1). Since (0, 1)·(0, 1) = (−1, 0), we have found i by constructing it, not postulating it. Using these formal operations on R2, it is easy to check that we satisfy the field axioms (associativity, commutativity, identity, inverses, distributivity). In particular, R is a subfield of C.
Though this low-level construction does accurately describe the structure of the complex numbers, the definitions seem arbitrary, so secondly C can be considered algebraically. In algebra (the theory of group-like structures), this explicit definition of operations in fact turns out to be the mechanism behind the idea of constructing the algebraic closure of the reals, that is, adding in some elements to R to make a new field, of which R is a subfield, where every non-constant polynomial has a root. Finally, yet another way of characterising C is in terms of its topological properties. Details of these are given below.