# Real 1 not equal to complex 1 [duplicate]

We have the set-theoretic definition for pairs as: (x , y) = {{x}, {x, y}} Also we have the definition: complex 1 = (1, 0) So if real 1 = complex 1 we would have: 1 = (1, 0) = {{1}, {1, 0}} Which seems paradoxical. Am I missing a point?

• In the complex "space" the real $1$ is the complex $(1,0)$. Jan 31, 2020 at 13:32
• Nice question, +1. Ultimately I think the point you are 'missing' is that in reality no-one thinks about numbers (or other mathematical objects) in terms of their set-theoretic definitions. But of course this is not an answer to your question in the strict sense. Jan 31, 2020 at 13:36
• If we think in terms of sets, we have that the set of complex numbers has a subset that "looks like" the set of real, that is not exactly the same as considering the reals a subset of the complex. Jan 31, 2020 at 13:37
• Real $1$ is identified with complex $1$ but not identical Jan 31, 2020 at 13:44
• Real $1$ and complex $1+0i$ are elements of different fields so you cannot really compare them. Jan 31, 2020 at 13:58