Are real numbers a subset of the complex numbers?

Question

I am having an argument with a friend. I think that in a sense, the answer is no. My reasoning is that in linear algebra, a vector $(a, b)$ is not the same as a vector $(a, b, 0)$ because the first one is in $\mathbb{R}^2$, while the second is in $\mathbb{R}^3$. However I am not sure if a similar argument can be made for real vs complex numbers.

Answer 1

Real numbers are just complex numbers with no imaginary part. Linear algebra can get away with saying "a 2-vector is not the same as a 3-vector" because there is no sense of multiplication between vectors. However, real numbers have multiplication, and the complex numbers extend the reals by adding i.