$\mathbb{C}$ is a one-dimensional complex vector space. What is its dimension when regarded as a vector space over $\mathbb{R}$? $\mathbb{C}$ is a one-dimensional complex vector space. What is its dimension when regarded as a vector space over $\mathbb{R}$?
I don't understand how $\mathbb{C}$ is one-dimensional. Please help me understand that.
Also, I'm pretty sure that when the field is reals we have $\dim(\mathbb{C})=2$. Since when $\alpha$ is real and $z=a+bi$ is complex we have $\alpha z=\alpha a+\alpha bi=\alpha a(1,0)+\alpha b(0,i)$.
How does this look? Any solutions or help is greatly appreciated.

The complex numbers as a vector space over the field of real numbers is of dimension $2$. The two vectors $1$ and $i$ form a basis and any complex vector $a+ib$ is a linear combination of the two vectors $1$ and $i$, multiplied by real scalars $a$ and $b$ and added.
The complex numbers as a vector space over the field of complex numbers is of dimension $1$. Chose any vector in this space. You can multiply it with a scalar $a+bi$ (scalar because the field is the field of complex numbers) to get the arbitrary vector $a+bi$ (now $a+bi$ is any vector in the complex vector space, got by multiplying a unit vector $1$ with the scalar $a+bi$).
