elementary set theory - Cardinal arithmetic confusion: What is $|Bbb R|+|Bbb N|$?

I do not understand how to calculate addition two cardinals.
I know that the formula as follows:

if $\alpha$ and $\beta$ are two cardinals, then

$\alpha + \beta= |\{(a,0):a\in \alpha\}\cup\{(b,1):b\in\beta\}|$.

If $\alpha=\mathbb{R}$ and $\beta=\mathbb{N}$, what's cardinals of $\alpha+\beta$?