Assuming that the entire line you mean $\mathbb R$ (the real numbers) and the line segment you mean some interval, e.g. $[0,1]$, then there are the same number of points on the two sets.
That is to say there is a bijection between $\mathbb R$ and $[0,1]$. This is discussed in Bijection from finite (closed) segment of real line to whole real line.
I should add that the cardinality (size of infinity) of $\mathbb R$ is strictly greater than $\aleph_0$, and in fact can be calculated to be $2^{\aleph_0}$. Namely, given that $\mathbb N$ has size $\aleph_0$ then the real numbers have the same size as the set $\{A\mid A\subseteq\mathbb N\}$, which can be calculated to be $2^{\aleph_0}$.