There is another definition, which is more general. A subset $F \subset E$ is dense in $E$ if the set of all limits of sequences of elements of $F$ is $E$.
Where does this come from?
A way to answer this question is from the closure of $F$. The closure of $F$ can be thought of as what I called the set of all limits of sequences of elements of $F$. It's generally defined as the smallest closed set containing $F$. Concretely, it's what can be reached by the elements of $F$ without really going "outside" of $F$. What really matters here is that you'll be able to reach the boundaries (as limits of some sequences).
Just picture the real interval $]0,1[$ (often written $(0,1)$ in English litterature). Let's consider the sequence $(a_n)_{n \geq 2}$ defined by $a_n = 1 - \frac{1}{n}$. For any $n \geq 2$, $a_n \in ]0,1[$. However, its limit is $1$, which does not belong to $]0,1[$. Here, 1 is an example of what I meant by "can be reached from elements of $F$" earlier. You can guess that the closure of $]0,1[$ is $[0,1]$.
So here we can say for instance that $]0,1[$ is dense in $[0,1]$. This definition holds in much more general contexts than $\mathbb{R}$ but that's really a way to look at what that is all about.
Hope this helps.