What exactly is a scalar? Particularly, referring to scalars I am only implying to individual numbers, for this instance, let's take the set R(real numbers).  
These numbers do lie of the real line, i.e. the x axis, which is perpendicular to the y-axis, together form the cartesian plane. So, how are they scalar?  
For instance, the number 5 can very well be represented by (5, 0) in the plane, which could also imply that it is a position vector.   
If so, let u & v be two vectors in the cartesian plane, then what would the dot product u.v imply? How can the value of it be interpreted? It's obviously not just a random number formed by multiplication of two vectors, what is it that the value of the dot product is so significant?

There's no such thing as a scalar. But there's a notion of field, and if $k$ is a field, we can speak of the vector spaces over $k$. Sometimes, we've fixed a field $k$, and we're studying vector spaces over it. In this case, we sometimes write "let $a$ denote a scalar" instead of the more formal "let $a$ denote an element of $k$." But that's all it is - a terminological gimmick, no more.
And you're completely correct that $k$ can itself be viewed as a vector space over $k$. This allows us to can say things like "every scalar can be viewed as a vector in a $1$-dimensional vector space" etc.
