Are real numbers vectors or scalars? I am trying to understand the difference between scalars and vectors. I know the basic definition that vectors are magnitude with direction. But we also call vectors those entities that belong to vector spaces and can be added.
I had previously asked this question,  a comment to this answer to the question states that real numbers can be both vectors and scalars, how is that even possible ?
If so then what is a vector and what is a scalar ? How can they be different but be represented simultaneously by the same mathematical concept ?
 A: A vector space is a quadruple $\{V,\mathbb{F},+ ,\odot\}$ where $V$ is a set whose elements are called vectors, $\mathbb{F}$ is a field whose elements are called scalars,  $+:V \times V \to V$ is a binary operation in $V$, called addition, and $\odot : \mathbb{F}\times V \to V$ is an operation called ''scalar multiplication'' (usually the symbol $\odot$ is omotted). These operations must satisfies some axioms that ''define'' what a vector space is.
So If we take $V=\mathbb {R}$ and define the addition as the usual addition in the field of real numbers, ve can build some vector spaces in which the real numbers are vectors.  
If we chose $\mathbb{F}=\mathbb{R}$ we have a vector space in which also the scalar are real numbers: the so called vector space $\mathbb{R}$ over $\mathbb{R}$. Ve can prove that this is a vector space of dimension $1$ and we can think, intuitively, to the vectors of this space as the point of a straight line.
If we chose $\mathbb{F}=\mathbb{Q}$ we have a vector space in which  the scalar are the rational numbers. This is the vector space $\mathbb{R}$ over $\mathbb{Q}$. and it is very different as a vector space because its dimension is uncountable infinite.
