Given a Hausdorff space $X$, the set of all vector bundles of finite dimensions $Vect(X)$ (up to isomorphism) with respect to the direct sum is a monoid. We can use the group completion construction to turn $Vect(X)$ into a group, generally denoted by $K(X)$. So now elements in $K(X)$ look like $E-F$, where $E,F$ are vector bundles over $X$. Furthermore, by using the fact that $X$ is Hausdorff, any element in $K(X)$ can be written as $E-\epsilon^n$, where $\epsilon^n$ is the $n$-dimensional trivial bundle over $X$.
Question: I can't imagine how the expression $E-F$ would carry any geometric meaning. How should we think of $E-F$, the 'difference' between vector bundles?
What I am trying to get to is that writing '-3 apples' doesn't make much sense. However in the context of money (debts), negative numbers do turn out to be useful and provides a new way for us to think of negative numbers.
So are there any ways we can look at $E-F$ in which we can assign some kind of meaning to $E-F$?
For example we can look at elements in a cyclic group as some kind of a rotation of some object. I can 'visualise' $Vect(X)$, but I don't know how I should look at $K(X)$. The elements $E-F$ are just symbols to me at the moment.
