When, as a child, I was told by my teacher that I had to be careful with
"named'' numbers and not to add apples and people, I remember asking her why in that case we can divide apples by people:
$latex 10\, \mbox{apples}\, :\, 5\, \mbox {people} = 2\, \mbox{apples}.$
Even worse: when we distribute 10 apples giving 2 apples to a…
"already at the
level of elementary school arithmetic children are working in a much more sophisticated structure, a graded ring
$\mathbb{Q}[x_1,x_1^{- 1},\dots, x_n,x_n^{-1}]$.
of Laurent polynomials in $n$ variables over $\mathbb{Q}$, where symbols $x_1,\dots, x_n$ stand for the names of objects involved in the calculation: apples, persons, etc. "
