“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. “

Mathematics under the Microscope

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
person, we have

$latex 10\, \mbox{apples}\, : \, 2\, \mbox{apples} = 5\,\mbox{people}$

Where do “people” on the right hand side of the equation come
from? Why does “people” appear and not, say, “kids”? There
were no “people” on the left hand side of the operation! How do
numbers on the left hand side know the name of the number on the
right hand side?

I did not get a satisfactory answer from my teacher and only
much later did I realize that the correct naming of the numbers…

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