“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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