Posted by: Alexandre Borovik | March 31, 2008

“Named” numbers

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:

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

Even worse: when we distribute 10 apples giving 2 apples to a
person, we have

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 should be

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

10\,\mbox{apples}\, : \, 2\, \frac{\mbox{apples}}{\mbox{people}} = 5 \,\mbox{people}.

It is a commonplace wisdom that the development of mathematical skills in a student goes alongside the gradual expansion of the realm of
numbers with which he or she works, from natural numbers to integers, then to rational, real, complex numbers:

\mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R} \subset \mathbb{C}.

What is missing from this natural hierarchy is that 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. This explains why educational psychologists confidently claim that the two operations above have little in common (P. Bryant and S. Squire, The influence of sharing on children’s initial concept of division, J. Experimental Child Psychology 81 no. 1 (January 2002) 1–43.). Indeed, the second operation involves operands of much more complex nature.

Usually, only Laurent monomials are interpreted as having physical (or real life) meaning. But the addition of heterogeneous quantities still makes sense and is done componentwise: if you have a lunch bag with (2 \mbox{ apples } + 1 \mbox{ orange}), and another bag, with (1 \mbox{ apple } +  1 \mbox{ orange}), together they make

(2 \mbox{ apples }+1\mbox{ orange}) + (1 \mbox{ apple } +  1 \mbox{ orange})

\qquad\qquad= (3\mbox{ apples } +  2 \mbox{ oranges}).

Notice that this gives a very intuitive and straightforward approach to vectors.

(By the way, this “lunch bag” approach to vectors allows a natural
introduction of duality and tensors: the total cost of a purchase of amounts g_1,g_2,g_3 of some goods at prices p^1,p^2, p^3 is a “scalar product”-type expression \sum g_ip^i. We see that the quantities g_i and $p_i$ could be of completely different nature. The standard treatment of scalar (dot) product in undergraduate linear algebra usually conceals the fact that dot product is a manifestation of duality of vector spaces, creating immense difficulties in the subsequent study of tensor algebra.)

Of course, there is no need to teach Laurent polynomials to kids; but it would not harm to teach them to teachers. I have an ally in Francois Viete who in 1591 wrote in his Introduction to the Analytic Art that

If one magnitude is divided by another, [the quotient] is heterogeneous to the former … Much of the fogginess and obscurity of the old analysts is due to their not paying attention to these [rules].

It pays to be attentive to the dimensions of quantities involved in a physical formula: the balance of names of units (dimensions) on the left and right hand sides may suggest the shape of the formula. Such dimensional analysis quickly leads to immensely deep results, like, for example, Kolmogorov’s celebrated “5/3 Law” for the energy spectrum of turbulence.

My book is full of examples which all lead to the same conclusions:

  • We should not underestimate the immense richness of basic elementary mathematics.
  • Glossing over difficulties presented by hidden structures may seriously imperil students’ progress.
  • The teacher has to be aware about the hidden structures and be able to guide pupils around dangerous spots—perhaps without needlessly alerting them every time.


  1. […] Graded Rings in Grade School Over at the new, improved Mathematics Under the Microscope (off of Blogger, onto WordPress), Alexandre makes an interesting point. word problems take place in a graded ring. […]

  2. […] I just read Alexandre Borovik’s article talking about something similar. I think the problem is resolved if proportionalities are written […]

  3. Congratulations with your new and improved blog.

    I was wondering if you can help me understand a similar issue relating to “named” equations.

    I wrote about it here.

    By the way, I saw the new website of the Mathematics Village. I hope you will have a less eventful stay this year.

  4. […] Posted by Alexandre Borovik in Uncategorized. trackback This is a follow-up to my earlier post on “named” numbers; the text is mostly cannibalised from my book; I refer the reader to the book (available for free […]

  5. […] that my worries were resolved by pedagogical guidance: I was taught to think about fractions as named numbers of special kind: quarter apples. Fractions like are not result of dividing 5 apples between 4 […]

  6. […] property are called graded algebras, and they show up surprisingly often in mathematics. As Alexandre Borovik notes, when schoolchildren work with units such as “apples” and “people” […]

  7. Or per “person”, rather. Thinking about why person or apple is singular rather than plural in the context you’re bringing it up is tripping some unexpected mental wires for me…

    As is this question: We know ∃ 10 apples and ∃ 5 people. But how do we know they’re in ratio? And how do we know which direction the ratio goes? Particularly with your second equation.

  8. Reblogged this on Human Mathematics and commented:
    “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. “

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