## Sunday, July 13, 2014

### What to call the top and bottom part of (n choose k)

In my last post I asked for candidates for names for the top and bottom part of (n choose k) . Here are the candidates and my comments on them and ... the winner!

1. Top part: Degree, Bottom part: Index.
2. Top part: Bino, Bottom part: Mial
3. Top part: Numerator, Bottom part: Denominator
4. Top part: Outcomes, Bottom part: Possibilities
5. Top part: Binomerator, Bottom part: I've got nothing
6. Top part: *, Bottom part: *
7. Top part: Biponendo, Bottom part: Bividendo
8. Top part: Choosand, Bottom part: choosee
9. Top part: Set size, Bottom part: Subset size.
I leave out the explanations for these since one criteria is that they be self explanatory.
While choices 4,8,9 are tempting along those lines, the winner is

Numerator/Denominator

Why? One of the people who suggested it gave a pointer. The pointer actually went to calling the top and bottom part of the Legendre symbol Numerator and Denominator. And thats just it- there are several
other math things that have a top and bottom part. We could try to find a name for each one, OR
just use Numerator and Denominator for all of them. That seems to work. SO- next time you write a paper and need to refer to the top part of a bin coeff or a leg symbol or something else, use the terms
Numerator and Denominator.

CODA: `The Denominator' could be an Arnold Schwarzenegger character. A Math teacher by day, a crimefighter by night. Catchphrase: I'm going to put you under!

1. The typos in this post (in item 5 and the winner) are more confusing than usual.

2. Math notation for many is already too complex. Many of the suggested names fail to take into any account the actual pedagogy for future understanding of the mathematics involved.

For such a simple and fundamental notation which has broad use in "lower level" mathematics and is therefore accessible to a much broader range of people, calling them numerator and denominator seems highly disingenuous! Though on some level, the written notation for \binom {a}{b} might seem to logically "suggest" calling them numerator and denominator, I would warn against it - particularly as this logic is more likely to come from neophytes and is far more likely to confound them when defining the notation with respect to the actual definition which really does have a traditionally defined numerator and a denominator.

The alternate category of notations in the vein of C(n,k) screams out to call both n and k INDICES! The notation really stands for an actual numerator and denominator, namely: \frac{n!}{k!(n-k)!}. Thus there is already a numerator here: n! and a denominator k!(n-k)!. Calling n a numerator and k the denominator is likely to throw off FAR too many students starting out in what is an important area of mathematics. If forced to make a choice (pun intended) other than indices, the concept of calling them something like "outcomes" and "possibilities" makes far more logical sense in terms of what is actually taking place. Otherwise, perhaps a nomenclature like: "top partition"/"bottom partition" would make more logical sense for the processes that these notations are used within?

Mathematics if far too logical a field for us to be so careless with our nomenclature. Most of its point is to be increasingly more specific in our thought while still allowing broad enough generalizations to categorize like things as like. Calling n and k numerator and denominator runs counter to both of these goals.

1. You raise an interesting distinction. My intent was what to use for the top and bottom part in our papers that we write. I've never had to refer to the top or bottom part in a classroom. However, once something is common in papers it might end up getting spread into the classroom. And I certainly agree that this would be a terrible terminology for students.

3. Well, the poll may be closed, but I would suggest: the top index, the bottom index. This requires forgetting the old C^k_n ... ^^

4. Thanks for this post. Seems like there’s always something new I learn even after being in the field for 25 years.