Mathematical Notation: Summation

In response to David Wees’ post about summation notation, I’d like to suggest that the terseness of mathematical notation is a godsend when working on long calculations, but it should, perhaps, be collapsible for experts and expandable for beginners.  What I mean is that while

\sum\limits_{i=1}^{6} i^2 or

\sum\limits_{i\in1\ldots6} i^2

may suffice for the expert, beginners may prefer

\mathop{\Sigma\text{um}}\limits_{i\text{ from }1}^{\text{to }6} i^2 or

\mathop{\text{Sum}}\limits_{i\text{ from }1\text{ to }6} i^2 .

I hate the linear form that David Wees mentions,

Summation (i, 3, 6, i2) = 32 + 42 + 52 + 62 = 86,

for it loses the spatial memory aspect of the original summation convention.  The real problem seems to be that this last expression serializes for computers well, but the other mathematics is hard to type.  I would prefer “smaller bits” of mathematics, like Sum and Sequence:

Sum[Sequence[Lambda[i,i^2], 1..6]] or even Sum[(i->i^2)[1..6]]

It would be nice if computers would do f[A] if f[a] is defined for every a in A without some kind of function like Map or Apply.  Here’s a longer version of the last expression:

Sum[Apply[Lambda[i,i^2],1..6]]

In fancy LaTeX form that might be:

\sum (i\mapsto i^2)[1..6] or

\sum \left[(i\mapsto i^2)[1..6]\right]

if we want to make the operator precedence completely clear.  Here, “1..6” is some Ruby-like syntactic sugar to mean the set (really: sequence) {1,2,3,4,5,6}.

 

 

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One thought on “Mathematical Notation: Summation

  1. I have many problems with mathematical notation that leads to confusion in education, but the Sigma notation is I think ok. I like the connection that can be made between Sigma and the stretched S of the integral sign in showing the connection between Rieman Sums and integrals. The student has the learn to “read” Sigma notation and I typically write out an English language sentence to shows how it reads and therefore what it means. I really like your “beginners notation” above, not so much as a notation for beginning students to use, but as a pedagogical tool to teach the notation. This method could be adopted to help teach other notations where an operator has a subscript and superscript that correspond to “from” and “to” values, like the definite integral and the vertical bar that represents evaluation of an expression between two value.

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