Why TeX and LaTeX are the defacto standard for writing math equations

I was curious how troff/eqn and TeX syntaxes compared, and didn't find any nice side-by-side examples, so here's a quick translation of one from the troff manual (http://www.troff.org/prog.html#eqn):

    .EQ
    G(z) ~=~ e sup { ln ~ G(z) }
         ~=~ exp left ( sum from k>=1 { S sub k z sup k } over k right )
         ~=~ prod from k>=1 e sup { S sub k z sup k /k }
    .EN

In TeX mathematical notation:

    $$G(z) = e^{\ln G(z)}
           = \exp \left( \sum_{k \geq 1} \frac{S_k z^k}{k} \right)
           = \prod_{k \geq 1} e^{S_k z^k / k}$$

The TeX is a bit more terse, especially in using ^ and _ for super/subscript instead of sup/sub. That can make a big readability difference in things like simple polynomials; a^2 + b^2 = c^2 is just that in TeX, but a lot more wordy in eqn. The troff/eqn looks a bit less like line noise, though, especially in things like "from" in the summation, and >= for \geq.

Though really I'd guess that TeX's math notation caught on as the de-facto standard for "ASCII math" because TeX as a whole caught on, rather than because of its ASCII-math representation being better.

Existing papers do not really force new papers to be typeset in LaTex, since end products are just PDFs.

I don't know of any alternative solution that is as easy to use and produces a high quality document.

This is one best directions with the available technology. We're trying to do this at Hoot with what we call "Smart Chat." We've taken the markdown editor and adapted backticks (`) to wrap a simplified subset of Latex, so that the language is more accessible to a wider audience by providing immediate rendering and feedback. In our case the uses are much more simple than the troff example above (https://apps.facebook.com/hootapp/?sid=5Yyt797Jg), but are still very applicable. I think our most common use is by highschoolers and college students for calculus, where ideas such as integrals and summations are easily described in plain english:

`integral cos(x)*e^(2x) dx` or `sum (x^n) / (n!)`

edit: added simpler examples

I use python_markdown with MathJaX, and it works pretty well.

see pandoc http://johnmacfarlane.net/pandoc/