It can search in handwritten notes, and can convert them to text (including converting handwritten math).

I don't do enough math writing or note taking to justify the cost of a Surface Pro, so don't personally know if it really works as well as it appears to in the video.

[1] https://www.youtube.com/watch?v=C7OROShRVzQ

Symbols are for the most part the same as in TeX, e.g. \alpha, etc.

The nicest thing of all that for interactive note-taking (at least for me) is that you see what you're entering. While you're typing you always get to see how it looks. This avoided (again, for me) many situations where I'd leave syntax errors in an equation just to get home and not remember where I'd forgot a closing brace.

The worst thing to enter via the plain-text format are large matrices, though that's no different from TeX, IMHO. In this case using the GUI facilities to enter them can actually be faster.

A (preliminary) description of the format they use can be found in Unicode Technical Note #28 [2], although that's not in all cases still the current implementation.

Fun thing is, there is even a Math Input Panel in Windows which does pretty much the same as that web app: Convert hand-written math into MathML. You can then insert it into various applications (Word works fine, Mathematica too, I think – I guess anything that can handle Presentation MathML on paste or drag/drop). [3]

[1] Sample: http://hypftier.de/files/uni/04/math/Vorlesungsmitschriften....

[2] http://unicode.org/notes/tn28/

[3] Excuse my horrible writing, but I have no pen on this machine: http://hypftier.de/temp/2016-01-13_080128.jpg – Clicking "Insert" while Word is focused, will then yield this: http://hypftier.de/temp/2016-01-13_080219.jpg (it even used the correct d (\dd) which has a small gap in the front and, depending on the font, can be upright)

To make LaTeX writing a bit less 'heavyweight', I maintain a .sty file for packages I use often + a template where I can just start writing + vim setup for editing the files + compile with latexmk.

When it comes to syntax and stuff like that, after writing my math B.Sc. thesis and who-knows-how-many homework reports with more or less same combo of packages and tools, I already know it and learning to use LaTeX no longer feels like an obstacle (..it's a cost I've already paid).

Whether a similar solution would be worth the effort for you depends on how much things LaTeX is good at you anticipate writing in the future. Academic papers, homework, stuff like that? Yes, it might be useful. Some random notes? Maybe not.

Later, I export the Quip doc to Markdown and make it searchable by adding it to Algolia.

I'll comment as someone who does hand-write and scan my notes. I date and number every page, then put a pile through a sheet-feeding scanner every so often. Sometimes I put date and page numbers in my code comments, so I can find my notes on where the code came from. I also enter a line of text (or more if necessary) about each scanned page so I can search that. Something I do in a brain-dead moment every month or two. It can be interesting to look back through the notes and see if I've forgotten about anything interesting.

Often I'll write out the math for something multiple times before I really understand it, and using a pen works much better for me. Only a small fraction of what I write out is worth TeXing up.

Use Pandoc; it basically abstracts all of the pain of LaTeX and allows you to write Markdown with equations. Writing macros is also useful to save typing cramps and clarify meaning -- a typical example is

    \newcommand{\implies}{\Rightarrow}

Lately I've just found it easiest to use a text editor that has good keyboard support for entering unicode and just type everything in plain text. Vim's digraphs or Emacs's TeX input method are what I have in mind. It works fine for most personal notes (with a few unevaluated LaTeX macros on occasion) and it's not hard to go back later and convert it into a pandoc markdown document for nicer formatting.

(sample)

http://write-live.com/d/2aa4a21f-7bb0-4e02-929f-484dc8b7af10

(also check my other comment)

There really isn't much need for the multiplication symbol when we have side-by-side multiplication notation, and if x is a vector then you can underline it to distinguish it from the cross product (or just not name your vectors x). The "two c's" x has always looked bad to me and ends up getting confused with α and other things.

I tried to write the equivalent of \widetilde{u} which it interpreted as u^n. On reflection, what I wrote did look like an n above the u. But if you work in the same field as me, you would immediately assume it was a widetilde.

Update -- it does work on an Android tablet, where hand gestures do count as drags. Very nice -- if only my tablet were twice as big so my fingers wouldn't get in each others' way.

Pretty slick. Now to get it as an open-source app.

https://dev.myscript.com/pricing/atk/

If I had to do it now I would use LaTeX though. For me, open always trumps proprietary when available, and because of their proprietary sort of walled garden nature, Adobe is not a company I would like to support.

Afterwards in the outside world, Office became the rule as I never needed to type mathematics again.

When we did a technical documentation project, DITA and Docbook were used instead, with WYSISWYG editors like oXygen.