Is something like this sufficient as a proof? It's so natural and easy, yet convincing and accurate, that I have hard time believing it. :) Even if it's just an explanation it's still awesome. How I wish I could find materials teaching math ideas without relying heavily on math language! (And no, because I'm not going to contribute to the field I don't really need to know that language... Except I need, because it's the only way to obtain what is useful to me, however uncomfortable it is for me...)

That is a completely valid and rigorous proof.

The difficulty with reading more "formal" proofs is that they try to make each statement as consise as possible, because it makes it easier to manipulate the results, and hold more of the proof in your head at a time.

For perspective, consider the quadratic equation. This equation was first discovered in 628 AD, India. An English translation (from 1817) reads: "To the absolute number multiplied by four times the [coefficient of the] square, add the square of the [coefficient of the] middle term; the square root of the same, less the [coefficient of the] middle term, being divided by twice the [coefficient of the] square is the value"

Or, as modern mathematicians would right, x=(sqrt(4ac+b^2)-b)/2a.

The equation is slightly different for two reasons. First, it applied to equations of the from ax^2+bx=c, as apposed to ax^2+bx+c=0, so the sign of 4ac is inverted. It also misses the second answer provided by the '±'.

The original paragraph form is probably easier for someone unfamiliar with anything beyond arithmetic, however the modern form is far easier to reason about, and use to those who invest the time to learn the notation.

> The original paragraph form is probably easier for someone unfamiliar with anything beyond arithmetic

Isn't this because of the fact that at the time (628 AD as you say) there wasn't that much to be familiar with beyond arithmetic? (I don't really know the history of maths, just curious.)

> and hold more of the proof in your head at a time

Is it really the case? I mean that no matter with which notation you start with, you need to parse and evaluate it mentally to convert it into terms you actually think with.

I can see that for maths 'natives' it is helpful, because their mental models are very closely tied to the formal notation. But for me mathematics is a foreign language, it seems that I think in slightly different terms and this succinct notation makes it harder for me to actually understand what it is really about. (I don't mean your example specifically - it's still well inside my comfort zone).

As a programmer I do struggle for "math literacy" because I have to. I need to be able to read proofs, understand some concepts to be able to use them and to know when I need them. The problem is it's not natural for me. Yesterday I found here on HN a book that just may be exactly what I'm looking for: http://greenteapress.com/thinkstats/html/index.html (didn't have time to read it yet, but it looks promising).

> who invest the time to learn the notation

I think I said it before but maybe it's worth noting explicitly. I am somewhat familiar with the notation, I consider myself 'literate'. But I'm far from fluent and I probably never will be. Similarly, I can read and write cyrillic script (with pen and paper). But it takes me more than twice the time to read russian in cyrillic than reading the same text written with latin letters. I don't need someone to translate russian to english for me. It's just the character set that is a problem. I think it's very similar to what I experience with math.