sigmaleph

Unrelated history-of-mathematics observation: Apparently for a pretty long time it was a fairly widespread view that 1 isn't a number (see this pdf about historical answers to the question of what the first prime number is, arguing that often the question of whether 1 is prime doesn't even come up, because the author in question doesn't even think 1 is a number). And by "a pretty long time" I mean up to the 16th century and among both European and Islamic mathematicians.

In some sense one can see the appeal (it's even preserved in idiomatic expressions! If I say something is true for "a number of reasons" I don't mean to include the possibility of the number being one) and ultimately it's a choice of definitions and presumably they wouldn't argue that the expression "2+1=3" is nonsense because I'm adding a non-number to a number. Still, it sounds so weird in a modern context. The obvious argument I'd make would be something about fractions, but presumably the relevant notion of 'number' is about natural numbers.

how people used to think about math is fascinating and kinda weird

The first proof of the irrationality of pi is from 1761 (due to Johann Heinrich Lambert). Here's what I'm curious about: did people think it might be irrational before that?

There's a long history of people trying to approximate pi, first with geometric methods and then with infinite series, and by the time of Lambert's proof the best one had over a hundred decimal places. What did the people doing those calculations think? The concept of irrational numbers was known for a while. Did they think there was a specific fraction they were approximating, had they guessed it might be irrational, did they just not have an opinion on the matter?

Wikipedia claims Euler had conjectured pi was irrational before Lambert's proof. Euler and Lambert were contemporaries; was he the first one to suggest it? Was there a debate on the subject among mathematicians?

(Wikipedia also suggests that the Indian mathematician Aryabhata might have conjectured it circa 500 BC, but the argument given seems pretty thin and even if so I'm more interested in whether it was a widely-held idea among some mathematical community than about whether some specific person thought so)

Check my math?

You flip a coin until it comes up tails. If you get tails immediately, you win $1. If you get one head and then lose on the second round, you win $2. If you get (n-1) heads and lose on the nth round, you win $ 2^(n-1).

Probability of you losing on the first round is 1/2. Probability of you losing on the nth round is 2^-n. Expected value of W (how much you win in USD) is therefore 1/2 + 2*1/4 + ... + 2^(n-1)*2*(^-n) + ... which is a sum of infinitely many terms all equal to 1/2. So the expected value of how much you win is infinite. Is that correct?

(this is not a puzzle or anything, I'm fairly confident I'm right but could be making a mistake)

so since writing that post this morning i learned that

1) MathML has a plugin called MathPlayer that’s meant for this, some screenreaders work with it. it has settings for both low vision and blind people, and can output stuff to refreshable braille displays (there are various standards for representing math in braille)

2) beyond just being usable for MathML, mathjax has its own set of accessibility tools. I think they allow you to navigate subsets of mathematical expressions for clarity? not super clear how they work

3) MathSpeak is a standard meant for unambiguous rendering of math in speech. MathPlayer can work with it, but presumably it can also be useful if you’re hand-annotating your equations.

(thanks to ilzolende and my fiancée, who saw the original post and brought various parts of this to my attention)

here's something most likely some people have thought more about than me and hopefully have better solutions: how do you represent math in a way accessible to screen readers?

a lot of the time, if you're writing math any more complicated than basic addition, substraction, and multiplication, you'll want to go beyond putting unicode characters in a line (a recent popular tumblr post about inconsistent order-of-operations standards illustrates why even just representing division can get tricky if you are confined to a single line). some websites use scripts to parse LaTeX into images, which is very convenient if you can see and entirely useless if you can't. and i'm sure trying to parse LaTeX formatting by hearing a bunch of backslashes and brackets and stuff is not pleasant.

so, if you had to provide alt text to an equation as an image, is there something better than just trying to spell it out in plain English and trying very hard to disambiguate all the things we usually rely on actually drawing the equation for instead?

S	M	T	W	T	F	S
			1	2	3	4
5	6	7	8	9	10	11
12	13	14	15	16	17	18
19	20	21	22	23	24	25
26	27	28	29	30

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