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About the irrationality of e=2.718... (1840)
- ydah 10y agoThe article is about e^1 tho
- tjalex 10y agoCheck out the annotations on the side of the document for more details -- you can extend the proof to e^2 easily with c = 0.
- siegelzero 10y agoThis a very terse proof. It is stated "The same method can also be used to prove that e cannot be a root of a second degree equation with rational coefficients", but if I recall correctly, the generalization for the irrationality of $e^2$ isn't entirely straightforward, mainly due to the presence of powers of 2 in the terms of the expansion.
- conistonwater 10y agoAs pointed out in the annotation, isn't the case $e^2$ just exactly the same proof but for the special case $c=0$? And in any case, if $e^2$ were rational, then $e$ could only be rational or a quadratic irrationality.
- siegelzero 10y agoYeah, this proof seems to work. I was thinking of the case where you attack $e^2$ directly, instead of going after both $e$ and $e^{-1}$.
- pmalynin 10y agoIt can be shown by clever application of Taylor Expansion that e^2 is irrational.
- arglebarnacle 10y agoOff topic, but can you tell me whether you're using a browser extension to render $e^2$ (and if so which one!), or if you're simply using the $ character as a delimiter for humans who read and write a lot of LaTeX code?
- siegelzero 10y agoIt's simply a habit from LaTeX :-)
- ajross 10y agoMath people can read TeX without compilation.
- stablemap 10y agoI don't have any recommendations, but they do exist: http://mathoverflow.net/questions/22141/how-do-i-see-latex-math-on-any-web-page-and-in-email http://mathoverflow.net/questions/22141/how-do-i-see-latex-m...
- simplicio 10y agoI always found it weird that despite being developed at CERN and most widely used in the early years by universities and research labs, HTML never developed a decent standard for transmitting/rendering math equations (I know that there's a math tag now, but it doesn't seem to get much use). Especially given that Latex would seem to provide a pretty widely known de-facto standard for the syntax.
- nikdaheratik 10y ago1) HTML is derived directly from SGML which was used to handle similar markup issues in printed texts. One of the reasons why HTML was successful was because there were already similar tools in place from SGML that could be quickly applied to handle processing and correctness. Adding math would require building a number of additional tools. 2) TeX by itself doesn't resolve very well into the DOM tree which is what parsers use to organize the markup data internally. You need a markup language that is similar to HTML if you want to make it easy for browsers, parsers, and libraries to adopt the new standard.
- stablemap 10y agoMaybe it's better to write his equation as ae² - ce + b = 0. The proof would say that e isn't degree ≤ 2 over the rationals.
- fred256 10y agoI love how matter-of-factly the article ends with "so, etc."
- sizzzzlerz 10y agoFunny how my geometry teacher back in high school didn't offer that up as a means of completing any of the proofs we had to do. It would have made finishing my homework so much faster.
- conistonwater 10y agoIt's curious how "when b is <0" was considered okay scientific French, but in English these days it looks terrible and is written as "when b<0".
- klodolph 10y agoThe way math papers and proofs are written is fascinating, because it reveals so much about how people read mathematical notation. Much of the older notation was more explicitly used as shorthand for words or phrases, rather than being a language in its own right (as it is today). (And old papers were written out in natural language, which seems so hard to read by today's standards.)
- taserian 10y agoI especially like how certain symbols we regularly use nowadays weren't always known that way. For example, the set of integers is now written as \mathbb{Z} , but used to be written as a fraktur "Z" with a bar above.
- aroberge 10y ago"was" considered?... In French, traditionally, the inequality signs do not include the verb to be; that is how I learned it some 40 years ago. Thus "<" is "plus petit que" although some people nowadays would take it to include the verb to be as in "est plus petit que", mimicking the English usage. See https://cegepadistance.ca/sous-site/En_route_vers_la_reussite/pdf/fiche_symboles_utiles.pdf https://cegepadistance.ca/sous-site/En_route_vers_la_reussit... to see that it is still how it is used in French today.
- stablemap 10y agoI still see this, particularly when it would be awkward to create a new symbol: "when the discriminant is ≤ 1". I feel that Halmos [1] wouldn't have liked it, but it does lessen the distance between what is written and the equation the reader will want to have in his mind. [1] http://www.math.washington.edu/~lind/Resources/Halmos.pdf http://www.math.washington.edu/~lind/Resources/Halmos.pdf
- khana 10y agoe. Irrational -and- Transcendental.
- waynecochran 10y agoFor the transcendental part, wouldn't you have to prove that e is not the root of any polynomial with rational coefficients (not just a quadratic)?