MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], ['\\(','\\)']], displayMath: [['$$','$$'], ['\\[','\\]']], processEscapes: true, processEnvironments: true, skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'], TeX: { equationNumbers: { autoNumber: "AMS" }, extensions: ["AMSmath.js", "AMSsymbols.js"] } } }); I recently found another calculation that I would consider reasonable for my students to perform that their calculators cannot. On my Calc II students’ final exam, I asked them to evaluate the infinite series $\displaystyle \sum_{n = 1}^\infty \frac{4}{(n + 3)(n+5)}$.
I enjoy finding calculations that I would consider reasonable for my students to perform that their calculators cannot.
MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], ['\\(','\\)']], displayMath: [['$$','$$'], ['\\[','\\]']], processEscapes: true, processEnvironments: true, skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'], TeX: { equationNumbers: { autoNumber: "AMS" }, extensions: ["AMSmath.js", "AMSsymbols.js"] } } }); Using a double-angle identity and the pythagorean identity, it’s pretty straightforward to show that $\sin\left(2\sin^{-1}\left(-\frac{4}{5}\right)\right) = -\frac{24}{25}.$ However, my TI-89 returns an unhelpful $-\sin(2\sin^{-1}(4/5))$ instead.
According to the Arizona Daily Star, the Arizona Senate Education Committee passed a bill that would prohibit the state from implementing the Common Core standards. Quoth the Daily Star (emphasis mine):
[Gubernatorial candidate and state Senator Al] Melvin said he understands “some of the reading material is borderline pornographic.” And he said the program uses “fuzzy math,” substituting letters for numbers in some examples.
Later on in the same article, Senator Melvin is quoted as expressing concern about the rigor of American academic standards, arguing that “We have cheated several generations of Americans out of a decent education.
For a long time, I have been a user of the math software system Sage, and for a longer time, I have been a user of LaTeX. So, it’s with some embarrassment that I report that I only recently discovered the awesomeness that is SageTeX, a LaTeX package that allows your LaTeX document to run Sage code and include the results.
I often use Sage to create plots for use in my lectures or printed class materials.
I learned about this Tumblr blog where people try to summarize their PhD dissertations in the style of xkcd’s “Up Goer Five” comic—that is, “using only the ten hundred words people use most often”. There’s an online text editor that checks whether you’ve strayed from these thousand most-common words.
Inspired by this blog, here’s my attempt to summarize my dissertation (Koszul and generalized Koszul properties for noncommutative graded algebras) in “Up Goer Five” style:
This comes from Saunders Mac Lane in his textbook Categories for the Working Mathematician (p. 108, emphasis mine), but I contend it applies to more than just mathematics:
MathJax.Hub.Config({ tex2jax: { inlineMath: [['$','$'], ['\\(','\\)']], displayMath: [['$$','$$'], ['\\[','\\]']], processEscapes: true, processEnvironments: true, skipTags: ['script', 'noscript', 'style', 'textarea', 'pre'], TeX: { equationNumbers: { autoNumber: "AMS" }, extensions: ["AMSmath.js", "AMSsymbols.js"] } } }); One may also speculate as to why the discovery of adjoint functors was so delayed.
[M]athematics only exists in a living community of mathematicians that spreads understanding and breaths life into ideas both old and new. The real satisfaction from mathematics is in learning from others and sharing with others. All of us have clear understanding of a few things and murky concepts of many more. There is no way to run out of ideas in need of clarification. The question of who is the first person to ever set foot on some square meter of land is really secondary.
Today is the 100th birthday of one of the 20th century’s most under-appreciated people, British mathematician, computer scientist, and cryptoanalyst Alan Turing.
It’s impossible to live in modern society without coming into the consequences of Turing’s work. Alan Turing was a pioneer computer scientist, laying the theoretical framework for the information age. He also made key contributions to the Allies’ code-breaking efforts during World War II. It’s been estimated that his contributions sped up the defeat of Hitler by as much as two years.