From 3 to 2

Funnily enough, tan(arcsin(cos(arctan(cos(arctan(sqrt(3))))))) = 2.

mathblab:

The pure mathematician, like the musician, is a free creator of his world of ordered beauty.

Bertrand Russell

The mortal matrix problem

Today I learned that there does not exist an algorithm for the following task: given a finite set of (say 3 × 3) matrices, can you multiply these together in some way and obtain the zero matrix?

Note, it’s not that no one ever cared to implement such an algorithm, or that it would just be an unreasonably inefficient brute-force-ish method. The interesting thing is, such an algorithm cannot exist—it is doomed to (sometimes) output wrong results or not give results at all.

Moessner’s magic

Start with a list of all nonzero natural numbers. Cross out every second one—the even numbers—and take the cumulative sum of the resulting sequence. The resulting list is 1, 3 + 1 = 4, 5 + 4 = 9, 7 + 9 = 16… and these numbers should look familiar: they’re precisely the square numbers!

In 1951, Alfred Moessner discovered the following similar procedure. Start with the same nonzero natural numbers, and cross out every third one. Add the numbers as before, and now cross out every second one. Then you’re left with the third powers of the natural numbers.

Starting with every fourth number results in the fourth powers, and Moessner conjectured (well, said that his not-so-easy proof would follow later) that this holds in general: starting with crossing out every k-th number, summing, crossing out every (k–1)-th number, summing… finally gives you the k-th powers.

What happens if we start with different numbers? Funnily enough, if we start with crossing out the triangular numbers 1, 1 + 2 = 3, 1 + 2 + 3 = 6, 1 + 2 + 3 + 4 = 10… (every time incrementing the step size with 1) then we find the factorial numbers 1! = 1, 2! = 2 × 1 = 2, 3! = 3 × 2 × 1 = 6, 4! = 4 × 3 × 2 × 1 = 24…

If we increment the increment in the step size with 1 every time, so that we cross out 1, 1 + 3 = 4, 1 + 3 + 6 = 10, 1 + 3 + 6 + 10 = 20… the resulting sequence may not look that familiar. These numbers are the superfactorials 1!! = 1, 2!! = 2! × 1! = 2, 3!! = 3! × 2! × 1! = 12, 4!! = 4! × 3! × 2! × 1! = 288…

Finally, crossing out the square numbers gives us another unfamiliar sequence. However, notice that the squares are given by 1, 1 + 2 + 1, 1 + 2 + 3 + 2 + 1, 1 + 2 + 3 + 4 + 3 + 2 + 1… and that the resulting numbers are precisely 1, 1 × 2 × 1, 1 × 2 × 3 × 2 × 1, 1 × 2 × 3 × 4 × 3 × 2 × 1…

Anonymous asked: hey; could you explain LaTeX to an absolute newbie? where and how can I learn it?

the-real-numbers:

\title{Real’s Answer}

\begin{answer}

\makeTitle

A good introduction will get you on your feet fast. You will probably just need to do what I did and learn the rest as you go once you get familiar with it.

\section{Getting LaTeX}

First thing you want to do is get something that supports LaTeX that you can play around in. All of the following should be good enough for getting started. Some options are:

\begin{itemize}

•  Overleaf online and mostly free and functional with a sign-in. I think you have to pay to store like over 2gb of documents on the cloud or something, not like that’ll ever be an issue really. I use this one. It is pretty good. Start with this one for online stuff but move on to something else if you don’t like it.
•  ShareLaTeX, I’ve never used this but it’s online and free.
• TeXmaker,This is free software, it’s the one I use and I like it.
• MiKTeX, This is also free software but I can’t seem to get it to work on my laptop.
• BakomaTeX, I’m told God uses this software. It’s not free, but it looks worth it.
  
• If I recall correctly, Microsoft Word will support latex symbols and notation in “equation mode”.
• LaTeX Base is an online compiler i just found with decent functionality for non-paying users.
• And here’s a few more if none of those make you happy.

\end{itemize}

\section{LaTeX tutorial speedrun}

Let’s let ”YCH” stand for “your content here” for the sake of demonstration.

\subsection{Environments}

LaTeX has these things called “environments” which tell latex how to display the text. there are the various math environments which display math in different ways, the “itemize” environment that builds lists with the “\item” command, and more. To use an an environment, you need three lines of code, ususally.

\begin{environmentName}

YCH

\end{environmentName}

You will usually need to open the “document” environment and put everything you want shown inside of it, even your other environments. Like so:

\begin{document}

    \begin{itemize}

       \item here’s a bulletpoint.

    \end{itemize}

    Here’s just some plain text.

    \begin{equation}

        e^{ix} = \cos(x) + i \sin(x)

    \end{equation}

    Below will give us the same thing. It’s easier to type:

    $$ e^{ix} = \cos(x) + i \sin(x) $$

    Note that in order to start a new paragraph in normal text you need to hit enter twice, not just start a new line.

\end{document}

There’s more about this below.

\subsection{Structure}

Use LaTeX’s commands to put in titles of sections, chapters, and whatnot wherever you type them, like these: 

• \chapter{YCH}
  
• \section{YCH}
  
• \subsection{YCH}
  
• \subsubsection{YCH}
  
• \paragraph{YCH}
  

That will probably be enough for now.

\subsection{Math}

For an equation that shows up inline, you use a single “$” on either side of your equation, like this: $YCH$. Some people use “\( YCH \)” for inline math or something but that’s confusing. Don’t do it. For an equation that shows up on it’s own line, you use “$$” on both sides, like this: $$YCH$$. You can also try the “equation” environment like this:

\begin{equation}

YCH

\end{equation}

Next, we’ll need to know what to put in there. Here’s a a handy dandy little chart of things (that barely scratches the surface, really):

• YCH_{subscript for your content}
  
• YCH^{superscript for your content}
• \mathbb{R} (the real numbers symbol)
• \lim (the Lim symbol)
• \sin, \cos, \tan … (You name the function, they got it.)
• \frac{inline numerator}{inline denominator}
• \dfrac{numerator that’s whatever fucking size it wants}{ denominator that’s whatever fucking size it wants }
• \sqrt{guess what this is, lol}
• \left( YCH \right)  This makes parentheses that are the right size.
• \yourgreekletterhere lowercase whatever greek letter that is
• \Yourgreekletterhere Uppercase whatever greek letter that is
• \int the integral symbol
• \sum the big sigma/sum symbol
• \forall the quantifier that is the upside down A
• \exists the existential quantifier that is the backwards E
• \in the “in” symbol, \ni the “in” symbol backwards, and \notin
• \leq \geq, \neq  the less than or equal to, greater than or equal to, and not equal to symbols.
• \begin{cases} YCH1 & condition \ YCH2 & condition \end{cases} make a piecewise function.

\section{conclusion}

PRINT OUT THIS CHEAT SHEET AND KEEP IT WITH YOU, IT IS A GOOD REFERENCE.

For more comprehensive information, please see the LaTex 2e manual (PDF), a few youtube videos, stackExchange, and the internet at large.

$\blacksquare$

Hope this helps. - Real

P.S. It’s pronounced Lay-Tech, not latex. In my experience, people will make fun of you if you say it wrong. They’re weirdly defensive about how LaTeX is pronounced.

\end{answer}