Tool choice for math number and symbol manipulation

Of course while learning math I have to do a lot of exercises using brain, pen and paper. When doodling and maybe also sometimes while learning 😉 I want some of the work done by a computer, therefore I tried several programs.

Some of the programs I tried before I decided:

  • Python libraries: Matplotlib, NumPy, SciPy, SymPy
  • Octave
  • WolframAlpha Pro, already for a couple of years I had an account, just to doodle, check homework for some online MIT courses I did, or help me get ahead when I got stuck

See Wikipedia for a list of more open-source software for mathematics.

I’m quite a fan of open source, but the above tried options could not live up the experience I had with WolframAlpha Pro. From there I had look at Matlab and Mathematica, Wolfram|One. What I read is that the focus of Matlab is more on the numeric aspect and that of Mathematica more on the symbol side. That and because I already had some experience with WolframAlpha Pro I have chosen for Wolfram|One, pricey but very powerful and intuitive, at least for the simple things I have done up until now. What I read is that Wolfram has an enormous amount of possibilities and therefore a steep learning curve … If you want to know more about Wolfram One you can read it here.

Funny detail is that Mathematica is a program initially written by Stephen Wolfram based on a program called Schoonschip (Dutch for clean sweep) by Martinus Veltman, winner of the Nobel Prize in Physics, both had an itch that computer support needed a step ahead.

Also nice is that there is a Wolfram Language plugin for my favourite development IDE from Jetbrains, see here.

Now let’s start the steep learning curve into Wolfram Language.

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Tool choice for math note taking, summaries, home work …

Note taking with pen and paper, with a stylus on a tablet, using keyboard and note taking app on a tablet … what did I do and where am I going for now.

No requirements study, just a couple of wishes:

  • LaTeX support of course. Why not LaTeX itself? To elaborate for note taking, producing summaries and home work you don’t have to share with others and if you share it speed of producing is more important than perfect layout.
  • Cross platform, so if you produce something and by any chance need to share it that it is possible, so Apple, Linux and Microsoft.
  • Usable on MacOS, iPadOS and iOS.
  • Works with iCloud for file usage and storage.
  • Nice to have would be an export to a LaTeX environment, for further polishing if needed or wanted. 😉
  • Should produce clean, simple but good readable .pdf’s for eventual printing and sharing with non specific editor / markdown users.
  • Simple, to use.

What did I try so far:

  • Apple notes, no LaTeX support.
  • Microsoft OneNote, in a Mac environment no LaTex support.
  • MWeb, checks a lot of boxes, but too complicated. It is Markdown based, later on some thoughts on Markdown in this context.
  • MacDown, just works, very simple, straight forward MacOS markdown editor, with superb LaTeX support. The simplicity almost made me forget about ‘iPadOS and cross platform’ demands.
  • iaWriter, pricey, but looked very good, if I remember it correct there were some printing issues with pdf’s with horizontal chopped last lines on page.
  • snip.mathpix.com, snip notes is a simple online editor powered by Mathpix Markdown. Not only can you write and publish documents using both Markdown and LaTeX, and much more. And last but not least you can export direct to Overleaf (online LaTeX collaborative IDE, another time more on this). Since 2020 I’m a paid user of Snip for translating screenshots, photo’s of handwritten formulas or or on the iPad handwritten into LaTeX, which works flawlessly, even for complex formulas.
    • I still have to try some of there features like translating scanned handwritten documents and pdf’s into LaTeX for example.

While playing with iaWriter when hitting on the printing problems, seen with also other Markdown editors I started to have enough of Markdown and reverting to handwriting on either paper of tablet. The reason for producing pdf’s is that Markdown doesn’t know a fixed standard, so what you produce in one editor might or more probably will look completely different in another editor, so sharing notes in Markdown is a no go.

My choice, for now:

snip.mathpix.com from the makers of Mathpix. When collaborating with other mathy’s I think they either already use Mathpix tools or easily switch to start using them, you can use them for free or intensively using them is possible for a very reasonably price.

To do:

  • Try all export features, especially the connection with overleaf.

If I would change my mind, my enthusiasm for the Snip editor I’ll come back here and share my further experience with it.

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Fermat’s Last Theorem – Simon Singh

Joy from all angles. With titbits and tidbits, which you might or might not have known, like for example we still didn’t proof whether or not there are slightly excessive numbers. Useful for understanding numbers, useless according to dandy Hardy, touching the why off mathematics.

no man’s land is strewn with numbers

kr=, a couple of seconds ago (or I’m unaware of citing somebody; if so, please enlighten me)

Another angle is the history of Fermat’s last theorem from the ancient times to its proof, growing the construction of the proof. And the growing pains of proofing.

It also shows the sometimes nearly invisible line between hero and zero, where the prover thinks zero and the writer rightly, and little seen, screams in silence how great Wiles’ contribution would have been even if he hadn’t been able to take the last step in the proof.

Thank you Simon Singh for this great book.

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Weapons of Math Destruction – Cathy O’Neil

What it is about, Wikipedia, with a bit of kris, says it clearly enough: O’Neil, a mathematician or how she calls herself math babe (https://mathbabe.org), analyses how the use of big data and algorithms in a variety of fields, including insurance, advertising, education, and policing, can lead to decisions that harm the poor, reinforce racism, and amplify inequality.

My 2 cents: it is a must must read for people working in the fields of big data, algorithms, machine learning and AI, to learn to take into account the social impact of the systems they are building. Or ambiguously, take care they deliver good systems.

Math students should read it before graduation.

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So sad, young mathematician killed

This really makes my stomach turn.

Julia Zdanoska killed by Russians

You can’t really trust news, but this comes in from several sources and this source seams trustworthy to me https://matholymp.com.ua/en/2022/03/08/yulya-zdanovska-2/

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AI gives mathematicians new insight into knots

Dutch: https://www.nrc.nl/nieuws/2022/02/18/na-miljoenen-knopen-heeft-de-computer-het-door-a4090715

English: https://www.nature.com/articles/d41586-021-03593-1

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Parabolas

I was wondering what the focus point and directrix of a general parabola ax^2+bx+c would be. So I took pen and paper for an interlude.

I’m working through Adams Calculus a Complete Course ninth edition; on page 19 the parabola with focus point $F(0,p)$ and directrix $y=-p$ is given as $4py = x^2$, the vertex is $(0,0)$.

First realise that $4p(y-k) = (x-h)^2$ has $(h,k)$ as vertex. Writing $ax^2+bx+c$ in the form of $4p(y-k)=(x-h)^2$ will help us find its focus point, directrix and vertex, all expressed in $a, b$ and $c$.

$$
\begin{aligned}
y &=a x^{2}+b x+c \\
y-c &=a x^{2}+b x \\
\frac{y-c}{a} &=x^{2}+\left(\frac{b}{a}\right) x \\
\frac{y-c}{a}+\left(\frac{b}{2 a}\right)^{2} &=x^{2}+\left(\frac{b}{a}\right) x+\left(\frac{b}{2 a}\right)^{2} \\
\frac{1}{a} \left(y-c+\frac{b^{2}}{4 a}\right) &=\left(x+\frac{b}{2 a}\right)^{2} \\
4 \cdot \underbrace{\frac{1}{4 a}}_{p} (y-\underbrace{\left(c-\frac{b^{2}}{4 a}\right)}_{k}) &=(x-\underbrace{\left(\frac{-b}{2 a}\right)}_{h})^{2}
\end{aligned}
$$

If you don’t understand the adding of $\left(\frac{b}{2a}\right)^2$, just think of $x^2+6x+9 = (x+3)^2$, where $9 = \left(\frac{6}{2}\right)^2$.

The focus point is:
$
\begin{aligned}
(h, k+p) &= \left(-\frac{b}{2 a}, c-\frac{b^{2}}{4 a}+\frac{1}{4 a}\right) \\
&= \left(-\frac{b}{2 a}, c+\frac{1-b^{2}}{4 a}\right)
\end{aligned}
$

The directrix is:
$
\begin{aligned}
y &= k-p \\
&= c-\frac{b^{2}}{4 a}-\frac{1}{4 a} \\
&= c-\frac{1+b^{2}}{4 a}
\end{aligned}
$

The vertex is: $(h,k) = \left(-\frac{b}{2 a}, c-\frac{b^{2}}{4 a}\right)$.

The distance between the focal point and the vertex is $p$, which is $\frac{1}{4a}$. Why would you want to know this? Because you might want to know where to put the pan in the solar oven, or the light bulb in a parabola shaped mirror. The funny thing is that a mathematician might be interested in it just because it is possible to calculate this distance, so for no practical reason.

Just in case that you wonder if it is wright that $4p(y-k) = (x-h)^2$ is a parabola with $(h,k)$ as vertex. Let us take focus point $(h,k+p)$ and line $y=k-p$, now let us see where the points $(x,y)$ are that have the same distance to the line and the focus point:
$
\require{cancel}
\begin{aligned}
\sqrt{(x-x)^{2}+(y-(k-p))^{2}} &= \sqrt{(x-h)^{2}+(y-(k+p))^{2}} \\
\left(\sqrt{(x-x)^{2}+(y-(k-p))^{2}}\right)^2 &= \left(\sqrt{(x-h)^{2}+(y-(k+p))^{2}}\right)^2 \\
(x-x)^{2}+(y-(k-p))^{2} &= (x-h)^{2}+(y-(k+p))^{2} \\
0^{2}+(y-(k-p))^{2} &= (x-h)^{2}+(y-(k+p))^{2} \\
\cancel{y^2}-2(k-p)y+(k-p)^2 &= (x-h)^{2}+\cancel{y^2}-2(k+p)y+(k+p)^2 \\
-2(k-p)y+(k-p)^2 &= (x-h)^{2}-2(k+p)y+(k+p)^2 \\
-\cancel{2ky} +2py +\cancel{k^2}-2kp+\cancel{p^2} &= (x-h)^{2} -\cancel{2ky} -2py +\cancel{k^2} +2kp+\cancel{p^2} \\
2py -2kp &= (x-h)^{2} -2py +2kp \\
2py +2py -2kp -2kp &= (x-h)^{2} \\
4py – 4kp &= (x-h)^{2} \\
4p(y – k) &= (x-h)^{2}
\end{aligned}
$

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