Friday, July 16, 2021

Sizes of Infinity

 

I am floored. Here is a new mathematical result that sounds pretty important. I'm surprised I hadn't heard of it sooner. It was published online in April.
 
This Quanta article explains it pretty well. But if the article doesn't make sense to you, I can explain more. This is the field I had planned to go into when I was thinking I'd get a PhD. I loved my two logic courses at Eastern Michigan University.  But the one I took at UCSD was not fun. I think because it was too far above me, and I couldn't stay grounded.
 
The one problem with the article is that it made it sound like the big question was resolved. But it's not. I thought it was saying that the continuum hypothesis is false. The continuum hypothesis is about sizes of infinity. The smallest infinity is what you get when you count out all the infinite whole numbers (or all the fractions), and it is called the countable infinity. The continuum hypothesis says that the next size up is what you'd get "counting" the real numbers (like the number line). But there may be a size in between. 
 
I hope there is a way to get a meaningful example of that in-between size of infinity. (The are bigger and bigger infinities, but the two things grounded in numbers we know well, integers and real numbers, are the most interesting to me.)
 
A fun way to start thinking about infinity is a book that's accessible even to young kids. It's a five chapter picture book titled The Cat in Numberland. Sadly, it doesn't seem to be available (unless you want to pay ridiculous prices). My publisher, Natural Math, tried to help the author get it reprinted, but Cricket books (Carus publishing) wouldn't give up their rights, and won't republish. (Maybe we should look into that again...)

[The Quanta article links to the proof that was published online in April. I don't expect to understand that, but I'll try reading it. I might quit very quickly.]

Friday, June 18, 2021

More Tech: Sue Finally Learns How to do Screencasts

I broke my ankle a few months ago, and could no longer use my whiteboard. I asked my college for an iPad and got it within a week. I asked in the Math Mamas group on Facebook for software recommendations -  goodnotes and one other both got high recommendations. I went with goodnotes and fell in love.

Teaching online is significantly more work than teaching in person, and this just added to my workload. But I love that students can easily get my notes on Canvas. And this week I made my first screencast. And then my second. It took me a few hours to get the hang of it for the first one. I may have done the second one in under 20 minutes. Both of them are for a basic geometry course I'm teaching at my college, in which most of the students are high school students.


Indirect Proof (aka Proof by Contradiction)




A Direct Proof

 

I think I could do a few of these a week. Before posting on Youtube, I'd like to find a way to have my face in the corner if possible... Once I feel like I know what I'm doing, the Math Mama's channel gets underway!

Sunday, January 3, 2021

LaTex, a curse and a blessing

I've been making teaching materials on computers for over 25 years. Maybe 15 years ago, I was introduced to MathType, and it made my equations so much nicer. Now it doesn't work with Word, and you have to pay a yearly fee. No thanks. It seems crazy to me that MS Word doesn't have a better equation editor. (I don't really remember what I don't like about it, but I think it has annoyed me lots over the years.)

I got a new computer in the Spring, and since then, whenever I need to make a formula, I've been using my old computer with an old version of Word, and my very old copy of MathType. Today I wondered if it was time to bite the bullet, and make a quiz using LaTex.

I've tried to learn a bit of Latex a number of times before, and it just felt overwhelmingly weird. I especially hated that I couldn't see what I was doing. This time was better in a number of ways. First, my colleague showed me overleaf, where I can see what I'm doing. You can choose split screen, and hit recompile after every little change.

The next thing that helped was that I got a bunch of materials from the author of the book I'll be using. (Oscar Levin, Discrete Mathematics: An Open Introduction.) I used those as templates for my own work. I deleted what I didn't want, and began to add what I did want. (If you want to learn LaTEx (or TEx), and you don't have a bunch of materials someone else made that you can modify, this quiz template might be helpful.)

The reason I was using LaTex was the equations, but that was one of the things I didn't know how to do. This site, codecogs, came to the rescue!

I also needed to include an image of a Venn diagram. I read up (googled latex image), tried to do what they said, and my image ended up in a weird place, next to the questions. I guessed, and added a line that I saw in other places in my documents from Levin (\vskip 1em). I figure that's a vertical skip. I have no idea what the 1em is. (I tried 5em for more space. Nope.) It worked!

But the image was still too big. Read up again, use [scale=0.5], put it in the wrong place, so it doesn't work. Figure out the right position, it works! And now the image doesn't look right hanging out on the left. I read up, use "the centered environment," and it is all just prefect!

Here's the centering:

\begin{center}
    \includegraphics[scale=0.5]{venn10}
\end{center}

That took me over an hour. (Maybe two.) I made a second version of that quiz in ten minutes.

 

I'm learning...



Summary

Does LaTex seem way too complicated, but it still might be the answer to your problems?

  • Use a simple environment like overleaf where the split screen lets you see what you've done.
  • Start with a template you can modify.
  • Use something simple like codecogs to build your equations.
  • google your questions.

Good luck!

 
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