Monday Morning Math: María Andresa Casamayor

Good morning!  This Thursday, November 30, marks the 303rd birthday of María Andresa Casamayor de la Coma, the first woman to publish a science book in Spanish.  This math book was published in 1738 when María was only 17 years old, and (quick check of some dates) 10 years *before* Maria Gaetana Agnesi published the Instituzioni analitiche ad uso della gioventù italiana.  Many sources (such as Scientific American) claim that Agnesi’s book is the first math textbook published by a woman in Europe, so either those sources are wrong  —  very possible, since not much was published about María Andresa Casamayor de la Coma until just a few years ago  —  or there is a distinction made because Casamayor’s book focused more on arithmetic, and Agnesi’s on calculus.

But I’m getting ahead of myself.  Let’s return to María Andresa. 

Portrait by Eulogia Merle, permission to use given by the Fundación Española para la Ciencia y la Tecnología

Maria was born on November 30, 1720, in Zaragoza, Spain; she was the seventh of nine children in her family.  Her father and her mother’s father were textile merchants, and while it was common for sons in families such as hers to be educated, it seems less common for daughters.  When she published her first book, Tyrocinio arithmético (as a teenager!) she dedicated it to her teacher Escuela Pía.  This is more surprising than it might seem, since he taught at a school for boys, which she presumably couldn’t have attended.  The site MacTutor (where I got most of my info) suggests this was aspirational, that she possibly wished that Escuela Pía had been her teacher, but since María Andresa published under a pseudonym  Casandro Mamés de La Marca y Araioa, I’m reminded of a later mathematician, Sophie Germain, who wrote to another mathematician (Gauss) using a male pseudonym.  Or maybe her older brothers taught her?  Really, I wish I knew more about how she came to write this book. 

(The other piece that intrigues me is that her textbook (which you can read here) seems designed for self-study, and it specifically seems like it was written in order to make mathematics accessible to people who couldn’t learn it otherwise.  Possibly other girls.)

The same year that María Andresa’s book was published, her father died.  María Andresa became a teacher at this point, teaching girls, which she appears to have continued doing for at least 30 years, according to local records. She wrote another book that was never published, and she passed away in 1780 just shy of her 60th birthday.

In 2020 there was a celebration of her life and work, including a series of lectures  and the release of a documentary.   

I recommend the  MacTutor site, too, which also talks about how it was only recently that people were able to verify where she was born, by making an educated guess as to her birthday, based on her middle name.

Happy birthday María Andresa! 

Monday Afternoon Math: Fractional Calculus

It’s…Monday afternoon!  I forgot to send anything out this morning!  This is certainly because this week is strange – classes are still going on, but only for part of the week, and while there are people around campus there aren’t as  many as usual.  So in honor of this fractional week, we’ll talk about fractional calculus.

Fractional calculus is what occurs when you think to yourself: “Since the first derivative of is , and the second derivative of is , I wonder what the one-and-a-half derivative of would be.”

The answer, it turns out, involves the Gamma Function.  The gamma function is what occurs when you learn about factorials and think to yourself “Since  is and is , I wonder what would be.”  It turns out that the answer to *that* question gets you  .    Some playing around shows that and and in general , so you can use the Gamma Function to find .  The answer is approximately 11.6. 

Fractional derivatives and fractional integrals use the Gramma Function as part of their formulas.  Interestingly, they also both use integrals, and also interestingly, the fractional integral seems more straightforward than the fractional derivative.  One kind of Fractional Integral, called the Riemann-Liouville Integral, looks like:

It is actually this integral that motivated this post, since I can see it on the whiteboard where it was written by a student (thanks Q!) as a suggestion for a Monday Morning Math

This is the part where I should give an example, and I think it’s best to find  of , which would involve integrating  twice.  The integral says it should be:

which simplifies to 

and this gives

and THAT simplifies to  for some constants  and .  Sure enough, that’s what you’d get if you integrate  twice. The cool thing about  Bernhard Riemann and Joseph Liouville’s almost 200 year old integral is that there’s nothing that requires you to use a number like 1, 2, 3, etc. for , so you can do fractional integration.

Neat, right?  Is that neat enough to overlook that I kind of skipped over fractional derivatives, which is how I started this email?  I had expected one clear formula for those, and that isn’t how it seems to work.  But I did run across some references to how fractional derivatives do have some application, including this video describing how fractional calculus can be used to model crowd behavior:  

Enjoy, and have a Happy Thanksgiving everyone!

Monday Morning Math: the obelus

Good morning!  Today’s MMM is short and sweet (why yes, it *is* mid-November).  Do you know what an obelus is?  If you do, I’m guessing you might enjoy crossword puzzles.  Here’s an obelus: 

That’s right, it’s the division symbol!  It’s also used in ancient manuscripts to mark questionable passages, according to Merriam-Webster.   And in between those uses it was used for subtraction, which seems confusing in a way similar to the exponent “-1” which can mean both the reciprocal of a number or expression or the more general inverse function.

And in case you were wondering, yes, the word obelus is related to obelisk!  They both come from the Greek obelós, which itself seemed to carry multiple meanings relating both to manuscript marking and to a pillar.

Monday Morning Math: Indigenous Mathematicians

Good morning! November is Native American History Month, and a great opportunity to highlight the website indigenousmathematicians.org, which features the mathematical contributions of Native Peoples. There are two 2023 honorees: Jared Field and Kainoa Dahlin. Jared Field uses mathematics to analyze the Gamilaraay kinship systems, in which people are born into one of four groups based on the group of their mother, and to show how this system was developed in an intentional and scientific way. Kainoa Dahlin, who also goes by Kyle, creates mathematical models of how temperature impacts the spread of mosquito borne diseases, taking into account the mitigation strategies (like wearing bug spray) that can be used.

And once you’ve read those head over to the podcast Native Stories! There are more than 100 episodes, and several of the most recent episodes are discussions with mathematicians. (Right now I’m listening to the Kori Czuy, who is speaking about the power of relationship and stories in mathematics, and I want to tell everyone HEY LISTEN TO THIS! So I’m telling you, friend to friend =))