390 Quick Answers 7 April

Now we wait for drafts.  It’s my turn to work.  It will take time.  For those who think you are presenting at GREAT Day, if you are impatient for written feedback, I am happy to talk with you if you come to visit during office hours.   Make sure that your rehearsals are 10-12 minutes long.

While your project is in my hands - this is a good time to work on starting final exam preparations.  Remember, it is like the midterm only twice so.  2-3 topics post 1600, and 2-3 topics that do span across the divide.  

I think today’s reading may be the longest of the semester.  Congratulations - you did it. 
 

Lecture Reactions 

Please remember - Bernoullis _need_ a first name.  You can't avoid it. 

e is not aptly called Euler’s number.  (wow, but I find it often so online, ugh!)  Euler’s constant is the limit as n goes to infinity of the sum of the harmonic series to n terms - the natural log of n.  e was named by Euler, either for "exponent" or just because it was the next letter.

Euler published so much, if he _had_ a proof that π is irrational, he surely would have published it, proudly. 
 
There are lots and lots of unknown conjectures in mathematics.  Goldbach’s conjecture is one of the oldest and simplest to state.  There are endlessly many.  It is what research mathematicians work on - asking and answering new questions.  Primes are the fundamental building blocks of natural numbers by multiplication - this is the content of the fundamental theorem of arithmetic:  unique prime factorisation of positive natural numbers.  Why is it hard?  Basic idea - prime numbers aren’t really about adding.  Are there older unsolved problems - the oldest I know - “there are no odd perfect numbers” is roughly 2500 years old.  The benefit for testing a conjecture like this with computers is that they can find counterexamples. 

Agnesi’s challenges:  had to be tutored at home, and debate scholars at home.  She was paraded as a spectacle by her father.   She needed to use her father’s money to publish her book.  She asked to become a nun, and her father denied her, but she convinced him to agree to her living a simple life and avoiding social engagements.  Appointed to an academic chair, but could not teach. 

Do I think there are other “secret” women in mathematics around this time?  Not in Europe is my guess.  Not who were contributing to new mathematics.  That were thinking about mathematics - sure.  Our course has mostly been about new mathematics.  And the reason no one was is only because the system prevented them.  

Reminder:  post the proof that π is irrational.  Here are several.  Here is a proof for e and π.  The one for π is a summary of Niven's proof.

Reading Reactions

One thing that's growing more and more noteworthy is "but … there was other mathematics either by this person or at this time."  There is most and more being done and neither Jeff nor I can discuss it all.  Keep the general trend in mind - always more mathematics over time. 

I will not talk about it given the vast amount of material today, but here is Clairaut's paper submitted when he was 12.  I saw that he learned to read by reading Euclid.  He is also best known for proving mixed partial derivatives are equal, e.g. f_xy = f_yx. 

The Cauchy-Riemann equations are assessing whether a function from the complexes to the complexes is differentiable.  This is an important result in complex analysis, familiar to all who take 371. 

d’Alembert’s probabilty error is common and tempting.  For those of you who have made the mistake of thinking there are three outcomes when flipping two coins.  Yes, it is incorrect, but it is understandable.  Before d’Alembert’s limit definition, we have seen some ideas in Newton, also struggling. 

Vaguely related, Geneseo was originally a normal school when we opened in 1871 (you might think about that date as we go forward in the next couple of chapters).  Here is a bit more, and I have more than this for any who wish to know. 

What did Châtelet do?  Best known for translating Newton’s Principia (published after her death - for many years the only translation into French).  Wrote a general audience explanation with Voltaire about Newton’s works.  She wrote on the propagation of fire.  “She attempted to integrate Cartesian, Newtonian, and Leibnizian ideas. On the philosophic side the themes she discusses are free will, God's power and role, and the nature of space, matter, and force.”  I also believe Châtelet predates Agnesi slightly.   So, if I were to pick a “first”, I would pick Châtelet.  Châtelet surely used her connections with high-placed academic men.  I do believe discussing Châtelet’s personal life _is_ relevant to show how she used them to fight through the barriers.  There are different paths.  Her path was different than Agnesi’s, very, but some path was necessary.  Could she have accomplished what she did without?  It’s not clear, and it’s _not_ her fault, but the system.  When I say “could she”, I mean would the system allow her? 

We will see a decimal clock.  Why not decimal time?  People are too hesitant to make logical choices for sentimental reasons.  The same goes for the world calendar which would unquestionably be better and more stable (same calendar each year [with leap year accommodation as needed]).  There is absolutely no technical reason to not use decimal time.   The French calendar was about breaking from tradition and cultural associations.  A deep theme of the revolution.

Jeff clearly shows us that Gauß is his favourite.  This time I won’t follow him, it’s too far afield.

We talked about this with Alcuin.  Please do not continue to tell the Gauß summing 1 to 100 story (which Alcuin _did_ talk about).  It is almost certainly false.  Feel free to tell my colleagues the same.  This is not known as Gauß’s formula by anyone who knows the history.  The story likely comes from ET Bell, who we will discuss later who is infamous for writing fictional stories about mathematicians.  (And other more honest fiction.) 

Also, when using close to Latin alphabets, I try to spell names as the people would have (I didn't for the Arabic, Chinese, or Indian, and I won't for Russian), so I use l'Hôpital, Châtalet (for whom the spelling was a source of pride), and Gauß. 

Remember that we're getting to mathematics that you haven't learned yet.  This is going to happen for the rest of the semester.  Keep your mind and eyes open, and all of it is something you can learn more about if you wish.