390 Quick Answers 18 March

Thank you again for those who joined us ten days ago.  I sincerely appreciate it.

Let me point you to the exam discussion on mylearning and some more information about the exam ... please come talk to me about your plans.  Any last question shout outs? 

Technical point:  Is the cut-off 1600 or Chapter 6?  I'll say you can sneak in some of Chapter 7 if it's before 1600. 

Make sure your reactions are thorough.   I notice several without Descartes and Leibniz for today's reading, and they are central. 

Thank you again for all who completed the library survey.  No more using it for reactions.  Thank you. 

Oh, here we go again.  You could complete Monday's reactions today and not worry next weekend. 


Lecture Reactions

The _Pope_ said "this is the new calendar" and Catholic countries said "ok".  Those that were not said "you can't tell us what to do" but then others said "yeah, ok, it would be useful to have a calendar that matches the seasons … and agrees with others".  There are places where other calendars are used (most prominently Islamic and Jewish calendars), but pretty much everyone sees the value of having a common reference.  Will some people be surprised in 2100 that there isn't a leap year?  Yes.  Will all of humanity forget?  Definitely not.  Will people learn more about the calendar along the way?  Yes!  This is the calendar that we currently use.  Could there be better?  Definitely. 

As we go forward, watch our sources to see how notation stabilises.  We're rapidly approaching things looking very familiar. 

Basically, yes, Fermat's coordinates (and Descartes today) will be the same as you use in graphing, just expressed a little differently.   This is important - to be able to recognise mathematics as the same as yours just appearing slightly differently.

It’s an interesting question for why Fermat’s Last Theorem is such a big deal.  I think the best reasons are that 1. it’s easy to state, thus tempting, but 2. it’s really hard to prove, so it takes time, and lots of mathematics is created to address it.  

The largest known prime number (found in 2018) is 2^82,589,933 − 1 (it’s a Mersenne prime!), a number which has 24,862,048 digits when written in base 10.  Finding large prime numbers is what keeps your internet information safe.  Sometime out of class ask me why [it’s not history].    The fact that there are infinitely many primes is not why they are difficult to find, we can find all the squares, nor is it that they are more sparse as numbers get larger, but there is no function like f(n)=n^2 for primes.  





Reading Reactions


Talk about the _Nether_lands and Holland.

“Between any two different quantities there may be found a quantity less than their difference.” - this must be a mistake from Suzuki.  I think he means the simple important fact that between any two different real (or rational) numbers there is a third real (or rational) number.

Remember we do history by region.  We’ll do England next.  The calculus controversy will come to you in pieces, and the fallout will be discussed in chapter 8.   Remember also that we're jumping back in time. 

My guess is that the ideas of focus and directrix go back to Apollonius, but that DeWitt coined the terms.  

Remember finding π approximations are finding perimeters of circles with a very large number of sides.  Remember they are regular polygons, so you only need one of the sides.  Each time the sides are doubled by using a half-angle formula.  The ideas are not sophisticated, but keeping track of all the work is computationally intense.  The concepts  used are basically the same as Archimedes.  
Huygens 11 or 14 on 3-dice problem likely involves repeated binomial probability analysis.   11 is more likely with 3 dice (where the mean is 10.5), so it makes sense that B is more likely to win.  

Definitely the interaction of mathematicians is growing and becoming crucial and central to our story.  

Insightful question - was it known the difference between convergent and divergent series?  No, not really.  This leads to some very interesting sums for the naturals, for example.  We will see more of this.