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.