390 Quick Answers 24 March
Important announcement: On Friday (28 March), I will be
attending the . (In fact, I will be running
a workshop for new faculty at the time of our class.)
Here's what this means for you: 1. reactions are due by
WEDNESDAY night 26 March 11:59p. This is so that I can process
them Thursday. 2. On Friday before I leave, I will update our
course materials. There will be a link to the new quick
answers, and a link to a lecture video from me for the day. 3.
Use these as your replacement lecture and use them as a basis for
your next set of lecture reactions. This will not
happen again. The meeting is unusually early this year, so as
a consequence - I will not miss a day a month from now.
Maybe for those who couldn't get reactions done in two weeks it will
be better if you have less time?
Having not read reactions once since 10 March, I’m glad to have the
time with you again. It’s a different form of interaction, but
I like that I get to hear from you each day. I do appreciate
this form of learning. (Much more than the exams.)
Refocus on your paper draft. 7 April is 2 weeks from today.
It’s a fine time to say - your final will be like the midterm, only
twice as much. 2-3 questions on 1600+ and 2-3 questions on the
entire course. You might take a chance at this point to look
over the midterm questions to give some inspiration for final
topics. Your "actual current average" is now updated and it is
what it says it is. Trust that not another column.
FYI: There were 14 As, 4 Bs, 2 Cs, and 3 Es on the exam.
You can do well if you prepare. I say again - this class is
not difficult, but it is a choice.
It’s also a fine time to be grateful that 1. you can type your essay
exams [and to lament for your predecessors for whom this was not the
case], 2. I’m only asking you to keep the tiniest bit of all that we
do. Learn what you like. Please see that engagement is
what’s most valued here.
Another story: on 14 March (the day of your exam) we had
in-service secondary math teachers here (23, I believe). I ran
a workshop in which they asked questions about anything they wanted
about history. There were familiar topics, π, complex, MCRTT,
irrationals, and more. My biggest concern for the day was that
I would bore our alumni (I think there were 5, maybe more) who took
this class with me. Afterward, I asked them about that.
They said "it's more valuable now because I know why I need
it." That's all well and good, and this includes some of our
best alumni, but … can we just skip the step and you try believing
when I tell you that it is good to be able to know where the
mathematics come from, not just to know the facts?
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.
Fermat's Last Theorem: There are plenty of integers such
that a^2+b^2=c^2. There are none for a^n + b^n = c^n and n
> 2. Fermat may possibly have proven the n=3 or n=4 case
(we'll see that one soon). There's no way he had more.
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
first Fermat number to not be prime is 2^(2^5)+1 = 4294967297= 641
x 6700417. That's not easy to find. It was discovered
later by Euler.
The largest known prime number (found last October 2024) is
2^136,279,841 − 1 (it’s a Mersenne prime!), a number which has
41.024.320 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.
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.
Reading
Reactions
Talk
about the _Nether_lands and Holland.
Remember Stevin was teaching people decimals for the first
time. We'll see his work, and it seems like a good teaching
strategy.
“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.
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.
Tschrinhaus transformations are replacing x = by+c for helpful
values of b and c (hence a sheer translation
geometrical). More variables gives more control, but
finding the good values for those variables leads to the same
problems that the Renaissance cubists found.
The tulip mania is true history. It would be great if
people could learn from it an stop speculative investment,
seems unlikely, alas.
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.
My guess is that the ideas of focus and directrix go back to
Apollonius, but that DeWitt coined the terms.
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.